Finite Group Databasebeta

← #21452144

Subgroups — #21452144

Order 7808.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14519758C211yesmaximal, characteristic, complemented
2C2×#14519749C211yesmaximal, characteristic, complemented
3C2×#14519746C211yesmaximal, characteristic, complemented
4#16839690C211yesmaximal, characteristic, complemented
5#16839685C211yesmaximal, characteristic, complemented
6#16839678C211yesmaximal, characteristic, complemented
7#16839677C211yesmaximal, characteristic, complemented
8#16839670C211yesmaximal, characteristic, complemented
9#16839670C211yesmaximal, characteristic, complemented
10#16839665C211yesmaximal, characteristic, complemented
11#16839662C211yesmaximal, characteristic, complemented
12#16839656C211yesmaximal, characteristic, complemented
13#16839642C211yesmaximal, characteristic, complemented
14#16839618C211yesmaximal, characteristic, complemented
15C2×#120×C61C211yesmaximal, characteristic, complemented
16C22×C22⋊D61C2211yescharacteristic
17C22×C22⋊D61C2211yescharacteristic
18C22×C22⋊D61C2211yescharacteristic
19C22×C22⋊D61C2211yescharacteristic, complemented
20C23×Dic61C2211yescharacteristic, complemented
21#14519758—24no—
22#14519749—24no—
23#14519746—24no—
24C2×C4⋊Dic61C2211yescharacteristic, complemented
25C2×#11758196C2211yescharacteristic, complemented
26C2×#11758196C2211yescharacteristic, complemented
27C2×#11758196C2211yescharacteristic, complemented
28C2×C22⋊Dic61C2211yescharacteristic
29C2×C22⋊Dic61C2211yescharacteristic
30C2×C22⋊Dic61C2211yescharacteristic
31C2×C22⋊Dic61C2211yescharacteristic
32C2×C22⋊Dic61C2211yescharacteristic
33C2×C22⋊Dic61C2211yescharacteristic
34C2×C22⋊Dic61C2211yescharacteristic
35C2×C22⋊Dic61C2211yescharacteristic
36C2×C22⋊Dic61C2211yescharacteristic
37C2×C22⋊Dic61C2211yescharacteristic, complemented
38C2×C22⋊Dic61C2211yescharacteristic, complemented
39C2×#11758198C2211yescharacteristic, complemented
40C2×#11758198C2211yescharacteristic, complemented
41C2×#11758198C2211yescharacteristic, complemented
42C2×C4×Dic61C2211yescharacteristic, complemented
43#14519684C2211yescharacteristic, complemented
44#14519684C2211yescharacteristic, complemented
45#14519684C2211yescharacteristic, complemented
46#14519684C2211yescharacteristic, complemented
47C25×C61C2211yescharacteristic
48C22×D4×C61C2211yescharacteristic
49C22×D4×C61C2211yescharacteristic
50C22×D4×C61C2211yescharacteristic
51#120×C61—24no—
52C2×C22⋊C4×C61C2211yescharacteristic
53C2×C22⋊C4×C61C2211yescharacteristic
54C2×C22⋊C4×C61C2211yescharacteristic
55C24×C61—22no—
56C24×C61—22no—
57C24×C61—22no—
58C24×C61—41no—
59C24×C61—41no—
60C24×C61—41no—
61C24×C61—41no—
62C24×C61C2311yescharacteristic
63C24×C61C2311yescharacteristic
64C24×C61C2311yescharacteristic
65C24×C61C2311yescharacteristic
66C23×D61C2311yescharacteristic
67C2×D4×C61—22no—
68C2×D4×C61—22no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61D414yescomplemented
76C22×C4×C61C2311yescharacteristic
77C22×C4×C61C2311yescharacteristic
78C22×C4×C61C2311yescharacteristic
79C2×C22⋊D61—22no—
80C2×C22⋊D61—22no—
81C2×C22⋊D61—22no—
82C2×C22⋊D61—22no—
83C2×C22⋊D61—22no—
84C2×C22⋊D61—22no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C22×Dic61—22no—
92C22×Dic61—22no—
93C22×Dic61—41no—
94C22×Dic61C2311yescharacteristic
95C22×Dic61C2311yescharacteristic
96C22×Dic61C2311yescharacteristic
97C22×Dic61C2311yescharacteristic
98C22×Dic61C2311yescharacteristic
99C22×Dic61C2311yescharacteristic
100C22×Dic61C2311yescharacteristic
101C22⋊C4×C61—22no—
102C22⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C22⋊Dic61—22no—
105C22⋊Dic61—22no—
106C22⋊Dic61—22no—
107C22⋊Dic61—22no—
108C22⋊Dic61—22no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111C22⋊Dic61—41no—
112C22⋊Dic61—41no—
113C22⋊Dic61—41no—
114C22⋊Dic61D414yescomplemented
115#11758198—22no—
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—41no—
119#11758196—22no—
120#11758196—22no—
121#11758196—22no—
122C4×Dic61—41no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—44no—
148C23×C61—44no—
149C23×C61—44no—
150C23×C61—44no—
151C23×C61—44no—
152C23×C61—44no—
153C23×C61—44no—
154C23×C61C2×D412yes—
155C23×C61C2×D412yes—
156C23×C61C2×D412yes—
157C23×C61#4112yes—
158C23×C61C2411yescharacteristic
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166D4×C61—44no—
167D4×C61—44no—
168D4×C61—44no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61C2×D412yes—
178C22⋊D61—44no—
179C22⋊D61—44no—
180C22⋊D61—44no—
181C22⋊D61—44no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—44no—
202C2×Dic61C2×D412yes—
203C2×Dic61C2×D412yes—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—22no—
210C22×C61—22no—
211C22×C61—22no—
212C22×C61—22no—
213C22×C61—22no—
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—416no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61C22×D411yescharacteristic
234C22×C61C22×D411yescharacteristic
235C22×C61C2×#4111yescharacteristic
236C22×C61H(5,𝔽2)11yescharacteristic
237C22×C61H(5,𝔽2)11yescharacteristic
238C22×C61H(5,𝔽2)11yescharacteristic
239C22×C61H(5,𝔽2)11yescharacteristic
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—41no—
247C4×C61—22no—
248C4×C61—41no—
249C4×C61—41no—
250Dic61—22no—
251Dic61—22no—
252Dic61—41no—
253Dic61—41no—
254Dic61—41no—
255Dic61—41no—
256Dic61—41no—
257#2678—611nomaximal, Sylow
258C2×C61—42no—
259C2×C61—42no—
260C2×C61—42no—
261C2×C61—42no—
262C2×C61#53411yescharacteristic
263C2×C61#53511yescharacteristic
264C2×C61#53811yescharacteristic
265C2×C61D4211yescharacteristic
266C2×C61#54611yescharacteristic
267C2×C61#54611yescharacteristic
268C2×C61#56011yescharacteristic
269D61—81no—
270C2×#123—611no—
271C2×#121—611no—
272C2×#120—611no—
273C2×#120D6111yescharacteristic, complemented
274#397—611no—
275#396—611no—
276#394—611no—
277#394—611no—
278#392—611no—
279#392—611no—
280#390—611no—
281#390—611no—
282#388—611no—
283#386—611no—
284#379—611no—
285C61#267811yescharacteristic, complemented, Sylow
286C25C2×D6111yescharacteristic
287C22×D4—611no—
288C22×D4—611no—
289C22×D4—611no—
290C22×D4—611no—
291C22×D4C2×D6111yescharacteristic
292C22×D4C2×D6111yescharacteristic
293C22×D4C2×D6111yescharacteristic
294C23×C4—611no—
295#123—1224no—
296#121—1224no—
297#120—24no—
298#120—1224no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—611no—
303C2×C22⋊C4—611no—
304C2×C22⋊C4—611no—
305C2×C22⋊C4—611no—
306C2×C22⋊C4—611no—
307C2×C22⋊C4—611no—
308C2×C22⋊C4—611no—
309C2×C22⋊C4—611no—
310C2×C22⋊C4—611no—
311C2×C22⋊C4—611no—
312C2×C22⋊C4—611no—
313C2×C22⋊C4—611no—
314C2×C22⋊C4—611no—
315C2×C22⋊C4—611no—
316C2×C22⋊C4—611no—
317C2×C22⋊C4C2×D6111yescharacteristic
318C2×C22⋊C4C2×D6111yescharacteristic
319C2×C22⋊C4C2×D6111yescharacteristic
320C2×C42—611no—
321#95—611no—
322#95—611no—
323#95—611no—
324#95—611no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—611no—
333C24C22×D6111yescharacteristic
334C24C22×D6111yescharacteristic
335C24C22×D6111yescharacteristic
336C24C22×D6111yescharacteristic
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—1222no—
346C2×D4—1222no—
347C2×D4—1222no—
348C2×D4—1222no—
349C2×D4—1222no—
350C2×D4—1222no—
351C2×D4—2441no—
352C2×D4—2441no—
353C2×D4—2441no—
354C2×D4—2441no—
355C2×D4—2441no—
356C2×D4—2441no—
357C2×D4C22⋊D6114yescomplemented
358C22×C4—611no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—2441no—
368C22×C4C22×D6111yescharacteristic
369C22×C4C22×D6111yescharacteristic
370C22×C4C22×D6111yescharacteristic
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4—2441no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—614no—
379C22⋊C4—1222no—
380C22⋊C4—1222no—
381C22⋊C4—1222no—
382C22⋊C4—1222no—
383C22⋊C4—1222no—
384C22⋊C4—1222no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C42—2441no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—1221no—
425C23—1221no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23D4×D6112yes—
432C23#1175822412yes—
433C23C2×C22⋊D6112yes—
434C23C2×C22⋊D6112yes—
435C23C23×D6111yescharacteristic
436D4—44no—
437D4—44no—
438D4—44no—
439D4—2444no—
440D4—2444no—
441D4—2444no—
442D4—2444no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—2444no—
473C2×C4C2×C22⋊D6112yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—416no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22—2441no—
509C22—2441no—
510C22C2×D4×D6111yescharacteristic
511C22C22×C22⋊D6111yescharacteristic
512C22C2×#1175822411yescharacteristic
513C22#1451982211yescharacteristic
514C22#1451982211yescharacteristic
515C22#1451982211yescharacteristic
516C22#1451982211yescharacteristic
517C4—22no—
518C4—41no—
519C4—41no—
520C4—1222no—
521C4—1222no—
522C4—2441no—
523C4—2441no—
524C4—2441no—
525C4—2441no—
526C4—2441no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—42no—
531C2—4881no—
532C2#1684032111yescharacteristic
533C2#1684032711yescharacteristic
534C2#1684034711yescharacteristic
535C2D4×C22⋊D6111yescharacteristic
536C2#1684039911yescharacteristic
537C2#1684040611yescharacteristic
538C2#1684050111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#120×C61C2
hypercenterC2×#120D61
hypocenterC61#2678
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14519758C2semidirect
2. C2×#14519749C2semidirect
3. C2×#14519746C2semidirect
4. #16839690C2semidirect
5. #16839685C2semidirect
6. #16839678C2semidirect
7. #16839677C2semidirect
8. #16839670C2semidirect
9. #16839670C2semidirect
10. #16839665C2semidirect
11. #16839662C2semidirect
12. #16839656C2semidirect
13. #16839642C2semidirect
14. #16839618C2semidirect
15. C2×#120×C61C2semidirect
19. C22×C22⋊D61C22semidirect
20. C23×Dic61C22semidirect
24. C2×C4⋊Dic61C22semidirect
25. C2×#11758196C22semidirect
26. C2×#11758196C22semidirect
27. C2×#11758196C22semidirect
37. C2×C22⋊Dic61C22semidirect
38. C2×C22⋊Dic61C22semidirect
39. C2×#11758198C22semidirect
40. C2×#11758198C22semidirect
41. C2×#11758198C22semidirect
42. C2×C4×Dic61C22semidirect
43. #14519684C22semidirect
44. #14519684C22semidirect
45. #14519684C22semidirect
46. #14519684C22semidirect
75. C2×D4×C61D4semidirect
114. C22⋊Dic61D4semidirect
273. C2×#120D61semidirect
285. C61#2678semidirect
357. C2×D4C22⋊D61semidirect