Finite Group Databasebeta

← #72262002

Subgroups — #72262002

Order 2048.

At least 6092 conjugacy classes of subgroups; only the 1364 normal subgroups are shown.

The subgroup lattice has 454 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72261994C211yesmaximal, characteristic, complemented
2#72261982C214yesmaximal, complemented
3#72261979C214yesmaximal, complemented
4C2×#498991C211yesmaximal, characteristic, complemented
5#72261966C211yesmaximal, characteristic, complemented
6#72261960C211yesmaximal, characteristic, complemented
7#72261949C214yesmaximal, complemented
8#72261939C214yesmaximal, complemented
9C2×#7711868C211yesmaximal, characteristic, complemented
10#72261738C211yesmaximal, characteristic, complemented
11#72260855C212yesmaximal, complemented
12C2×#498909C211yesmaximal, characteristic, complemented
13#72250630C212yesmaximal, complemented
14C2×#494622C212yesmaximal, complemented
15#72238533C212yesmaximal, complemented
16C2×#62811C2211yescharacteristic
17C22×#3021C2212yes—
18C2×#46318C2212yescomplemented
19C2×#46099C2212yescomplemented
20C2×#60048C2212yescomplemented
21C2×#59885C2212yescomplemented
22C2×#37297C2211yescharacteristic
23C2×#37306C2211yescharacteristic
24C2×#35250C2212yes—
25C2×#34415C2212yes—
26C22×#2297C2211yescharacteristic
27C22×#2134C2212yes—
28C22×#2134C2212yes—
29C2×#20956C2212yes—
30C2×#20618C2212yes—
31C2×#20615C2211yescharacteristic
32C2×#20615C2211yescharacteristic
33C2×#20591C2212yes—
34C2×#20591C2211yescharacteristic
35C2×#13379C2214yescomplemented
36C2×#13378C2214yescomplemented
37C2×#13357C2214yescomplemented
38C2×#13357C2214yescomplemented
39C2×#13057C2218yescomplemented
40C2×#13055C2218yescomplemented
41C2×#12587C2212yes—
42C2×#12587C2212yes—
43C2×#12585C2212yes—
44C2×#12585C2211yescharacteristic
45C2×#12585C2211yescharacteristic
46#393970C2218yescomplemented
47#393969C2218yescomplemented
48#359566C2212yescomplemented
49#359566C2212yescomplemented
50#359564C2212yescomplemented
51#359564C2212yescomplemented
52#353440C2212yes—
53#353437C2211yescharacteristic
54#353437C2211yescharacteristic
55#149486C2214yescomplemented
56#149485C2214yescomplemented
57#147792C2214yescomplemented
58#147792C2214yescomplemented
59#147564C2218yescomplemented
60#147563C2218yescomplemented
61#146631C2218yescomplemented
62#146629C2218yescomplemented
63#146470C2212yescomplemented
64#146470C2212yescomplemented
65#146469C2212yescomplemented
66#146469C2212yescomplemented
67C22×#585C2311yescharacteristic
68C24×#41C2311yescharacteristic
69C2×#3487C2312yes—
70C2×#3486C2312yes—
71C22×#535D412yes—
72C22×#535C2312yes—
73C23×#121D412yes—
74C2×#3428C2311yescharacteristic
75C23×#117D411yescharacteristic
76C24×C42D411yescharacteristic
77C22×#454C2318yes—
78C22×#413C2311yescharacteristic
79C22×#413C2311yescharacteristic
80C22×#413C2312yes—
81C22×#413C2312yes—
82C2×#2811C2312yes—
83C2×#2809C2312yes—
84C2×#2733C2312yes—
85C2×#2733C2312yes—
86#37306D418yescomplemented
87#37297D418yescomplemented
88C2×#2431D414yes—
89C2×#2431C2314yes—
90C2×#2436C2314yes—
91C2×#2433D414yes—
92#34415D4116yescomplemented
93C2×#2379C2314yes—
94C2×#2372C2314yes—
95C2×#2311D412yes—
96C2×#2311C2312yes—
97C2×#2297D411yescharacteristic
98C2×#2297D411yescharacteristic
99C2×#2297D411yescharacteristic
100C2×#2297D411yescharacteristic
101C2×#2297C2311yescharacteristic
102C2×#2297C2311yescharacteristic
103C2×#2297C2311yescharacteristic
104C2×#2297C2311yescharacteristic
105C22×#391C2311yescharacteristic
106C22×#391C2311yescharacteristic
107C22×#390D411yescharacteristic
108C22×#390D411yescharacteristic
109C2×#2280D412yes—
110C22×#355C2318yes—
111C22×#355C2318yes—
112C2×#1888C2318yes—
113C2×#1793C2312yes—
114C2×#1793C2318yes—
115C2×#1793C2312yes—
116C2×#1793C2312yes—
117C2×#1793C2312yes—
118C2×#1785C2314yes—
119C2×#1785C2312yes—
120C2×#1784C2314yes—
121C2×#1784C2312yes—
122C2×#1394C2314yes—
123C2×#1394C2318yes—
124C2×#1394C2314yes—
125C2×#1394C2312yes—
126C2×#1394C2312yes—
127C2×#1394C2312yes—
128C2×#1394C2312yes—
129C2×#1394C2318yes—
130#10750C2312yes—
131#10750C2312yes—
132#10750C2312yes—
133#10750C2312yes—
134#10750C2318yes—
135C2×#585#4114yescomplemented
136C22×2−1+4C2412yes—
137C23×#41C2×D412yes—
138C23×#41C2×D411yescharacteristic
139C23×#41C2×D412yes—
140C23×#41C2×D411yescharacteristic
141C23×#41C2411yescharacteristic
142C23×#41C2411yescharacteristic
143C24×Q8C2411yescharacteristic
144C24×D4C2×D411yescharacteristic
145C24×D4C2×D412yes—
146C24×D4C2×D412yes—
147C25×C4C2×D411yescharacteristic
148C25×C4C2×D412yes—
149C2×#535C2×D418yes—
150C2×#535C2×D418yes—
151C22×#127C2×D414yes—
152C22×#127C2×D414yes—
153C22×#124C2×D414yes—
154C22×#124C2414yes—
155C22×#121C2×D414yes—
156C22×#121C2×D414yes—
157#3428#4114yescomplemented
158C2×#519C2412yes—
159C22×#117C2×D412yes—
160C22×#117C2×D412yes—
161C22×#117C2×D411yescharacteristic
162C22×#117C2×D411yescharacteristic
163C22×#117C2×D411yescharacteristic
164C22×#117C2×D411yescharacteristic
165C23×C4⋊C4C2×D411yescharacteristic
166C23×C4⋊C4C2×D411yescharacteristic
167C23×C4⋊C4C2×D412yes—
168C23×C42C2×D411yescharacteristic
169C23×C42C2×D411yescharacteristic
170C23×C42C2×D411yescharacteristic
171C23×C42C2×D411yescharacteristic
172C23×C42C2×D411yescharacteristic
173C23×C42C2×D411yescharacteristic
174C2×#413#4118yescomplemented
175C22×#101C2414yes—
176C22×#101C2412yes—
177C22×#101C2412yes—
178C22×#101C2414yes—
179#2811C2×D418yescomplemented
180#2809C2×D418yescomplemented
181#2733C2×D418yescomplemented
182#2733C2×D418yescomplemented
183#2379C2×D4116yescomplemented
184#2372C2×D4116yescomplemented
185#2311C2×D418yes—
186#2311C2×D418yes—
187#2297C2×D418yes—
188#2297C2×D414yes—
189#2297C2×D414yes—
190C2×#391C2×D411yescharacteristic
191C2×#391C2×D411yescharacteristic
192C2×#391C2×D411yescharacteristic
193C2×#391C2×D411yescharacteristic
194C2×#391C2×D414yes—
195C2×#391C2411yescharacteristic
196C2×#391C2414yes—
197C2×#391C2411yescharacteristic
198C2×#391C2411yescharacteristic
199C2×#391C2411yescharacteristic
200C2×#390C2×D411yescharacteristic
201C2×#390C2×D414yes—
202C2×#390C2×D411yescharacteristic
203C2×#390C2×D411yescharacteristic
204C2×#390C2×D411yescharacteristic
205C2×#390C2×D414yes—
206C2×#390C2×D411yescharacteristic
207C2×#390C2×D411yescharacteristic
208C2×#390C2×D411yescharacteristic
209C2×#390C2×D411yescharacteristic
210C2×C4×C22⋊C4C2×D414yes—
211C2×C4×C22⋊C4C2×D414yes—
212C26#12011yescharacteristic
213C26#12012yes—
214C2×2−1+4C2×#4114yes—
215C22×#41C22×D414yes—
216C22×#41C22×D412yes—
217C22×#41C22×D412yes—
218C22×#41C2×#4112yes—
219C22×#41C2×#4112yes—
220C22×#41C2×#4114yes—
221C22×#41H(5,𝔽2)14yes—
222C22×#41H(5,𝔽2)12yes—
223C22×#412−1+412yes—
224C23×Q8C22×D412yes—
225C23×Q8C22×D411yescharacteristic
226C23×Q8C2511yescharacteristic
227C23×D4#12012yes—
228C23×D4#12012yes—
229C23×D4#12012yes—
230C23×D4#12018yes—
231C23×D4C22×D412yes—
232C23×D4C22×D411yescharacteristic
233C23×D4C22×D414yes—
234C23×D4C22×D411yescharacteristic
235C23×D4C22×D412yes—
236C24×C4#12012yes—
237C24×C4#12011yescharacteristic
238C24×C4#12011yescharacteristic
239C24×C4#12012yes—
240C24×C4#12011yescharacteristic
241C24×C4C22×D411yescharacteristic
242C24×C4C22×D411yescharacteristic
243C24×C4C22×D412yes—
244C24×C4C22×D412yes—
245C24×C4C22×D411yescharacteristic
246C24×C4C22×D412yes—
247C2×#127#120116yes—
248C2×#124C22×D418yes—
249C2×#124C22×D418yes—
250#519C2×#4114yes—
251C2×#117#12014yes—
252C2×#117#12014yes—
253C2×#117#12018yes—
254C2×#117#12018yes—
255C22×C4⋊C4#12018yes—
256C22×C4⋊C4C22×D411yescharacteristic
257C22×C4⋊C4C22×D411yescharacteristic
258C22×C4⋊C4C22×D414yes—
259C22×C4⋊C4C22×D411yescharacteristic
260C22×C4⋊C4C22×D411yescharacteristic
261C22×C42#12014yes—
262C22×C42#12014yes—
263C22×C42C22×D414yes—
264C22×C42C22×D411yescharacteristic
265C22×C42C22×D411yescharacteristic
266C22×C42C22×D411yescharacteristic
267C22×C42C22×D411yescharacteristic
268C22×C42C22×D414yes—
269C2×#101C2×#4118yes—
270#391C22×D418yes—
271#391C22×D418yes—
272#391C22×D414yes—
273#391C22×D414yes—
274#391C22×D412yes—
275#391C22×D412yes—
276#391C22×D412yes—
277#391C22×D412yes—
278#390#120116yes—
279#390#12014yes—
280#390#12014yes—
281#390#12014yes—
282#390#12014yes—
283C25C2×#12011yescharacteristic
284C25C2×#12012yes—
285C25C2×#12011yescharacteristic
286C25C2×#12012yes—
287C25C2×#12011yescharacteristic
288C2×#41#52518yes—
289C2×#41#53218yes—
290C2×#41#53518yes—
291C2×#41#53718yes—
292C2×#41#54614yes—
293C2×#41#54618yes—
294C2×#41#54714yes—
295C2×#41#54814yes—
296C2×#41#55018yes—
297C2×#41#55014yes—
298C22×Q8C23×D412yes—
299C22×Q8C23×D411yescharacteristic
300C22×Q8C23×D411yescharacteristic
301C22×Q8C22×#4111yescharacteristic
302C22×Q8C22×#4111yescharacteristic
303C22×Q8C22×#4112yes—
304C22×Q8C2×H(5,𝔽2)12yes—
305C22×Q8C2×H(5,𝔽2)11yescharacteristic
306C22×Q8C2×2−1+411yescharacteristic
307C22×D4C2×#12018yes—
308C22×D4C2×#12012yes—
309C22×D4C2×#12014yes—
310C22×D4C2×#12012yes—
311C22×D4C2×#12014yes—
312C22×D4#52512yes—
313C22×D4#53212yes—
314C22×D4#53512yes—
315C22×D4#53712yes—
316C22×D4#54614yes—
317C22×D4#54614yes—
318C22×D4#54614yes—
319C22×D4#54714yes—
320C23×C4B(4,𝔽2)12yes—
321C23×C4B(4,𝔽2)12yes—
322C23×C4B(4,𝔽2)14yes—
323C23×C4C2×#12012yes—
324C23×C4C2×#12011yescharacteristic
325C23×C4C2×#12011yescharacteristic
326C23×C4C2×#12012yes—
327C23×C4C2×#12014yes—
328C23×C4C2×#12014yes—
329C23×C4C2×#12012yes—
330C23×C4C2×#12011yescharacteristic
331C23×C4#52512yes—
332C23×C4#53212yes—
333C23×C4#53512yes—
334C23×C4#53712yes—
335C23×C4#54614yes—
336C23×C4#54614yes—
337C23×C4#54614yes—
338C23×C4#54714yes—
339C23×C4C23×D411yescharacteristic
340C23×C4C23×D412yes—
341C2×C4⋊C4C2×#12012yes—
342C2×C4⋊C4C2×#12012yes—
343C2×C4⋊C4C2×#12012yes—
344C2×C4⋊C4C2×#12012yes—
345C2×C4⋊C4C2×#12018yes—
346C2×C42C2×#12018yes—
347C2×C42C2×#12018yes—
348C2×C42C2×#12012yes—
349C2×C42C2×#12012yes—
350C2×C42C2×#12012yes—
351C2×C42C2×#12012yes—
352C24#240414yes—
353C24#240412yes—
354C24#243014yes—
355C24#243012yes—
356C24#284614yes—
357C24#284612yes—
358C24C2×B(4,𝔽2)12yes—
359C24C2×B(4,𝔽2)12yes—
360C24C22×#12011yescharacteristic
361C2×Q8C2×#52512yes—
362C2×Q8C2×#53212yes—
363C2×Q8C2×#53512yes—
364C2×Q8C2×#53712yes—
365C2×Q8C2×#54611yescharacteristic
366C2×Q8C2×#54612yes—
367C2×Q8C2×#54711yescharacteristic
368C2×Q8C2×#54811yescharacteristic
369C2×Q8C2×#55012yes—
370C2×Q8C2×#55011yescharacteristic
371C2×D4#238814yes—
372C2×D4#240418yes—
373C2×D4#240714yes—
374C2×D4#240718yes—
375C2×D4#243018yes—
376C2×D4#243114yes—
377C2×D4#243118yes—
378C2×D4#261514yes—
379C22×C4#238814yes—
380C22×C4#240418yes—
381C22×C4#240412yes—
382C22×C4#240718yes—
383C22×C4#240714yes—
384C22×C4#243018yes—
385C22×C4#243012yes—
386C22×C4#243118yes—
387C22×C4#243114yes—
388C22×C4#261514yes—
389C22×C4#284612yes—
390C22×C4C2×B(4,𝔽2)12yes—
391C22×C4C2×B(4,𝔽2)12yes—
392C22×C4C2×B(4,𝔽2)12yes—
393C22×C4C2×B(4,𝔽2)11yescharacteristic
394C22×C4C2×B(4,𝔽2)11yescharacteristic
395C22×C4#302612yes—
396C22×C4#302612yes—
397C22×C4C22×#12011yescharacteristic
398C22×C4C22×#12012yes—
399C22×C4C22×#12011yescharacteristic
400C22×C4C22×#12012yes—
401C22×C4C2×#52511yescharacteristic
402C22×C4C2×#53211yescharacteristic
403C22×C4C2×#53511yescharacteristic
404C22×C4C2×#53711yescharacteristic
405C22×C4C2×#54612yes—
406C22×C4C2×#54612yes—
407C22×C4C2×#54612yes—
408C22×C4C2×#54712yes—
409C23#1488518yes—
410C23#1488514yes—
411C23#3357014yes—
412C23C2×#240411yescharacteristic
413C23C2×#243011yescharacteristic
414C23C2×#284611yescharacteristic
415C23C22×B(4,𝔽2)11yescharacteristic
416C23C22×B(4,𝔽2)11yescharacteristic
417C23C2×#302611yescharacteristic
418C23C2×#302611yescharacteristic
419C2×C4#1488514yes—
420C2×C4#2398412yes—
421C2×C4#2398412yes—
422C2×C4#2398714yes—
423C2×C4#2556312yes—
424C2×C4#2556312yes—
425C2×C4#2556314yes—
426C2×C4#3289812yes—
427C2×C4#3289812yes—
428C2×C4#3289814yes—
429C2×C4C2×#238811yescharacteristic
430C2×C4C2×#261511yescharacteristic
431C2×C4C2×#240712yes—
432C2×C4C2×#240711yescharacteristic
433C2×C4C2×#243112yes—
434C2×C4C2×#243111yescharacteristic
435C2×C4C2×#240412yes—
436C2×C4C2×#243012yes—
437C22C2×#1488511yescharacteristic
438C22C2×#2398411yescharacteristic
439C22C2×#2398711yescharacteristic
440C22C2×#2556311yescharacteristic
441C22C2×#2556311yescharacteristic
442C22C2×#3289811yescharacteristic
443C22C2×#3289811yescharacteristic
444C22C2×#3357012yes—
445C22#762290312yes—
446C4#40395214yes—
447C4#40395214yes—
448C2#7226044412yes—
449C2#7226100511yescharacteristic
450C2#7226119011yescharacteristic
451C2#7226123112yes—
452C2C2×#40395211yescharacteristic
subgroupG/H
center, socleC23C22×B(4,𝔽2)
commutator, Frattini, cosocleC23×Q8C25
LCSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×#41 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72261994C2semidirect
2. #72261982C2semidirect
3. #72261979C2semidirect
4. C2×#498991C2semidirect
5. #72261966C2semidirect
6. #72261960C2semidirect
7. #72261949C2semidirect
8. #72261939C2semidirect
9. C2×#7711868C2semidirect
10. #72261738C2semidirect
11. #72260855C2semidirect
12. C2×#498909C2semidirect
13. #72250630C2semidirect
14. C2×#494622C2semidirect
15. #72238533C2semidirect
18. C2×#46318C22semidirect
19. C2×#46099C22semidirect
20. C2×#60048C22semidirect
21. C2×#59885C22semidirect
35. C2×#13379C22semidirect
36. C2×#13378C22semidirect
37. C2×#13357C22semidirect
38. C2×#13357C22semidirect
39. C2×#13057C22semidirect
40. C2×#13055C22semidirect
46. #393970C22semidirect
47. #393969C22semidirect
48. #359566C22semidirect
49. #359566C22semidirect
50. #359564C22semidirect
51. #359564C22semidirect
55. #149486C22semidirect
56. #149485C22semidirect
57. #147792C22semidirect
58. #147792C22semidirect
59. #147564C22semidirect
60. #147563C22semidirect
61. #146631C22semidirect
62. #146629C22semidirect
63. #146470C22semidirect
64. #146470C22semidirect
65. #146469C22semidirect
66. #146469C22semidirect
86. #37306D4semidirect
87. #37297D4semidirect
92. #34415D4semidirect
135. C2×#585#41semidirect
157. #3428#41semidirect
174. C2×#413#41semidirect
179. #2811C2×D4semidirect
180. #2809C2×D4semidirect
181. #2733C2×D4semidirect
182. #2733C2×D4semidirect
183. #2379C2×D4semidirect
184. #2372C2×D4semidirect

Maximal subgroups

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

subgroupindex
#722619942
#722619822
#722619792
C2×#4989912
#722619662
#722619602
#722619492
#722619392
C2×#77118682
#722617382
#722608552
C2×#4989092
#722506302
C2×#4946222
#722385332