Finite Group Databasebeta

← #29709445

Subgroups — #29709445

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758202C211yesmaximal, characteristic, complemented
2C2×#16839815C211yesmaximal, characteristic
3C4×#14519756C211yesmaximal, characteristic
4#21447315C211yesmaximal, characteristic, complemented
5#21447250C211yesmaximal, characteristic
6#21447214C211yesmaximal, characteristic, complemented
7#21447211C211yesmaximal, characteristic, complemented
8#21447141C211yesmaximal, characteristic, complemented
9#21447022C211yesmaximal, characteristic
10#21447019C211yesmaximal, characteristic
11#21446827C211yesmaximal, characteristic
12C4×#14519710C211yesmaximal, characteristic, complemented
13C4×#14519700C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C61C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C4×#11758202—24no—
20#16839815—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839646—21no—
39#16839635—21no—
40#16839622C2211yescharacteristic
41C4×#11758196—21no—
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C22⋊Dic61—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C22⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C22×C42×C61C2211yescharacteristic
53C61×#391C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#385—21no—
57C61×#380C2211yescharacteristic
58C4×C4⋊C4×C61—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C22⋊C4×C61—21no—
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519756—22no—
70#14519756—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83#14519710—22no—
84#14519710—22no—
85#14519710—22no—
86C4⋊#92418—22no—
87C4⋊#92418—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519700—22no—
92#14519700—22no—
93#14519700—22no—
94C22⋊#92418—22no—
95C22⋊#92418—22no—
96C4×#92418—22no—
97C4×#92418D414yes—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61C2311yescharacteristic
104C23×C4×C61D411yescharacteristic
105C23×C4×C61D411yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—22no—
120C2×C42×C61—22no—
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61#4112yes—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61Q1614yescomplemented
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—42no—
182C42×C61—22no—
183C42×C61—22no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C42×C61—42no—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190C22⋊Dic61—24no—
191C22⋊Dic61—42no—
192#11758202—28no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61#4112yes—
199#11758196—24no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D412yes—
215C2×#92418C2×D412yes—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61C2×Q1612yes—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—28no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—44no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C2×Q1612yes—
259C2×C4×C61C2×Q1612yes—
260C2×C4×C61#13512yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283#92418—22no—
284#92418—22no—
285#92418—41no—
286#92418—41no—
287#23698—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61C2×#12111yescharacteristic
296C22×C61#52511yescharacteristic
297C22×C61#52511yescharacteristic
298C22×C61C22×Q1611yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57511yescharacteristic
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—41no—
306C4×C61—42no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61#46714yes—
311C4×C61#52511yescharacteristic
312C4×C61#52911yescharacteristic
313C4×C61#52911yescharacteristic
314C4×C61#53211yescharacteristic
315Dic61—42no—
316Dic61—42no—
317Dic61—81no—
318Dic61—81no—
319C2×C4×Q16—611no—
320C2×#431—611no—
321C4×C22⋊Q8—611no—
322C4×C22⋊Q8D6111yescharacteristic
323#2087—611no—
324#2076—611no—
325#2068—611no—
326#2068—611no—
327#2056—611no—
328#2036—611no—
329#2036—611no—
330#2001—611no—
331C4×#107—611no—
332C4×#103—611no—
333C4×#103—611no—
334C2×C61—42no—
335C2×C61#239211yescharacteristic
336C2×C61C2×#46711yescharacteristic
337C2×C61#305411yescharacteristic
338C2×C61#310211yescharacteristic
339C2×C61#310211yescharacteristic
340C2×C61#310411yescharacteristic
341C2×C61#310411yescharacteristic
342C22×Q16—611no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8C2×D6111yescharacteristic
345C2×C4×Q8—611no—
346C2×C4×Q8C2×D6111yescharacteristic
347C22×C42C2×D6111yescharacteristic
348C4×Q16—1224no—
349#431—1224no—
350C2×#107—611no—
351C2×#107—611no—
352C2×#107—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×C4×C8—611no—
364C2×C4×C8—611no—
365#391—611no—
366#391C2×D6111yescharacteristic
367#389—611no—
368#389C2×D6111yescharacteristic
369#387—21no—
370#387—1221no—
371#385—21no—
372#385—1221no—
373#380—611no—
374#380C2×D6111yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4—1221no—
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387C61#2369811yescharacteristic, complemented, Sylow
388C22×Q8—611no—
389C22×Q8C22×D6111yescharacteristic
390C23×C4C22⋊D6111yescharacteristic
391C23×C4C22⋊D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C2×Q16—1222no—
394C2×Q16—1222no—
395C2×Q16—1222no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—22no—
425C2×C42—22no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42C22⋊D6111yescharacteristic
431C2×C42C22⋊D6111yescharacteristic
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434#107—1222no—
435#107—1222no—
436#107—1222no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—1222no—
444#103—1222no—
445C22⋊C8—1222no—
446C22⋊C8—1222no—
447C4×C8—614no—
448C4×C8—1222no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24C2×C22⋊D6111yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8#1175822212yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4#1175820214yescomplemented
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498Q16—1228no—
499C2×C8—612no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—612no—
520C4⋊C4—1224no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822212yes—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—1224no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—2442no—
540C42—2442no—
541C42D4×D6112yes—
542C42#1175822412yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23C2×#1175820212yes—
550C23C22×C22⋊D6111yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—41no—
554Q8—41no—
555Q8—1222no—
556Q8—1222no—
557Q8—2441no—
558Q8—2441no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4#1451974912yes—
600C2×C4#1451974912yes—
601C2×C4C2×#1175820212yes—
602C2×C4C2×#1175820212yes—
603C2×C4#1451979712yes—
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4#41×D6111yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C8—1222no—
611C8—1222no—
612C8—2441no—
613C8—2441no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22C2×#1451974911yescharacteristic
622C22#1684026911yescharacteristic
623C22#1684027311yescharacteristic
624C22C22×#1175820211yescharacteristic
625C22C2×#1451979711yescharacteristic
626C22#1684056011yescharacteristic
627C22#1684056011yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—4881no—
637C4—4881no—
638C4—2442no—
639C4—2442no—
640C4#1684000114yes—
641C4#1684027111yescharacteristic
642C4#1684029211yescharacteristic
643C4#1684029411yescharacteristic
644C4#1684030911yescharacteristic
645C2—42no—
646C2#2144906111yescharacteristic
647C2C2×#1684000111yescharacteristic
648C2#2145542511yescharacteristic
649C2#2145583111yescharacteristic
650C2#2145583711yescharacteristic
651C2#2145585411yescharacteristic
652C2#2145585811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC4×C22⋊Q8D61
hypocenterC61#23698
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11758202C2semidirect
4. #21447315C2semidirect
6. #21447214C2semidirect
7. #21447211C2semidirect
8. #21447141C2semidirect
12. C4×#14519710C2semidirect
13. C4×#14519700C2semidirect
14. C4×#14519701C2semidirect
154. C22×C4×C61Q16semidirect
387. C61#23698semidirect
491. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582022
C2×#168398152
C4×#145197562
#214473152
#214472502
#214472142
#214472112
#214471412
#214470222
#214470192
#214468272
C4×#145197102
C4×#145197002
C4×#145197012
C4×C22⋊Q8×C612
#2369861