Finite Group Databasebeta

← #72260926

Subgroups — #72260926

Order 2048.

At least 7542 conjugacy classes of subgroups; only the 726 normal subgroups are shown.

The subgroup lattice has 324 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72260837C211yesmaximal, characteristic, complemented
2#72249504C211yesmaximal, characteristic, complemented
3#72234169C212yesmaximal, complemented
4#72234159C212yesmaximal, complemented
5#72233608C211yesmaximal, characteristic
6#72233199C212yesmaximal, complemented
7#72233089C212yesmaximal, complemented
8#72232777C214yesmaximal, complemented
9#9063254C412yescomplemented
10#9061593C412yescomplemented
11#8196961C411yescharacteristic, complemented
12#8195815C412yescomplemented
13#8195808C411yescharacteristic, complemented
14#8181078C2212yes—
15#7804411C2212yes—
16#7671933C2211yescharacteristic
17C2×#59126C2212yes—
18C23×#353C2211yescharacteristic, complemented
19#496225C2214yescomplemented
20#494713C2212yescomplemented
21C2×#13086C2214yescomplemented
22C2×#12693C2211yescharacteristic, complemented
23#374038C2214yescomplemented
24#359892C2212yescomplemented
25#359458C2212yescomplemented
26#201664C2214yescomplemented
27#201336C2212yescomplemented
28#190139C2211yescharacteristic, complemented
29#181344C2211yescharacteristic, complemented
30C2×#3508D411yescharacteristic
31C22×D42D412yes—
32C23×#127C2311yescharacteristic
33C23×#99C2312yes—
34#59370D414yes—
35#59126C2×C414yes—
36#59126C2×C412yes—
37#59126C2×C412yes—
38#59126D412yes—
39#59126D414yes—
40#59126D412yes—
41#58990D414yes—
42C2×#2669D414yes—
43#52004C2×C412yes—
44#51940C2×C414yes—
45#51902C2×C412yes—
46#47563C2×C412yes—
47#46996D412yes—
48#46387C2×C414yes—
49#46387D412yes—
50#46384C2×C412yes—
51#46384D414yes—
52#45727D414yes—
53#45439D414yes—
54#37475D412yes—
55#36625D411yescharacteristic
56#35554C2×C411yescharacteristic
57#35554D411yescharacteristic
58#35528C2×C411yescharacteristic
59#35528D411yescharacteristic
60#35285C2×C411yescharacteristic
61#35272C2×C411yescharacteristic
62#35222D411yescharacteristic
63#35222C2311yescharacteristic
64C2×#2403D412yes—
65C2×#2403C2314yes—
66#34210D411yescharacteristic
67#34171D411yescharacteristic
68#34171D414yes—
69#34171C2312yes—
70#34171C2311yescharacteristic
71C2×#2366D412yes—
72C4×#525D411yescharacteristic
73C2×#1783C2312yes—
74C2×#1781C2312yes—
75C24×D4C2×D414yes—
76C24×D4C2×D411yescharacteristic
77C24×D4C2411yescharacteristic
78C2×H(3,𝔽4)C22⋊C412yescomplemented
79C2×D42C22⋊C414yescomplemented
80C2×D42C22⋊C412yescomplemented
81C2×D42C22⋊C412yescomplemented
82C2×D42C22⋊C412yescomplemented
83C2×#535C22⋊C411yescharacteristic, complemented
84C2×#535C22⋊C411yescharacteristic, complemented
85C2×#535C22×C411yescharacteristic
86C2×#535C2×D411yescharacteristic
87C2×#533C22⋊C411yescharacteristic, complemented
88C22×#127C22×C411yescharacteristic
89C22×#127C22×C411yescharacteristic
90C22×#127C2×D412yes—
91C22×#127C2×D411yescharacteristic
92C22×#127C2×D412yes—
93C22×#127C2×D411yescharacteristic
94C22×#127C2×D412yes—
95C22×#127C2×D412yes—
96C2×#525C2×D411yescharacteristic
97C22×#120C2×D414yes—
98C22×#120C2×D414yes—
99C2×#518C22⋊C411yescharacteristic, complemented
100C2×#518C22⋊C411yescharacteristic, complemented
101C2×#518C2×D411yescharacteristic
102C2×#518C2×D411yescharacteristic
103C2×C4×#41C22⋊C411yescharacteristic, complemented
104C23×C42C2×D411yescharacteristic
105#2842C22⋊C412yescomplemented
106#2842C22⋊C414yescomplemented
107#2840C22⋊C412yescomplemented
108#2748C22⋊C414yescomplemented
109#2684C22⋊C414yescomplemented
110#2679C22⋊C414yescomplemented
111#2669C22⋊C414yescomplemented
112#2669C22⋊C414yescomplemented
113#2669C22⋊C414yescomplemented
114#2669C22⋊C414yescomplemented
115#2669C22⋊C414yescomplemented
116#2435C2×D414yes—
117#2433C2×D414yes—
118#2403C22⋊C412yescomplemented
119#2403C22⋊C414yescomplemented
120#2403C22⋊C412yescomplemented
121#2403C22×C414yes—
122#2403C22×C414yes—
123#2403C2×D414yes—
124#2403C2×D414yes—
125#2403C2×D414yes—
126#2403C2×D414yes—
127#2403C2×D414yes—
128#2403C2×D414yes—
129#2380C22⋊C412yescomplemented
130#2375C22⋊C414yescomplemented
131#2372C22⋊C412yescomplemented
132#2366C22⋊C412yescomplemented
133#2366C22⋊C412yescomplemented
134#2366C22⋊C414yescomplemented
135#2358C22⋊C414yescomplemented
136#2309C22⋊C414yescomplemented
137#2309C22⋊C414yescomplemented
138#2309C22⋊C411yescharacteristic, complemented
139#2309C22⋊C411yescharacteristic, complemented
140#2309C22⋊C411yescharacteristic, complemented
141#2309C22⋊C411yescharacteristic, complemented
142#2309C22×C411yescharacteristic
143#2309C22×C411yescharacteristic
144#2309C22×C411yescharacteristic
145#2309C2×D411yescharacteristic
146#2308C22⋊C414yescomplemented
147#2308C22⋊C414yescomplemented
148#2308C2×D412yes—
149#2308C2×D412yes—
150#2297C22⋊C411yescharacteristic, complemented
151#2296C22⋊C411yescharacteristic, complemented
152#2296C22⋊C411yescharacteristic, complemented
153#2296C22⋊C411yescharacteristic, complemented
154C2×#386C2×D414yes—
155C2×#386C2×D414yes—
156#2285C22⋊C411yescharacteristic, complemented
157#2280C2×D411yescharacteristic
158#2280C2×D411yescharacteristic
159#2280C2×D411yescharacteristic
160C2×#379C2×D414yes—
161C2×#379C2×D412yes—
162C2×#379C2×D412yes—
163C4×#117C22⋊C411yescharacteristic, complemented
164C26#12014yes—
165C26C22×D412yes—
166C22×#41C2×C22⋊C411yescharacteristic
167C23×D4C2×C22⋊C411yescharacteristic
168C23×D4C2×C22⋊C411yescharacteristic
169C23×D4C2×C22⋊C414yes—
170C23×D4C2×C22⋊C414yes—
171C23×D4#12012yes—
172C23×D4#12012yes—
173C23×D4#12014yes—
174C23×D4#12014yes—
175C23×D4C23×C411yescharacteristic
176C23×D4C22×D411yescharacteristic
177C23×D4C22×D412yes—
178C23×D4C22×D414yes—
179C23×D4C22×D412yes—
180C23×D4C22×D412yes—
181C24×C4#12014yes—
182C24×C4C22×D411yescharacteristic
183C2×#127C2×C22⋊C412yes—
184C2×#127C2×C22⋊C412yes—
185C2×#127C2×C22⋊C412yes—
186C2×#127C2×C22⋊C412yes—
187C2×#127C2×C22⋊C412yes—
188C2×#127#12011yescharacteristic
189C2×#127#12012yes—
190C2×#127#12011yescharacteristic
191C2×#127#12011yescharacteristic
192C2×#127#12011yescharacteristic
193C2×#127#12012yes—
194C2×#127#12012yes—
195C2×#124C2×C22⋊C412yes—
196C2×#124#12012yes—
197C2×C22⋊Q8#12012yes—
198C2×#121#12012yes—
199C2×#120C2×C22⋊C414yes—
200C2×#120C2×C22⋊C414yes—
201C2×#120C2×C22⋊C414yes—
202C2×#120C2×C22⋊C414yes—
203C2×#120#12014yes—
204C2×#120#12014yes—
205C2×#120#12014yes—
206C2×#120#12014yes—
207C2×#117C2×C22⋊C411yescharacteristic
208C2×#117C2×C22⋊C411yescharacteristic
209C2×#117C2×C22⋊C411yescharacteristic
210C2×#117C2×C22⋊C411yescharacteristic
211C2×#117C2×C22⋊C411yescharacteristic
212C2×#117C2×C22⋊C411yescharacteristic
213C2×#117C2×C22⋊C411yescharacteristic
214C22×C22⋊C4#12012yes—
215C22×C22⋊C4#12012yes—
216C22×C22⋊C4#12014yes—
217C22×C22⋊C4#12012yes—
218C22×C22⋊C4#12012yes—
219C22×C22⋊C4#12014yes—
220C22×C42C2×C22⋊C411yescharacteristic
221C22×C42C2×C22⋊C411yescharacteristic
222C22×C42#12012yes—
223C22×C42#12012yes—
224#390C2×C22⋊C414yes—
225#390#12014yes—
226#386C2×C22⋊C414yes—
227#386C2×C22⋊C414yes—
228#386C2×C22⋊C414yes—
229#386C2×C22⋊C414yes—
230#386#12014yes—
231#386#12014yes—
232#386#12014yes—
233#386#12014yes—
234#381#12011yescharacteristic
235#381#12011yescharacteristic
236#381#12012yes—
237#379C2×C22⋊C412yes—
238#379C2×C22⋊C412yes—
239#379C2×C22⋊C414yes—
240#379C2×C22⋊C414yes—
241#379C2×C22⋊C412yes—
242#379C2×C22⋊C412yes—
243#379#12014yes—
244#379#12014yes—
245#379#12012yes—
246#379#12011yescharacteristic
247#379#12011yescharacteristic
248C4×C22⋊C4#12014yes—
249C25#37914yes—
250C25#37914yes—
251C25C22×C22⋊C412yes—
252C25C2×#12014yes—
253C25C2×#12012yes—
254C25C2×#12014yes—
255C25C2×#12011yescharacteristic
256C22×Q8#37911yescharacteristic
257C22×D4#37911yescharacteristic
258C22×D4#37912yes—
259C22×D4#37911yescharacteristic
260C22×D4#37911yescharacteristic
261C22×D4C22×C22⋊C412yes—
262C22×D4C22×C22⋊C412yes—
263C22×D4C2×#12014yes—
264C22×D4C2×#12014yes—
265C22×D4C2×#12012yes—
266C22×D4C2×#12011yescharacteristic
267C22×D4C2×#12011yescharacteristic
268C23×C4#37914yes—
269C23×C4#37914yes—
270C23×C4C22×C22⋊C411yescharacteristic
271C23×C4C2×#12012yes—
272C23×C4C2×#12011yescharacteristic
273C23×C4C2×#12012yes—
274#127B(4,𝔽2)12yes—
275#127#45812yes—
276C2×C4⋊C4#37912yes—
277C2×C22⋊C4#37911yescharacteristic
278C2×C22⋊C4#37911yescharacteristic
279C2×C22⋊C4#37911yescharacteristic
280C2×C22⋊C4#37911yescharacteristic
281C2×C22⋊C4#37912yes—
282C2×C22⋊C4#37912yes—
283C2×C22⋊C4#37914yes—
284C2×C22⋊C4#37914yes—
285C2×C22⋊C4#37914yes—
286C2×C22⋊C4#37914yes—
287C2×C42#37911yescharacteristic
288C2×C42#37912yes—
289C2×C42#37911yescharacteristic
290C2×C42#37912yes—
291C2×C42#37912yes—
292C24C2×#37914yes—
293C24C2×#37911yescharacteristic
294C24#284612yes—
295C24#284612yes—
296C24#284612yes—
297C24#302512yes—
298C24#302711yescharacteristic
299C24#302811yescharacteristic
300C2×D4#179012yes—
301C2×D4C2×B(4,𝔽2)11yescharacteristic
302C2×D4C2×#45811yescharacteristic
303C2×D4#302612yes—
304C22×C4C2×#37911yescharacteristic
305C22×C4C2×#37911yescharacteristic
306C22×C4#284612yes—
307C42#179012yes—
308C23#1467911yescharacteristic
309C23#2060811yescharacteristic
310C23#2061011yescharacteristic
311C23#3289812yes—
312C23#3289912yes—
313C23#3290012yes—
314C23#3290112yes—
315C2×C4C2×#179011yescharacteristic
316C2×C4#2060911yescharacteristic
317C22#39212511yescharacteristic
318C22#39212611yescharacteristic
319C22#39213511yescharacteristic
320C22#51185314yes—
321C2#7224950012yes—
322C2#7226063811yescharacteristic
subgroupG/H
center, socleC22#392126
commutatorC23×D4C23×C4
Frattini, cosocleC24×D4C24
prime radical#72233608C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ #34171 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72260837C2semidirect
2. #72249504C2semidirect
3. #72234169C2semidirect
4. #72234159C2semidirect
6. #72233199C2semidirect
7. #72233089C2semidirect
8. #72232777C2semidirect
9. #9063254C4semidirect
10. #9061593C4semidirect
11. #8196961C4semidirect
12. #8195815C4semidirect
13. #8195808C4semidirect
18. C23×#353C22semidirect
19. #496225C22semidirect
20. #494713C22semidirect
21. C2×#13086C22semidirect
22. C2×#12693C22semidirect
23. #374038C22semidirect
24. #359892C22semidirect
25. #359458C22semidirect
26. #201664C22semidirect
27. #201336C22semidirect
28. #190139C22semidirect
29. #181344C22semidirect
78. C2×H(3,𝔽4)C22⋊C4semidirect
79. C2×D42C22⋊C4semidirect
80. C2×D42C22⋊C4semidirect
81. C2×D42C22⋊C4semidirect
82. C2×D42C22⋊C4semidirect
83. C2×#535C22⋊C4semidirect
84. C2×#535C22⋊C4semidirect
87. C2×#533C22⋊C4semidirect
99. C2×#518C22⋊C4semidirect
100. C2×#518C22⋊C4semidirect
103. C2×C4×#41C22⋊C4semidirect
105. #2842C22⋊C4semidirect
106. #2842C22⋊C4semidirect
107. #2840C22⋊C4semidirect
108. #2748C22⋊C4semidirect
109. #2684C22⋊C4semidirect
110. #2679C22⋊C4semidirect
111. #2669C22⋊C4semidirect
112. #2669C22⋊C4semidirect
113. #2669C22⋊C4semidirect
114. #2669C22⋊C4semidirect
115. #2669C22⋊C4semidirect
118. #2403C22⋊C4semidirect
119. #2403C22⋊C4semidirect
120. #2403C22⋊C4semidirect
129. #2380C22⋊C4semidirect
130. #2375C22⋊C4semidirect
131. #2372C22⋊C4semidirect
132. #2366C22⋊C4semidirect
133. #2366C22⋊C4semidirect
134. #2366C22⋊C4semidirect
135. #2358C22⋊C4semidirect
136. #2309C22⋊C4semidirect
137. #2309C22⋊C4semidirect
138. #2309C22⋊C4semidirect
139. #2309C22⋊C4semidirect
140. #2309C22⋊C4semidirect
141. #2309C22⋊C4semidirect
146. #2308C22⋊C4semidirect
147. #2308C22⋊C4semidirect
150. #2297C22⋊C4semidirect
151. #2296C22⋊C4semidirect
152. #2296C22⋊C4semidirect
153. #2296C22⋊C4semidirect
156. #2285C22⋊C4semidirect
163. C4×#117C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#722608372
#722495042
#722341692
#722341592
#722336082
#722331992
#722330892
#722327772