Finite Group Databasebeta

← #72266333

Subgroups — #72266333

Order 2048.

At least 9094 conjugacy classes of subgroups; only the 720 normal subgroups are shown.

The subgroup lattice has 338 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72266329C214yesmaximal, complemented
2#72255642C211yesmaximal, characteristic, complemented
3#72251570C211yesmaximal, characteristic, complemented
4#72241156C211yesmaximal, characteristic
5#72236837C211yesmaximal, characteristic, complemented
6#72236714C214yesmaximal, complemented
7#72235848C211yesmaximal, characteristic, complemented
8#72226038C211yesmaximal, characteristic, complemented
9#72225994C211yesmaximal, characteristic, complemented
10#8390796C414yescomplemented
11#8181078C2214yes—
12#8093110C411yescharacteristic, complemented
13#8093108C411yescharacteristic, complemented
14#7670333C411yescharacteristic, complemented
15#7670203C411yescharacteristic, complemented
16#7642273C2211yescharacteristic
17C2×#35346C2211yescharacteristic
18C2×#34980C2211yescharacteristic
19#555894C2211yescharacteristic, complemented
20#496225C2214yescomplemented
21#496225C2214yescomplemented
22#496194C2214yescomplemented
23#382422C2211yescharacteristic, complemented
24#379759C2212yescomplemented
25#379754C2212yescomplemented
26C2×#9497C2212yescomplemented
27C2×#9497C2212yescomplemented
28C2×#9495C2212yescomplemented
29#201322C2212yescomplemented
30C2×#8547C2211yescharacteristic, complemented
31C2×#8541C2211yescharacteristic, complemented
32C23×H(5,𝔽2)D411yescharacteristic
33C25×D4D411yescharacteristic
34C2×#3490D411yescharacteristic
35C23×#120C2311yescharacteristic
36#59370C2×C414yes—
37#59370D414yes—
38#58990C2×C414yes—
39#58990D414yes—
40#52004D414yes—
41#51940D414yes—
42#51940D414yes—
43#51902D414yes—
44#45727C2×C414yes—
45#45727D414yes—
46#45439C2×C414yes—
47#45439D414yes—
48#36703C2×C412yes—
49#36703D412yes—
50C2×#2849C2×C411yescharacteristic
51C2×#2849D411yescharacteristic
52C2×#2846C2×C411yescharacteristic
53C2×#2846D411yescharacteristic
54#35390C2×C411yescharacteristic
55#35390D411yescharacteristic
56#35367C2×C411yescharacteristic
57#35367D411yescharacteristic
58#35346C2×C411yescharacteristic
59#35346C2×C412yes—
60#35346C2×C411yescharacteristic
61#35346D412yes—
62#35346D411yescharacteristic
63#35346D411yescharacteristic
64#35222C2312yes—
65#35050D412yes—
66C2×#2405D412yes—
67C2×#2410C2×C412yes—
68C2×#2410D412yes—
69C2×#2403D412yes—
70C2×#2403C2312yes—
71C2×#2403C2312yes—
72#34176D411yescharacteristic
73C2×#2309C2311yescharacteristic, complemented
74C2×#2308D411yescharacteristic
75C2×#2308C2311yescharacteristic, complemented
76C22×#379D411yescharacteristic
77C2×#1783C2312yescomplemented
78C2×#1781C2312yescomplemented
79C2×#1781C2312yescomplemented
80C27C22⋊C411yescharacteristic, complemented
81C27C2×D411yescharacteristic
82C22×H(5,𝔽2)C22⋊C411yescharacteristic
83C22×H(5,𝔽2)C22⋊C411yescharacteristic
84C22×H(5,𝔽2)C22⋊C412yes—
85C24×D4C22⋊C412yescomplemented
86C24×D4C2×D412yes—
87C24×D4C2×D411yescharacteristic
88C24×D4C2×D411yescharacteristic
89C24×D4C2×D412yes—
90C24×D4C2411yescharacteristic
91C25×C4C22⋊C411yescharacteristic, complemented
92C2×H(3,𝔽4)C22⋊C414yescomplemented
93C2×D42C22⋊C414yescomplemented
94C2×#543C22⋊C411yescharacteristic, complemented
95C2×#538C22⋊C412yescomplemented
96C2×#535C2×D414yes—
97C2×#534C22⋊C411yescharacteristic, complemented
98C2×#525C2×D412yes—
99C22×#120C22×C412yes—
100C22×#120C22×C411yescharacteristic
101C22×#120C22×C411yescharacteristic
102C22×#120C2×D412yes—
103C22×#120C2×D414yes—
104C22×#120C2×D411yescharacteristic
105C22×#120C2×D411yescharacteristic
106C22×#120C2×D411yescharacteristic
107C22×#120C2×D411yescharacteristic
108C22×#120C2×D411yescharacteristic
109C22×#120C2×D411yescharacteristic
110C23×C22⋊C4C22⋊C412yescomplemented
111C23×C22⋊C4C2×D411yescharacteristic
112C23×C22⋊C4C2×D411yescharacteristic
113C23×C22⋊C4C2×D411yescharacteristic
114#2748C22⋊C414yescomplemented
115#2748C22⋊C414yescomplemented
116#2684C22⋊C414yescomplemented
117#2684C22⋊C414yescomplemented
118#2679C22⋊C414yescomplemented
119#2679C22⋊C414yescomplemented
120#2669C22⋊C414yescomplemented
121#2669C22⋊C414yescomplemented
122#2561C22⋊C412yescomplemented
123#2561C22⋊C412yescomplemented
124#2539C22⋊C412yescomplemented
125#2539C22⋊C412yescomplemented
126#2435C22×C412yes—
127#2435C2×D412yes—
128#2435C2×D414yes—
129#2433C22×C412yes—
130#2433C2×D412yes—
131#2433C2×D414yes—
132#2403C22⋊C412yes—
133#2403C22⋊C412yes—
134#2403C22⋊C412yes—
135#2403C22⋊C412yes—
136#2403C22×C412yes—
137#2403C22×C412yes—
138#2403C2×D414yes—
139#2403C2×D414yes—
140#2403C2×D414yes—
141#2403C2×D414yes—
142#2403C2×D414yes—
143#2403C2×D414yes—
144#2403C2×D412yes—
145#2403C2×D412yes—
146#2380C22⋊C414yescomplemented
147#2375C22⋊C414yescomplemented
148#2375C22⋊C414yescomplemented
149#2372C22⋊C414yescomplemented
150#2358C22⋊C414yescomplemented
151#2358C22⋊C414yescomplemented
152#2312C22⋊C412yescomplemented
153#2309C2×D414yes—
154#2308C22⋊C411yescharacteristic
155#2308C22⋊C412yes—
156#2308C22⋊C411yescharacteristic
157C2×#386C22×C412yes—
158C2×#386C2×D412yes—
159C2×#386C2×D414yes—
160C2×#385C22⋊C412yescomplemented
161C2×#385C22⋊C412yescomplemented
162#2280C2×D412yes—
163C2×#379C22⋊C411yescharacteristic, complemented
164C2×#379C22⋊C412yescomplemented
165C2×#379C22⋊C411yescharacteristic, complemented
166C2×#379C22⋊C412yescomplemented
167C2×#379C2×D412yes—
168C2×#379C2×D412yes—
169#2273C22⋊C411yescharacteristic, complemented
170#2273C22⋊C411yescharacteristic, complemented
171C26C2×C22⋊C411yescharacteristic
172C26C2×C22⋊C411yescharacteristic
173C26C2×C22⋊C412yes—
174C26#12011yescharacteristic
175C26#12011yescharacteristic
176C26#12011yescharacteristic
177C26#12011yescharacteristic
178C26#12012yes—
179C26C22×D411yescharacteristic
180C26C22×D411yescharacteristic
181C22×#41#12012yes—
182C23×D4C2×C22⋊C411yescharacteristic
183C23×D4C2×C22⋊C411yescharacteristic
184C23×D4C2×C22⋊C412yes—
185C23×D4C2×C22⋊C412yes—
186C23×D4C2×C22⋊C411yescharacteristic
187C23×D4C2×C22⋊C411yescharacteristic
188C23×D4#12011yescharacteristic
189C23×D4#12011yescharacteristic
190C23×D4#12012yes—
191C23×D4#12012yes—
192C23×D4#12012yes—
193C23×D4#12012yes—
194C23×D4#12012yes—
195C23×D4#12012yes—
196C23×D4#12011yescharacteristic
197C23×D4#12011yescharacteristic
198C23×D4C23×C411yescharacteristic
199C23×D4C22×D411yescharacteristic
200C23×D4C22×D414yes—
201C23×D4C22×D414yes—
202C23×D4C22×D411yescharacteristic
203C23×D4C22×D411yescharacteristic
204C24×C4C2×C22⋊C412yes—
205C24×C4#12011yescharacteristic
206C24×C4#12011yescharacteristic
207C24×C4#12012yes—
208C24×C4C22×D411yescharacteristic
209C2×#127C2×C22⋊C414yes—
210C2×#127#12014yes—
211C2×#124C2×C22⋊C414yes—
212C2×#124#12014yes—
213C2×C22⋊Q8C2×C22⋊C412yes—
214C2×C22⋊Q8#12012yes—
215C2×#121C2×C22⋊C412yes—
216C2×#121#12012yes—
217C2×#120C2×C22⋊C414yes—
218C2×#120C2×C22⋊C414yes—
219C2×#120C2×C22⋊C414yes—
220C2×#120C2×C22⋊C414yes—
221C2×#120#12014yes—
222C2×#120#12014yes—
223C2×#120#12014yes—
224C2×#120#12014yes—
225C2×#117#12012yes—
226C2×#117#12014yes—
227C22×C22⋊C4C2×C22⋊C411yescharacteristic
228C22×C22⋊C4C2×C22⋊C411yescharacteristic
229C22×C22⋊C4C2×C22⋊C411yescharacteristic
230C22×C22⋊C4C2×C22⋊C411yescharacteristic
231C22×C22⋊C4C2×C22⋊C411yescharacteristic
232C22×C22⋊C4C2×C22⋊C411yescharacteristic
233C22×C22⋊C4C2×C22⋊C412yes—
234C22×C22⋊C4C2×C22⋊C412yes—
235C22×C22⋊C4#12011yescharacteristic
236C22×C22⋊C4#12011yescharacteristic
237C22×C22⋊C4#12011yescharacteristic
238C22×C22⋊C4#12011yescharacteristic
239C22×C22⋊C4#12011yescharacteristic
240C22×C22⋊C4#12011yescharacteristic
241C22×C22⋊C4#12014yes—
242#390C2×C22⋊C414yes—
243#390C2×C22⋊C414yes—
244#390#12014yes—
245#390#12014yes—
246#386C2×C22⋊C414yes—
247#386C2×C22⋊C414yes—
248#386C2×C22⋊C414yes—
249#386C2×C22⋊C414yes—
250#386#12014yes—
251#386#12014yes—
252#386#12014yes—
253#386#12014yes—
254#379C2×C22⋊C412yes—
255#379C2×C22⋊C412yes—
256#379#12014yes—
257#379#12014yes—
258#379#12014yes—
259C4×C22⋊C4C2×C22⋊C412yes—
260C4×C22⋊C4C2×C22⋊C412yes—
261C4×C22⋊C4#12014yes—
262C25#37911yescharacteristic
263C25#37911yescharacteristic
264C25#37911yescharacteristic
265C25#37911yescharacteristic
266C25#37912yes—
267C25#37912yes—
268C25#37912yes—
269C25#37912yes—
270C25B(4,𝔽2)12yes—
271C25#45812yes—
272C25C22×C22⋊C411yescharacteristic
273C25C22×C22⋊C411yescharacteristic
274C25C2×#12012yes—
275C25C2×#12011yescharacteristic
276C25C2×#12011yescharacteristic
277C25C2×#12012yes—
278C25C2×#12012yes—
279C25C2×#12012yes—
280C25C2×#12011yescharacteristic
281C22×Q8#37912yes—
282C22×D4#37912yes—
283C22×D4C22×C22⋊C414yes—
284C22×D4C2×#12014yes—
285C22×D4C2×#12014yes—
286C22×D4C2×#12014yes—
287C23×C4#37911yescharacteristic
288C23×C4#37911yescharacteristic
289C23×C4#37911yescharacteristic
290C23×C4#37911yescharacteristic
291C23×C4#37912yes—
292C23×C4#37912yes—
293C23×C4#37912yes—
294C23×C4#37912yes—
295C23×C4C22×C22⋊C411yescharacteristic
296C23×C4C2×#12011yescharacteristic
297C23×C4C2×#12012yes—
298C23×C4C2×#12012yes—
299C2×C4⋊C4#37912yes—
300C2×C4⋊C4#37912yes—
301C2×C22⋊C4#37914yes—
302C2×C22⋊C4#37914yes—
303C2×C22⋊C4#37914yes—
304C2×C22⋊C4#37914yes—
305C2×C42#37914yes—
306C2×C42#37914yes—
307C24#179012yes—
308C24#179012yes—
309C24C2×#37911yescharacteristic
310C24C2×#37912yes—
311C24C2×#37912yes—
312C24#284611yescharacteristic
313C24#284612yes—
314C24#284612yes—
315C24#284611yescharacteristic
316C24C2×B(4,𝔽2)11yescharacteristic
317C24C2×#45811yescharacteristic
318C24#302611yescharacteristic
319C24#302611yescharacteristic
320C22×C4C2×#37912yes—
321C22×C4#284612yes—
322C23#1275314yes—
323C23#1467911yescharacteristic
324C23C2×#179011yescharacteristic
325C23#2060911yescharacteristic
326C23#3289811yescharacteristic
327C23#3289811yescharacteristic
328C23#3290011yescharacteristic
329C23#3290011yescharacteristic
330C22#39212511yescharacteristic
331C22#39213512yes—
332C22C2×#1275312yes—
333C22#51179012yes—
334C2#7221847511yescharacteristic
335C2#7221898011yescharacteristic
336C2#7225177911yescharacteristic
subgroupG/H
center, socleC22#392125
commutatorC23×D4C23×C4
Frattini, cosocleC24×D4C24
prime radical#72241156C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×#120 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72266329C2semidirect
2. #72255642C2semidirect
3. #72251570C2semidirect
5. #72236837C2semidirect
6. #72236714C2semidirect
7. #72235848C2semidirect
8. #72226038C2semidirect
9. #72225994C2semidirect
10. #8390796C4semidirect
12. #8093110C4semidirect
13. #8093108C4semidirect
14. #7670333C4semidirect
15. #7670203C4semidirect
19. #555894C22semidirect
20. #496225C22semidirect
21. #496225C22semidirect
22. #496194C22semidirect
23. #382422C22semidirect
24. #379759C22semidirect
25. #379754C22semidirect
26. C2×#9497C22semidirect
27. C2×#9497C22semidirect
28. C2×#9495C22semidirect
29. #201322C22semidirect
30. C2×#8547C22semidirect
31. C2×#8541C22semidirect
73. C2×#2309C23semidirect
75. C2×#2308C23semidirect
77. C2×#1783C23semidirect
78. C2×#1781C23semidirect
79. C2×#1781C23semidirect
80. C27C22⋊C4semidirect
85. C24×D4C22⋊C4semidirect
91. C25×C4C22⋊C4semidirect
92. C2×H(3,𝔽4)C22⋊C4semidirect
93. C2×D42C22⋊C4semidirect
94. C2×#543C22⋊C4semidirect
95. C2×#538C22⋊C4semidirect
97. C2×#534C22⋊C4semidirect
110. C23×C22⋊C4C22⋊C4semidirect
114. #2748C22⋊C4semidirect
115. #2748C22⋊C4semidirect
116. #2684C22⋊C4semidirect
117. #2684C22⋊C4semidirect
118. #2679C22⋊C4semidirect
119. #2679C22⋊C4semidirect
120. #2669C22⋊C4semidirect
121. #2669C22⋊C4semidirect
122. #2561C22⋊C4semidirect
123. #2561C22⋊C4semidirect
124. #2539C22⋊C4semidirect
125. #2539C22⋊C4semidirect
146. #2380C22⋊C4semidirect
147. #2375C22⋊C4semidirect
148. #2375C22⋊C4semidirect
149. #2372C22⋊C4semidirect
150. #2358C22⋊C4semidirect
151. #2358C22⋊C4semidirect
152. #2312C22⋊C4semidirect
160. C2×#385C22⋊C4semidirect
161. C2×#385C22⋊C4semidirect
163. C2×#379C22⋊C4semidirect
164. C2×#379C22⋊C4semidirect
165. C2×#379C22⋊C4semidirect
166. C2×#379C22⋊C4semidirect
169. #2273C22⋊C4semidirect
170. #2273C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#722663292
#722556422
#722515702
#722411562
#722368372
#722367142
#722358482
#722260382
#722259942