Finite Group Databasebeta

← #72340729

Subgroups — #72340729

Order 2048.

At least 9608 conjugacy classes of subgroups; only the 1008 normal subgroups are shown.

The subgroup lattice has 417 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72340727C214yesmaximal, complemented
2#72340726C212yesmaximal, complemented
3#72340725C212yesmaximal, complemented
4#72340699C211yesmaximal, characteristic, complemented
5#72340694C214yesmaximal, complemented
6C2×#134056C211yesmaximal, characteristic, complemented
7#72340667C212yesmaximal, complemented
8#72340660C212yesmaximal, complemented
9#72340425C212yesmaximal, complemented
10#72340394C212yesmaximal, complemented
11#72337681C212yesmaximal, complemented
12#72283702C214yesmaximal, complemented
13C2×#417798C211yesmaximal, characteristic, complemented
14#72213362C212yesmaximal, complemented
15C2×#60644C2211yescharacteristic, complemented
16#8934620C2211yescharacteristic, complemented
17#8933848C2211yescharacteristic, complemented
18#8933774C2212yescomplemented
19#8929995C2212yescomplemented
20#8929993C2212yescomplemented
21#7714980C2212yes—
22#7713426C2212yes—
23#7711375C2212yes—
24C2×#35527C2211yescharacteristic
25C2×#23445C2211yescharacteristic, complemented
26C2×#23406C2211yescharacteristic, complemented
27#6347124C2214yescomplemented
28#485668C2214yescomplemented
29#485667C2214yescomplemented
30#485648C2218yescomplemented
31#485634C2218yescomplemented
32#485615C2214yescomplemented
33#485614C2214yescomplemented
34#481702C2214yescomplemented
35#481680C2214yescomplemented
36C2×#11507C2212yescomplemented
37C2×#11503C2212yescomplemented
38#417798C2212yes—
39#417798C2212yescomplemented
40#417798C2212yescomplemented
41#417798C2212yescomplemented
42C2×#11006C2211yescharacteristic, complemented
43C2×#10939C2211yescharacteristic, complemented
44C2×#10959C2211yescharacteristic, complemented
45#246991C2218yescomplemented
46#246989C2214yescomplemented
47#246987C2214yescomplemented
48#200460C2214yescomplemented
49C2×#8522C2212yescomplemented
50#134056C2212yescomplemented
51#134056C2212yescomplemented
52#134056C2212yescomplemented
53#134056C2212yescomplemented
54#132002C2214yescomplemented
55#132001C2214yescomplemented
56#131999C2218yescomplemented
57#130079C2214yescomplemented
58#130078C2214yescomplemented
59#130076C2218yescomplemented
60#130059C2218yescomplemented
61#130057C2214yescomplemented
62#130056C2214yescomplemented
63C23×H(5,𝔽2)D411yescharacteristic
64C2×#3545D412yescomplemented
65C2×#3449C2311yescharacteristic
66C22×#535D411yescharacteristic
67C22×#535C2311yescharacteristic
68C23×#127D411yescharacteristic
69#60644C2312yes—
70#60644C2312yes—
71C2×#2873C2311yescharacteristic
72#60048C2312yes—
73#60047C2312yes—
74#59799D414yescomplemented
75#51978D412yescomplemented
76#51978C2312yes—
77#51903D414yescomplemented
78#51903C2314yes—
79#51902D412yescomplemented
80#51902C2312yes—
81#47563C2312yes—
82#47195C2312yes—
83#47026C2314yes—
84#46998D412yes—
85#46458D414yes—
86#46384D412yes—
87#46318D412yescomplemented
88#46316D412yescomplemented
89#46274C2314yes—
90#37614D414yescomplemented
91#37517D414yescomplemented
92#37475D412yes—
93#37436D412yes—
94#37194D412yescomplemented
95#37190D412yescomplemented
96#36725D414yes—
97#35527D411yescharacteristic
98#35527D411yescharacteristic
99#35527D411yescharacteristic
100#35527D411yescharacteristic
101#35527C2311yescharacteristic
102#35527C2311yescharacteristic
103#35527C2311yescharacteristic
104#35527C2311yescharacteristic
105#35118C2314yes—
106#34366D412yescomplemented
107C2×#2309C2311yescharacteristic
108C2×#2297D411yescharacteristic
109C2×#2297C2311yescharacteristic
110#23445C2312yes—
111#23445C2312yes—
112#23406C2312yes—
113#23406C2312yes—
114C2×#1742C2311yescharacteristic
115C2×#1740C2311yescharacteristic
116#18339C2318yescomplemented
117#18332C2318yescomplemented
118C2×#1464C2314yes—
119#11767C2318yescomplemented
120#11765C2318yescomplemented
121#11760C2318yescomplemented
122#11758C2318yescomplemented
123#11507C2314yes—
124#11507C2314yes—
125#11503C2314yes—
126#11503C2314yes—
127#11006C2312yes—
128#11006C2312yes—
129#10959C2312yes—
130#10959C2312yes—
131#10939C2312yes—
132#10939C2312yes—
133#9388C2318yescomplemented
134#9387C2318yescomplemented
135#8522C2314yes—
136#8522C2314yes—
137C22×H(5,𝔽2)C2×D411yescharacteristic
138C22×H(5,𝔽2)C2×D411yescharacteristic
139C22×H(5,𝔽2)C2×D411yescharacteristic
140C22×H(5,𝔽2)C2×D411yescharacteristic
141C23×#41C2×D411yescharacteristic
142C23×#41C2×D411yescharacteristic
143C24×D4C2×D411yescharacteristic
144C24×D4C2×D411yescharacteristic
145C2×#563C2×D412yescomplemented
146C2×#562C2×D414yescomplemented
147C2×H(3,𝔽4)C2×D412yescomplemented
148#3449C2412yes—
149C2×#535C2×D411yescharacteristic
150C2×#535C2×D411yescharacteristic
151C2×#535C2×D411yescharacteristic
152C2×#535C2×D411yescharacteristic
153C2×#535C2×D411yescharacteristic
154C2×#535C2×D411yescharacteristic
155C2×#535C2×D412yes—
156C2×#535C2×D412yes—
157C2×#535C2411yescharacteristic
158C2×#535C2411yescharacteristic
159C22×#127C2×D411yescharacteristic
160C22×#127C2×D411yescharacteristic
161C22×#127C2×D411yescharacteristic
162C22×#127C2×D411yescharacteristic
163C22×#127C2×D412yes—
164C2×#529C2×D414yes—
165C22×#124C2×D412yes—
166C22×#117C2×D411yescharacteristic
167C22×#117C2411yescharacteristic
168C23×C42C2×D411yescharacteristic
169#2873C2412yes—
170#2842C2×D412yes—
171#2840C2×D412yes—
172#2830C2×D414yescomplemented
173#2828C2×D414yescomplemented
174#2812C2×D414yes—
175#2811C2×D412yescomplemented
176#2809C2×D412yescomplemented
177#2799C2×D414yescomplemented
178#2736C2×D414yes—
179#2735C2×D414yes—
180#2734C2×D412yescomplemented
181#2733C2×D412yescomplemented
182#2728C2×D414yescomplemented
183#2670C2×D414yes—
184#2668C2×D414yes—
185#2657C2×D414yescomplemented
186#2657C2×D414yescomplemented
187#2641C2×D414yescomplemented
188#2641C2×D414yescomplemented
189#2627C2×D414yes—
190#2626C2×D414yes—
191#2618C2×D414yes—
192#2617C2×D414yes—
193#2611C2×D414yes—
194#2611C2414yes—
195#2610C2×D414yes—
196#2610C2414yes—
197#2599C2×D414yes—
198#2599C2×D414yes—
199#2598C2×D414yes—
200#2598C2×D414yes—
201#2593C2×D414yescomplemented
202#2555C2×D414yescomplemented
203#2483C2×D412yes—
204#2437C2×D414yes—
205#2403C2×D412yes—
206#2372C2×D412yescomplemented
207#2372C2×D412yescomplemented
208#2372C2×D414yescomplemented
209#2309C2×D412yes—
210#2309C2×D412yes—
211#2309C2×D411yescharacteristic
212#2309C2×D411yescharacteristic
213#2309C2411yescharacteristic
214#2309C2411yescharacteristic
215#2297C2×D412yes—
216#2297C2×D412yes—
217#2297C2×D411yescharacteristic
218#2297C2×D411yescharacteristic
219#2297C2×D411yescharacteristic
220#2297C2×D411yescharacteristic
221#2297C2×D411yescharacteristic
222#2297C2×D411yescharacteristic
223#2297C2411yescharacteristic
224#2297C2411yescharacteristic
225C2×#391C2×D412yes—
226C2×#390C2×D412yes—
227#2283C2×D414yes—
228C2×#379C2×D412yes—
229C2×#379C2×D412yes—
230#1742C2412yes—
231#1740C2412yes—
232#1464C2418yes—
233C26#12011yescharacteristic
234C26#12011yescharacteristic
235C22×#41#12011yescharacteristic
236C22×#41#12011yescharacteristic
237C22×#41#12011yescharacteristic
238C22×#41#12011yescharacteristic
239C22×#41C22×D411yescharacteristic
240C22×#41C22×D411yescharacteristic
241C22×#41C22×D411yescharacteristic
242C22×#41C22×D411yescharacteristic
243C23×Q8#12011yescharacteristic
244C23×Q8#12011yescharacteristic
245C23×D4#12011yescharacteristic
246C23×D4#12011yescharacteristic
247C23×D4#12011yescharacteristic
248C23×D4#12012yes—
249C23×D4#12012yes—
250C23×D4#12011yescharacteristic
251C23×D4C22×D411yescharacteristic
252C23×D4C22×D411yescharacteristic
253C23×D4C22×D411yescharacteristic
254C23×D4C22×D411yescharacteristic
255C24×C4C22×D411yescharacteristic
256C2×#127#12012yes—
257C2×#127#12012yes—
258C2×#127C22×D412yes—
259C2×#127C22×D412yes—
260C2×#125C22×D414yes—
261C2×#124#12012yes—
262C2×#124#12012yes—
263C2×#124C22×D412yes—
264C2×#124C22×D412yes—
265C2×#124C22×D414yes—
266C2×C22⋊Q8#12014yes—
267C2×#121#12014yes—
268C2×C4×Q8#12014yes—
269C2×C4×D4#12014yes—
270C2×#117#12012yes—
271C2×#117C22×D411yescharacteristic
272C2×#117C22×D411yescharacteristic
273C2×#117C22×D411yescharacteristic
274C2×#117C22×D412yes—
275C2×#117C22×D412yes—
276C2×#117C22×D412yes—
277C2×#117C2511yescharacteristic
278C22×C4⋊C4C22×D412yes—
279C22×C22⋊C4#12012yes—
280C22×C22⋊C4C22×D412yes—
281C22×C42#12012yes—
282C22×C42#12012yes—
283C22×C42C22×D412yes—
284C22×C42C22×D411yescharacteristic
285C22×C42C22×D411yescharacteristic
286#400#12014yes—
287#400C22×D414yes—
288#399#12014yes—
289#399C22×D414yes—
290#398#12014yes—
291#398C22×D414yes—
292#397#12014yes—
293#397C22×D414yes—
294#395#12014yes—
295#395C22×D414yes—
296#394#12014yes—
297#394C22×D414yes—
298#393#12014yes—
299#393C22×D414yes—
300#392#12014yes—
301#392C22×D414yes—
302#391#12012yes—
303#391#12012yes—
304#391C22×D412yes—
305#391C22×D412yes—
306#390#12012yes—
307#390#12012yes—
308#390C22×D412yes—
309#390C22×D412yes—
310#383#12014yes—
311#383#12014yes—
312#380#12014yes—
313#380#12014yes—
314#379#12012yes—
315#379#12012yes—
316#379#12012yes—
317#379#12012yes—
318#379C22×D412yes—
319#379C22×D412yes—
320#379C22×D412yes—
321#379C22×D412yes—
322#376C22×D414yes—
323#376C22×D414yes—
324C25#45314yes—
325C25C2×#12011yescharacteristic
326C25C2×#12011yescharacteristic
327C25C2×#12011yescharacteristic
328C25C2×#12011yescharacteristic
329C25C2×#12011yescharacteristic
330C22×Q8#45314yes—
331C22×Q8B(4,𝔽2)12yes—
332C22×Q8B(4,𝔽2)12yes—
333C22×Q8C2×#12011yescharacteristic
334C22×Q8C2×#12011yescharacteristic
335C22×Q8C2×#12011yescharacteristic
336C22×Q8C2×#12011yescharacteristic
337C22×Q8C2×#12014yes—
338C22×D4C2×#12012yes—
339C22×D4C2×#12012yes—
340C22×D4C2×#12011yescharacteristic
341C22×D4C2×#12011yescharacteristic
342C22×D4C2×#12011yescharacteristic
343C22×D4C2×#12011yescharacteristic
344C22×D4C2×#12014yes—
345C23×C4C2×#12011yescharacteristic
346C23×C4C2×#12011yescharacteristic
347C23×C4C2×#12011yescharacteristic
348C23×C4C2×#12011yescharacteristic
349C23×C4C2×#12014yes—
350C23×C4C2×#12011yescharacteristic
351C23×C4C23×D411yescharacteristic
352C2×C4⋊C4C2×#12014yes—
353C2×C4⋊C4C2×#12014yes—
354C2×C4⋊C4C2×#12014yes—
355C2×C4⋊C4C2×#12012yes—
356C2×C4⋊C4C23×D412yes—
357C2×C22⋊C4C2×#12012yes—
358C2×C22⋊C4C2×#12012yes—
359C2×C22⋊C4C2×#12012yes—
360C2×C22⋊C4C2×#12014yes—
361C2×C22⋊C4C23×D412yes—
362C2×C42C2×#12012yes—
363C2×C42C2×#12014yes—
364C2×C42C2×#12012yes—
365C2×C42C2×#12012yes—
366C2×C42C23×D412yes—
367#95C2×#12014yes—
368#95C2×#12014yes—
369#95C2×#12014yes—
370#95C2×#12014yes—
371C24#284611yescharacteristic
372C24#284611yescharacteristic
373C24C2×#45312yes—
374C24C2×#45312yes—
375C24C2×#45312yes—
376C24#301912yes—
377C24C22×#12011yescharacteristic
378C2×Q8C2×#45312yes—
379C2×Q8C2×#45312yes—
380C2×Q8C2×B(4,𝔽2)12yes—
381C2×Q8C2×B(4,𝔽2)12yes—
382C22×C4#284611yescharacteristic
383C22×C4#284611yescharacteristic
384C22×C4#284614yes—
385C22×C4C2×#45312yes—
386C22×C4#301912yes—
387C22×C4C2×B(4,𝔽2)11yescharacteristic
388C22×C4C2×B(4,𝔽2)11yescharacteristic
389C22×C4#302611yescharacteristic
390C22×C4#302611yescharacteristic
391C22×C4C22×#12011yescharacteristic
392C22×C4C22×#12011yescharacteristic
393C22×C4C22×#12014yes—
394C23#3288812yes—
395C23#3288812yes—
396C23#3288812yes—
397C23#3289811yescharacteristic
398C23#3289811yescharacteristic
399C23C2×#284611yescharacteristic
400C23C22×#45311yescharacteristic
401C23C2×#301911yescharacteristic
402C2×C4#3288812yes—
403C2×C4#3289811yescharacteristic
404C2×C4#3289811yescharacteristic
405C2×C4C22×#45311yescharacteristic
406C2×C4C2×#301911yescharacteristic
407C2×C4C22×B(4,𝔽2)11yescharacteristic
408C2×C4C2×#302611yescharacteristic
409C22#34644612yes—
410C22#34644612yes—
411C22C2×#3288811yescharacteristic
412C22C2×#3288811yescharacteristic
413C22C2×#3289811yescharacteristic
414C2C2×#34644611yescharacteristic
415C2#7234064412yes—
subgroupG/H
center, socleC22C2×#32898
commutator, Frattini, cosocleC2×#117C25
LCSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#535 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72340727C2semidirect
2. #72340726C2semidirect
3. #72340725C2semidirect
4. #72340699C2semidirect
5. #72340694C2semidirect
6. C2×#134056C2semidirect
7. #72340667C2semidirect
8. #72340660C2semidirect
9. #72340425C2semidirect
10. #72340394C2semidirect
11. #72337681C2semidirect
12. #72283702C2semidirect
13. C2×#417798C2semidirect
14. #72213362C2semidirect
15. C2×#60644C22semidirect
16. #8934620C22semidirect
17. #8933848C22semidirect
18. #8933774C22semidirect
19. #8929995C22semidirect
20. #8929993C22semidirect
25. C2×#23445C22semidirect
26. C2×#23406C22semidirect
27. #6347124C22semidirect
28. #485668C22semidirect
29. #485667C22semidirect
30. #485648C22semidirect
31. #485634C22semidirect
32. #485615C22semidirect
33. #485614C22semidirect
34. #481702C22semidirect
35. #481680C22semidirect
36. C2×#11507C22semidirect
37. C2×#11503C22semidirect
39. #417798C22semidirect
40. #417798C22semidirect
41. #417798C22semidirect
42. C2×#11006C22semidirect
43. C2×#10939C22semidirect
44. C2×#10959C22semidirect
45. #246991C22semidirect
46. #246989C22semidirect
47. #246987C22semidirect
48. #200460C22semidirect
49. C2×#8522C22semidirect
50. #134056C22semidirect
51. #134056C22semidirect
52. #134056C22semidirect
53. #134056C22semidirect
54. #132002C22semidirect
55. #132001C22semidirect
56. #131999C22semidirect
57. #130079C22semidirect
58. #130078C22semidirect
59. #130076C22semidirect
60. #130059C22semidirect
61. #130057C22semidirect
62. #130056C22semidirect
64. C2×#3545D4semidirect
74. #59799D4semidirect
75. #51978D4semidirect
77. #51903D4semidirect
79. #51902D4semidirect
87. #46318D4semidirect
88. #46316D4semidirect
90. #37614D4semidirect
91. #37517D4semidirect
94. #37194D4semidirect
95. #37190D4semidirect
106. #34366D4semidirect
116. #18339C23semidirect
117. #18332C23semidirect
119. #11767C23semidirect
120. #11765C23semidirect
121. #11760C23semidirect
122. #11758C23semidirect
133. #9388C23semidirect
134. #9387C23semidirect
145. C2×#563C2×D4semidirect
146. C2×#562C2×D4semidirect
147. C2×H(3,𝔽4)C2×D4semidirect
172. #2830C2×D4semidirect
173. #2828C2×D4semidirect
175. #2811C2×D4semidirect
176. #2809C2×D4semidirect
177. #2799C2×D4semidirect
180. #2734C2×D4semidirect
181. #2733C2×D4semidirect
182. #2728C2×D4semidirect
185. #2657C2×D4semidirect
186. #2657C2×D4semidirect
187. #2641C2×D4semidirect
188. #2641C2×D4semidirect
201. #2593C2×D4semidirect
202. #2555C2×D4semidirect
206. #2372C2×D4semidirect
207. #2372C2×D4semidirect
208. #2372C2×D4semidirect

Maximal subgroups

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

subgroupindex
#723407272
#723407262
#723407252
#723406992
#723406942
C2×#1340562
#723406672
#723406602
#723404252
#723403942
#723376812
#722837022
C2×#4177982
#722133622