Finite Group Databasebeta

← #30609292

Subgroups — #30609292

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461008C212yesmaximal
2#21461006C212yesmaximal
3#21461005C212yesmaximal
4#21461004C212yesmaximal
5#21460997C211yesmaximal, characteristic
6#21460995C211yesmaximal, characteristic
7#21460995C211yesmaximal, characteristic
8#21460995C211yesmaximal, characteristic
9#21460995C211yesmaximal, characteristic
10#21460995C211yesmaximal, characteristic
11C4×C4⋊C4×D61C211yesmaximal, characteristic
12C22×C4⋊#6923C2212yes—
13C2×(C2×#6923)⋊C4C2211yescharacteristic
14C2×(C2×#6923)⋊C4C2211yescharacteristic
15C2×(C2×#6923)⋊C4C2211yescharacteristic
16C2×(C2×#6923)⋊C4C2211yescharacteristic
17C2×(C2×#6923)⋊C4C2212yes—
18C2×(C2×#6923)⋊C4C2212yes—
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2212yes—
21C2×(C2×#6923)⋊C4C2212yes—
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2212yes—
26C2×(C2×#6923)⋊C4C2212yes—
27C2×(C2×#6923)⋊C4C2212yes—
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C42×D61C2211yescharacteristic
31C2×C42×D61C2211yescharacteristic
32C2×C42×D61C2211yescharacteristic
33C4⋊C4×Dic61C411yescharacteristic, complemented
34C4⋊C4×Dic61C411yescharacteristic, complemented
35C4×#11758196C411yescharacteristic, complemented
36C4×#11758196C411yescharacteristic, complemented
37C4×C4⋊Dic61C411yescharacteristic, complemented
38#16839613C411yescharacteristic, complemented
39#16839613C411yescharacteristic, complemented
40#95×D61C2211yescharacteristic
41#95×D61C2211yescharacteristic
42C4×C4⋊C4×C61C411yescharacteristic, complemented
43C23×#6923C2312yes—
44C23×#6923C2312yes—
45C23×#6923C2312yes—
46C23×#6923C2312yes—
47C2×C4⋊#6923—24no—
48C2×C4⋊#6923—42no—
49C2×C4⋊#6923—42no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—24no—
55(C2×#6923)⋊C4—24no—
56(C2×#6923)⋊C4—24no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4—42no—
61(C2×#6923)⋊C4—42no—
62(C2×#6923)⋊C4—42no—
63(C2×#6923)⋊C4—42no—
64(C2×#6923)⋊C4—42no—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C22×C4×D61C2311yescharacteristic
70C22×C4×D61C2311yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×#11758196C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C4⋊C4×D61—24no—
79C4⋊C4×D61—24no—
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61C2×C411yescharacteristic
83C2×C4×Dic61C2×C411yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C42×D61—22no—
90C42×D61—22no—
91C42×D61—42no—
92C42×D61—42no—
93#14519684C2×C411yescharacteristic
94#14519684C2×C411yescharacteristic
95#14519684C2×C411yescharacteristic
96#14519684C2×C411yescharacteristic
97#14519684C2×C411yescharacteristic
98#14519684C2×C411yescharacteristic
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2×C411yescharacteristic
101C2×C42×C61C2×C411yescharacteristic
102C2×C42×C61C2×C411yescharacteristic
103C2×C42×C61C2×C411yescharacteristic
104#95×C61C2×C411yescharacteristic
105#95×C61C2×C411yescharacteristic
106C23×D61C2411yescharacteristic
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923—22no—
118C22×#6923—22no—
119C22×#6923—22no—
120C22×#6923—22no—
121C22×#6923—22no—
122C22×#6923—22no—
123C22×#6923—22no—
124C22×#6923—22no—
125C22×#6923—22no—
126C22×#6923—22no—
127C22×#6923—22no—
128C22×#6923—22no—
129C22×#6923—22no—
130C22×#6923—22no—
131C22×#6923—22no—
132C22×#6923—22no—
133C22×#6923—22no—
134C22×#6923—22no—
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—22no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61#4112yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4114yes—
178C2×C4×D61#4114yes—
179C4⋊#6923—48no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61—41no—
184C42×C61—41no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—41no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197C4×Dic61—41no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C23×C61C23×C411yescharacteristic
201C22×D61C2×#4112yes—
202C22×D61C2×#4111yescharacteristic
203C22×D61C2×#4111yescharacteristic
204C22×D61C2×#4112yes—
205C22×D61H(5,𝔽2)11yescharacteristic
206C22×D61H(5,𝔽2)12yes—
207C22×D61H(5,𝔽2)11yescharacteristic
208C22×D612−1+411yescharacteristic
209C22×D612−1+412yes—
210C22×D612−1+411yescharacteristic
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—42no—
216C2×#6923—42no—
217C2×#6923—42no—
218C2×#6923—42no—
219C2×#6923—42no—
220C2×#6923—42no—
221C2×#6923—42no—
222C2×#6923—42no—
223C2×#6923—42no—
224C2×#6923—42no—
225C2×#6923—42no—
226C2×#6923—42no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—42no—
230C2×#6923—42no—
231C2×#6923—42no—
232C2×#6923—42no—
233C2×#6923—42no—
234C2×#6923—42no—
235C2×#6923—42no—
236C2×#6923—42no—
237C2×#6923—42no—
238C2×#6923—42no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61#11712yes—
258C2×C4×C61#11712yes—
259C2×C4×C61#11712yes—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61#11712yes—
279C2×Dic61#11712yes—
280C2×Dic61#11712yes—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—24no—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—41no—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—42no—
296C4×D61—42no—
297#14352—611nomaximal, Sylow
298C22×C61C2×#11711yescharacteristic
299C22×C61C2×#11711yescharacteristic
300C22×C61C2×#11711yescharacteristic
301C22×C61#51811yescharacteristic
302C22×C61#51911yescharacteristic
303C22×C61#52011yescharacteristic
304C22×C61#52011yescharacteristic
305C2×D61#52912yes—
306C2×D61#52912yes—
307C2×D61#53812yes—
308C2×D61#53912yes—
309C2×D61#53912yes—
310C2×D61#53912yes—
311C2×D61#54011yescharacteristic
312C2×D61#54012yes—
313C2×D61#54112yes—
314C2×D61#54112yes—
315C2×D61#54111yescharacteristic
316C2×D61#54211yescharacteristic
317C2×D61#54212yes—
318C2×D61#54212yes—
319C2×D61#54212yes—
320C2×D61#54211yescharacteristic
321#6923—82no—
322#6923—82no—
323#6923—82no—
324#6923—82no—
325C4×C61—22no—
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61—41no—
332Dic61—22no—
333Dic61—22no—
334Dic61—22no—
335Dic61—41no—
336Dic61—41no—
337Dic61—41no—
338Dic61—41no—
339C2×C4×C4⋊C4—611no—
340#1445—612no—
341#1443—612no—
342#1442—612no—
343#1441—612no—
344C4⋊#95—611no—
345#1434—611no—
346#1434—611no—
347#1434—611no—
348#1434—611no—
349#1434—611no—
350C2×C61#228311yescharacteristic
351C2×C61#231311yescharacteristic
352C2×C61#231411yescharacteristic
353C2×C61#231811yescharacteristic
354C2×C61#231911yescharacteristic
355C2×C61#232011yescharacteristic
356C2×C61#232011yescharacteristic
357D61#265012yes—
358D61#265112yes—
359D61#265412yes—
360D61#265512yes—
361C22×C4⋊C4—611no—
362C22×C4⋊C4—611no—
363C22×C4⋊C4—612no—
364C22×C42—611no—
365C22×C42—611no—
366C22×C42—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4#692311yescharacteristic, complemented
375C2×#95—611no—
376C2×#95—611no—
377C2×#95—611no—
378C2×#95—611no—
379C2×#95—611no—
380C2×#95—611no—
381C2×#95—612no—
382C2×#95—612no—
383C2×#95—612no—
384C2×#95—612no—
385C2×#95—612no—
386C2×#95—612no—
387C2×#95—612no—
388C2×#95—612no—
389C2×#95—612no—
390C2×#95—612no—
391C2×#95—612no—
392C61#1435211yescharacteristic, complemented, Sylow
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—612no—
401C23×C4—612no—
402C23×C4—612no—
403C23×C4—612no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1224no—
411C2×C4⋊C4—1224no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—2442no—
414C2×C4⋊C4—2442no—
415C2×C4⋊C4C2×#692311yescharacteristic
416C2×C4⋊C4C2×#692311yescharacteristic
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1222no—
427C2×C42—1222no—
428C2×C42—2442no—
429C2×C42—2442no—
430C2×C42C2×#692311yescharacteristic
431C2×C42C2×#692311yescharacteristic
432C2×C42C2×#692311yescharacteristic
433#95—611no—
434#95—611no—
435#95—611no—
436#95—611no—
437#95—611no—
438#95—611no—
439#95—1222no—
440#95—1222no—
441#95—1222no—
442#95—1222no—
443#95—1224no—
444#95—1224no—
445#95—1224no—
446#95—2442no—
447#95—2442no—
448#95—2442no—
449#95—2442no—
450#95—2442no—
451#95—2442no—
452#95—2442no—
453#95—2442no—
454#95C2×#692311yescharacteristic
455#95C2×#692311yescharacteristic
456C24—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4C22×#692311yescharacteristic
523C22×C4C22×#692311yescharacteristic
524C22×C4C22×#692311yescharacteristic
525C22×C4C22×#692311yescharacteristic
526C22×C4C22×#692311yescharacteristic
527C22×C4C22×#692311yescharacteristic
528C22×C4C22×#692311yescharacteristic
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—2448no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42—1222no—
542C42—1222no—
543C42—1222no—
544C42—2441no—
545C42—2441no—
546C42—2441no—
547C42—2441no—
548C42—2441no—
549C42—2441no—
550C23—611no—
551C23—611no—
552C23—612no—
553C23—612no—
554C23—612no—
555C23—612no—
556C23—611no—
557C23—611no—
558C23—611no—
559C23—611no—
560C23C23×#692311yescharacteristic
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4—2442no—
633C2×C4—2442no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4#1451985612yes—
645C2×C4#1451985612yes—
646C2×C4#1451985612yes—
647C22—612no—
648C22—612no—
649C22—611no—
650C22—611no—
651C22—611no—
652C22—611no—
653C22—612no—
654C22—612no—
655C22—612no—
656C22—612no—
657C22—612no—
658C22—612no—
659C22—612no—
660C22—612no—
661C22—612no—
662C22—612no—
663C22C2×#1451985611yescharacteristic
664C22C2×#1451985611yescharacteristic
665C22C2×#1451985611yescharacteristic
666C22#1684082911yescharacteristic
667C22#1684083011yescharacteristic
668C22#1684083111yescharacteristic
669C22#1684083111yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—1222no—
678C4—1222no—
679C4—1222no—
680C4—4882no—
681C4—4882no—
682C4—4882no—
683C4—4882no—
684C4—2441no—
685C4—2441no—
686C4—2441no—
687C4—2441no—
688C2—612no—
689C2—612no—
690C2—612no—
691C2—612no—
692C2#2146226611yescharacteristic
693C2#2146229611yescharacteristic
694C2#2146229711yescharacteristic
695C2#2146230111yescharacteristic
696C2#2146230211yescharacteristic
697C2#2146230311yescharacteristic
698C2#2146230311yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC4×C4⋊C4×C61C4
prime radical, cosocleC23×D61C24
hypercenterC4×C4⋊C4#6923
hypocenterC61#14352
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
33. C4⋊C4×Dic61C4semidirect
34. C4⋊C4×Dic61C4semidirect
35. C4×#11758196C4semidirect
36. C4×#11758196C4semidirect
37. C4×C4⋊Dic61C4semidirect
38. #16839613C4semidirect
39. #16839613C4semidirect
42. C4×C4⋊C4×C61C4semidirect
374. C4×C4⋊C4#6923semidirect
392. C61#14352semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214610082
#214610062
#214610052
#214610042
#214609972
#214609952
C4×C4⋊C4×D612
#1435261