Finite Group Databasebeta

← #29738870

Subgroups — #29738870

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840171C211yesmaximal, characteristic, complemented
2C2×#16839804C211yesmaximal, characteristic
3#21449532C211yesmaximal, characteristic
4#21449521C211yesmaximal, characteristic
5#21447297C211yesmaximal, characteristic, complemented
6#21447292C211yesmaximal, characteristic, complemented
7#21447158C211yesmaximal, characteristic, complemented
8#21447153C211yesmaximal, characteristic, complemented
9#21446557C211yesmaximal, characteristic, complemented
10#21446552C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic
12#21446014C211yesmaximal, characteristic
13#21445976C211yesmaximal, characteristic, complemented
14#21445852C211yesmaximal, characteristic
15C61×#1753C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519754—21no—
19C2×#14519736C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21#16840171—24no—
22#16839804—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519709C2211yescharacteristic
31C2×#14519709C2211yescharacteristic
32C2×#14519709C2211yescharacteristic
33#16839698—21no—
34#16839694—21no—
35#16839685—21no—
36#16839683—21no—
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839659—21no—
42#16839645—21no—
43#16839637—21no—
44#16839632C2211yescharacteristic
45#16839632C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63#14519777—22no—
64#14519777—41no—
65#14519769—22no—
66#14519769—41no—
67#14519754—44no—
68#14519736—22no—
69#14519736—41no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—41no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C4×Dic61—21no—
96#14519711—22no—
97#14519711—22no—
98#14519711—22no—
99#14519711D414yes—
100#14519709—22no—
101#14519709—22no—
102#14519709—22no—
103#14519709—22no—
104#14519709—41no—
105#14519709—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C23×C4×C61C2311yescharacteristic
113C2×M16×C61—22no—
114C2×M16×C61—22no—
115C2×M16×C61Q814yescomplemented
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C22⋊C4×C61D411yescharacteristic
121C2×C22⋊C4×C61D411yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C22⋊C8×C61Q814yescomplemented
125#97×C61—22no—
126C4×C8×C61—22no—
127#95×C61D411yescharacteristic
128#95×C61D411yescharacteristic
129C24×C61C2×D411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×Q812yes—
140C22×C4×C61C2×Q812yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—81no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—81no—
156M16×C61—28no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D412yes—
164C2×C8×C61C2×Q812yes—
165C2×C8×C61C2×Q812yes—
166C2×C8×C61C2×Q812yes—
167C2×C8×C61C2×Q812yes—
168C2×C8×C61#4112yes—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197C4⋊Dic61—81no—
198C4⋊Dic61C2×D412yes—
199C4⋊Dic61C2×D412yes—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—81no—
206C4×Dic61—42no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61C4⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61C4⋊Q812yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61#13712yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×Q811yescharacteristic
236C2×C4×C61C22×Q811yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—84no—
265Dic122—84no—
266C8×C61—22no—
267C8×C61—22no—
268C8×C61—22no—
269C8×C61C22⋊Q814yescomplemented
270#26078—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4⋊Q811yescharacteristic
279C22×C61D4211yescharacteristic
280C22×C61#54711yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57711yescharacteristic
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×C22⋊Q811yescharacteristic
291C4×C61#54411yescharacteristic
292C4×C61D4×Q811yescharacteristic
293C4×C61#55411yescharacteristic
294Dic61—42no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#501—611no—
303C2×#428—611no—
304#2441—611no—
305#2440—611no—
306#2084—611no—
307#2083—611no—
308#2059—611no—
309#2058—611no—
310#1943—611no—
311#1942—611no—
312#1854—611no—
313#1853—611no—
314#1844—611no—
315#1809—611no—
316#1753D6111yescharacteristic
317C2×C61—42no—
318C2×C61#244011yescharacteristic
319C2×C61#316311yescharacteristic
320C2×C61#316511yescharacteristic
321C2×C61#327211yescharacteristic
322C2×C61#327811yescharacteristic
323C2×C61#334411yescharacteristic
324C2×C61#334511yescharacteristic
325C22×M16C2×D6111yescharacteristic
326C2×C4⋊Q8—611no—
327C2×#125—611no—
328C2×C22⋊Q8—1221no—
329C2×#117—611no—
330C22×C4⋊C4—611no—
331#501—1224no—
332#428—1224no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107—611no—
336C2×#107—611no—
337C2×#106—611no—
338C2×#106—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C22⋊C8C2×D6111yescharacteristic
344C2×#97C2×D6111yescharacteristic
345C2×C4×C8C2×D6111yescharacteristic
346#399—1221no—
347#398—1221no—
348#396—1221no—
349#395—1221no—
350#389—611no—
351#389—611no—
352#389—611no—
353#389—611no—
354#389—1221no—
355#387—1221no—
356#385—1221no—
357#384—611no—
358#384—611no—
359#380—611no—
360#380—611no—
361C4×C22⋊C4C2×D6111yescharacteristic
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#336C2×D6111yescharacteristic
369#336C2×D6111yescharacteristic
370C61#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic12214yescomplemented
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—1222no—
382C4⋊Q8—2441no—
383#125—1222no—
384#125—2441no—
385C22⋊Q8—2444no—
386#117—1222no—
387#117—2441no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—2441no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4D24411yescharacteristic
414C2×C22⋊C4D24411yescharacteristic
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418#107—614no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#106—1222no—
423#106—1222no—
424#106—1222no—
425#106—1222no—
426#106—2441no—
427#106—2441no—
428C22⋊C8Dic12214yescomplemented
429#97—22no—
430C4×C8—22no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95D24411yescharacteristic
438#95D24411yescharacteristic
439C24C2×D24411yescharacteristic
440C2×Q8—4881no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4881no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4C2×Dic12212yes—
459C22×C4C2×Dic12212yes—
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×Dic12212yes—
474C2×C8C2×Dic12212yes—
475C2×C8C2×Dic12212yes—
476C2×C8C2×Dic12212yes—
477C2×C8D4×D6112yes—
478C2×C8#1175822412yes—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—4881no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—4881no—
510C22⋊C4—2442no—
511C22⋊C4—2442no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2442no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1451977712yes—
524C23C22×D24411yescharacteristic
525Q8—4884no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—4884no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4#1451977712yes—
568C2×C4#1451980512yes—
569C2×C4#1451981012yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C22×Dic12211yescharacteristic
572C2×C4C22×Dic12211yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4#1451982311yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1451975414yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1451977711yescharacteristic
588C22D4×D24411yescharacteristic
589C22#1684041611yescharacteristic
590C22C2×#1451980511yescharacteristic
591C22C2×#1451981011yescharacteristic
592C22#1684057411yescharacteristic
593C22#1684057411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—4881no—
600C4—4881no—
601C4—4881no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451975411yescharacteristic
608C4#1684039011yescharacteristic
609C4D4×Dic12211yescharacteristic
610C4#1684046911yescharacteristic
611C2—42no—
612C2#2144952111yescharacteristic
613C2#2145635011yescharacteristic
614C2#2145636511yescharacteristic
615C2#2145742111yescharacteristic
616C2#2145748711yescharacteristic
617C2#2145827611yescharacteristic
618C2#2145829111yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1753C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1753D61
hypocenterC61#26078
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16840171C2semidirect
5. #21447297C2semidirect
6. #21447292C2semidirect
7. #21447158C2semidirect
8. #21447153C2semidirect
9. #21446557C2semidirect
10. #21446552C2semidirect
13. #21445976C2semidirect
115. C2×M16×C61Q8semidirect
124. C22⋊C8×C61Q8semidirect
269. C8×C61C22⋊Q8semidirect
370. C61#26078semidirect
376. C2×M16Dic122semidirect
428. C22⋊C8Dic122semidirect
579. C8#14519754semidirect

Maximal subgroups

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

subgroupindex
C2×#168401712
C2×#168398042
#214495322
#214495212
#214472972
#214472922
#214471582
#214471532
#214465572
#214465522
#214460202
#214460142
#214459762
#214458522
C61×#17532
#2607861