Finite Group Databasebeta

← #72265775

Subgroups — #72265775

Order 2048.

At least 10072 conjugacy classes of subgroups; only the 758 normal subgroups are shown.

The subgroup lattice has 354 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72265766C211yesmaximal, characteristic, complemented
2#72259019C211yesmaximal, characteristic, complemented
3#72258234C214yesmaximal, complemented
4#72241156C211yesmaximal, characteristic
5C2×#133278C211yesmaximal, characteristic, complemented
6#72236754C211yesmaximal, characteristic, complemented
7#72235131C214yesmaximal, complemented
8#72219034C211yesmaximal, characteristic, complemented
9#72219022C211yesmaximal, characteristic, complemented
10#8357779C414yescomplemented
11#8093110C411yescharacteristic, complemented
12#8093108C411yescharacteristic, complemented
13#7763449C2214yes—
14#7670333C411yescharacteristic, complemented
15#7670203C411yescharacteristic, complemented
16#7642273C2211yescharacteristic
17C2×#35346C2211yescharacteristic
18C2×#34980C2211yescharacteristic
19C22×#1913C2212yescomplemented
20#515285C2211yescharacteristic, complemented
21#494713C2214yescomplemented
22C2×#13086C2214yescomplemented
23#493357C2214yescomplemented
24#392696C2212yescomplemented
25#392658C2212yescomplemented
26C2×#10972C2212yescomplemented
27C2×#10965C2211yescharacteristic, complemented
28C2×#9495C2212yescomplemented
29#201322C2212yescomplemented
30C2×#8547C2211yescharacteristic, complemented
31C2×#8541C2211yescharacteristic, complemented
32C23×H(5,𝔽2)D411yescharacteristic
33C25×D4D411yescharacteristic
34C2×#3490D411yescharacteristic
35C2×#3532D414yes—
36C22×#534C2312yes—
37C23×#120C2311yescharacteristic
38C22×#518C2311yescharacteristic, complemented
39C23×#99C2312yescomplemented
40#59191C2×C414yes—
41#58990C2×C414yes—
42#50856D414yes—
43#50646D414yes—
44#47848D414yes—
45#47827D414yes—
46#45455C2×C414yes—
47#45439C2×C414yes—
48#36737D414yes—
49#36714D414yes—
50#36703C2×C412yes—
51#36703D412yes—
52C2×#2849C2×C411yescharacteristic
53C2×#2849D411yescharacteristic
54C2×#2846C2×C411yescharacteristic
55C2×#2846D411yescharacteristic
56#35390C2×C411yescharacteristic
57#35390D411yescharacteristic
58#35367C2×C411yescharacteristic
59#35367D411yescharacteristic
60#35346C2×C411yescharacteristic
61#35346C2×C412yes—
62#35346C2×C411yescharacteristic
63#35346D412yes—
64#35346D411yescharacteristic
65#35346D411yescharacteristic
66#35222C2312yes—
67#35050D412yes—
68C2×#2403D412yes—
69C2×#2403C2312yes—
70C2×#2613C2×C412yes—
71C2×#2613D412yes—
72#34407D414yes—
73#34176D411yescharacteristic
74C2×#2308D412yes—
75C2×#2308D411yescharacteristic
76C2×#2308C2311yescharacteristic, complemented
77C22×#379D411yescharacteristic
78C2×#1783C2312yescomplemented
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×D412yes—
89C24×D4C2×D411yescharacteristic
90C24×D4C2411yescharacteristic
91C25×C4C22⋊C411yescharacteristic, complemented
92C2×H(3,𝔽4)C22⋊C414yescomplemented
93C2×#560C22⋊C414yescomplemented
94C2×#551C22⋊C414yescomplemented
95C2×D42C22⋊C414yescomplemented
96C2×#543C22⋊C411yescharacteristic, complemented
97C2×#538C22⋊C412yescomplemented
98C2×#535C2×D414yes—
99C2×#534C22⋊C411yescharacteristic, complemented
100C2×#534C2×D414yes—
101C2×#534C2×D414yes—
102C2×#534C2×D414yes—
103C2×#534C2×D414yes—
104C2×#525C2×D412yes—
105C22×#123C22×C412yes—
106C22×#123C2×D412yes—
107C22×#120C22×C412yes—
108C22×#120C22×C411yescharacteristic
109C22×#120C22×C411yescharacteristic
110C22×#120C2×D412yes—
111C22×#120C2×D414yes—
112C22×#120C2×D411yescharacteristic
113C22×#120C2×D411yescharacteristic
114C22×#120C2×D411yescharacteristic
115C22×#120C2×D411yescharacteristic
116C22×#120C2×D411yescharacteristic
117C22×#120C2×D411yescharacteristic
118C23×C22⋊C4C2×D411yescharacteristic
119C23×C22⋊C4C2×D411yescharacteristic
120C23×C22⋊C4C2×D411yescharacteristic
121C23×C22⋊C4C2×D412yes—
122#2717C22⋊C414yescomplemented
123#2709C22⋊C414yescomplemented
124#2697C22⋊C414yescomplemented
125#2697C22⋊C414yescomplemented
126#2684C22⋊C414yescomplemented
127#2679C22⋊C414yescomplemented
128#2679C22⋊C414yescomplemented
129#2669C22⋊C414yescomplemented
130#2561C22⋊C412yescomplemented
131#2561C22⋊C412yescomplemented
132#2539C22⋊C412yescomplemented
133#2539C22⋊C412yescomplemented
134#2435C22×C412yes—
135#2435C2×D412yes—
136#2435C2×D414yes—
137#2433C22×C412yes—
138#2433C2×D412yes—
139#2433C2×D414yes—
140#2403C22⋊C412yes—
141#2403C22⋊C412yes—
142#2403C22⋊C412yes—
143#2403C22⋊C412yes—
144#2403C22×C412yes—
145#2403C22×C412yes—
146#2403C2×D414yes—
147#2403C2×D414yes—
148#2403C2×D412yes—
149#2403C2×D412yes—
150#2376C22⋊C414yescomplemented
151#2375C22⋊C414yescomplemented
152#2374C22⋊C414yescomplemented
153#2358C22⋊C414yescomplemented
154#2312C22⋊C412yescomplemented
155#2309C2×D414yes—
156#2308C22⋊C411yescharacteristic
157#2308C22⋊C412yes—
158#2308C22⋊C411yescharacteristic
159C2×#386C2×D414yes—
160#2280C2×D412yes—
161C2×#381C22⋊C412yescomplemented
162C2×#381C22⋊C412yescomplemented
163C2×#379C22⋊C411yescharacteristic, complemented
164C2×#379C22⋊C412yescomplemented
165C2×#379C22⋊C411yescharacteristic, complemented
166C2×#379C22⋊C412yescomplemented
167C2×#379C22⋊C412yescomplemented
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#12012yes—
189C23×D4#12011yescharacteristic
190C23×D4#12012yes—
191C23×D4#12012yes—
192C23×D4#12012yes—
193C23×D4#12011yescharacteristic
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×#123C2×C22⋊C414yes—
214C2×#123C2×C22⋊C414yes—
215C2×#123#12014yes—
216C2×#123#12014yes—
217C2×C22⋊Q8C2×C22⋊C412yes—
218C2×C22⋊Q8#12012yes—
219C2×#121C2×C22⋊C412yes—
220C2×#121#12012yes—
221C2×#120C2×C22⋊C414yes—
222C2×#120C2×C22⋊C414yes—
223C2×#120C2×C22⋊C414yes—
224C2×#120C2×C22⋊C414yes—
225C2×#120#12014yes—
226C2×#120#12014yes—
227C2×#120#12014yes—
228C2×#120#12014yes—
229C2×#117#12012yes—
230C2×#117#12014yes—
231C22×C22⋊C4C2×C22⋊C411yescharacteristic
232C22×C22⋊C4C2×C22⋊C411yescharacteristic
233C22×C22⋊C4C2×C22⋊C411yescharacteristic
234C22×C22⋊C4C2×C22⋊C411yescharacteristic
235C22×C22⋊C4C2×C22⋊C411yescharacteristic
236C22×C22⋊C4C2×C22⋊C411yescharacteristic
237C22×C22⋊C4C2×C22⋊C412yes—
238C22×C22⋊C4C2×C22⋊C412yes—
239C22×C22⋊C4#12014yes—
240C22×C22⋊C4#12011yescharacteristic
241C22×C22⋊C4#12011yescharacteristic
242C22×C22⋊C4#12011yescharacteristic
243C22×C22⋊C4#12011yescharacteristic
244C22×C22⋊C4#12011yescharacteristic
245C22×C22⋊C4#12011yescharacteristic
246C22×C22⋊C4#12014yes—
247C22×C22⋊C4#12014yes—
248#390C2×C22⋊C414yes—
249#390C2×C22⋊C414yes—
250#390#12014yes—
251#390#12014yes—
252#386C2×C22⋊C414yes—
253#386C2×C22⋊C414yes—
254#386#12014yes—
255#386#12014yes—
256#379C2×C22⋊C412yes—
257#379C2×C22⋊C412yes—
258#379#12014yes—
259C4×C22⋊C4C2×C22⋊C412yes—
260C4×C22⋊C4C2×C22⋊C412yes—
261C4×C22⋊C4#12014yes—
262C25#37911yescharacteristic
263C25#37911yescharacteristic
264C25#37911yescharacteristic
265C25#37912yes—
266C25#37911yescharacteristic
267C25#37912yes—
268C25#37912yes—
269C25#37912yes—
270C25B(4,𝔽2)12yes—
271C25#45812yes—
272C25C22×C22⋊C411yescharacteristic
273C25C22×C22⋊C411yescharacteristic
274C25C2×#12011yescharacteristic
275C25C2×#12012yes—
276C25C2×#12012yes—
277C25C2×#12012yes—
278C25C2×#12011yescharacteristic
279C25C2×#12012yes—
280C25C2×#12011yescharacteristic
281C22×Q8#37912yes—
282C22×D4#37912yes—
283C22×D4B(4,𝔽2)14yes—
284C22×D4#45814yes—
285C22×D4C22×C22⋊C414yes—
286C22×D4C2×#12014yes—
287C22×D4C2×#12014yes—
288C22×D4C2×#12014yes—
289C23×C4#37911yescharacteristic
290C23×C4#37911yescharacteristic
291C23×C4#37912yes—
292C23×C4#37911yescharacteristic
293C23×C4#37911yescharacteristic
294C23×C4#37912yes—
295C23×C4#37912yes—
296C23×C4#37912yes—
297C23×C4C22×C22⋊C411yescharacteristic
298C23×C4C2×#12012yes—
299C23×C4C2×#12011yescharacteristic
300C23×C4C2×#12012yes—
301C2×C4⋊C4#37912yes—
302C2×C4⋊C4#37912yes—
303C2×C22⋊C4#37914yes—
304C2×C22⋊C4#37914yes—
305C2×C22⋊C4#37914yes—
306C2×C22⋊C4#37914yes—
307C2×C42#37914yes—
308C2×C42#37914yes—
309C24#179012yes—
310C24#179012yes—
311C24#179014yescomplemented
312C24C2×#37911yescharacteristic
313C24C2×#37912yes—
314C24C2×#37912yes—
315C24#284612yes—
316C24#284611yescharacteristic
317C24#284612yes—
318C24#284611yescharacteristic
319C24C2×B(4,𝔽2)11yescharacteristic
320C24C2×B(4,𝔽2)12yes—
321C24C2×#45811yescharacteristic
322C24C2×#45812yes—
323C24#302611yescharacteristic
324C24#302611yescharacteristic
325C24#302612yes—
326C24#302612yes—
327C22×C4#179014yescomplemented
328C22×C4C2×#37912yes—
329C22×C4#284612yes—
330C23#1275314yescomplemented
331C23#1467911yescharacteristic
332C23C2×#179011yescharacteristic
333C23C2×#179012yes—
334C23#2060911yescharacteristic
335C23#2060912yes—
336C23#3289811yescharacteristic
337C23#3289811yescharacteristic
338C23#3289812yes—
339C23#3289812yes—
340C23#3290011yescharacteristic
341C23#3290011yescharacteristic
342C23#3290012yes—
343C23#3290012yes—
344C22#34704211yescharacteristic
345C22#34704811yescharacteristic
346C22#34704812yes—
347C22#39212512yes—
348C22#39212511yescharacteristic
349C22C2×#1275312yes—
350C2#7221898011yescharacteristic
351C2#7221904511yescharacteristic
352C2#7223747211yescharacteristic
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. #72265766C2semidirect
2. #72259019C2semidirect
3. #72258234C2semidirect
5. C2×#133278C2semidirect
6. #72236754C2semidirect
7. #72235131C2semidirect
8. #72219034C2semidirect
9. #72219022C2semidirect
10. #8357779C4semidirect
11. #8093110C4semidirect
12. #8093108C4semidirect
14. #7670333C4semidirect
15. #7670203C4semidirect
19. C22×#1913C22semidirect
20. #515285C22semidirect
21. #494713C22semidirect
22. C2×#13086C22semidirect
23. #493357C22semidirect
24. #392696C22semidirect
25. #392658C22semidirect
26. C2×#10972C22semidirect
27. C2×#10965C22semidirect
28. C2×#9495C22semidirect
29. #201322C22semidirect
30. C2×#8547C22semidirect
31. C2×#8541C22semidirect
38. C22×#518C23semidirect
39. C23×#99C23semidirect
76. C2×#2308C23semidirect
78. C2×#1783C23semidirect
79. C2×#1781C23semidirect
80. C27C22⋊C4semidirect
85. C24×D4C22⋊C4semidirect
91. C25×C4C22⋊C4semidirect
92. C2×H(3,𝔽4)C22⋊C4semidirect
93. C2×#560C22⋊C4semidirect
94. C2×#551C22⋊C4semidirect
95. C2×D42C22⋊C4semidirect
96. C2×#543C22⋊C4semidirect
97. C2×#538C22⋊C4semidirect
99. C2×#534C22⋊C4semidirect
122. #2717C22⋊C4semidirect
123. #2709C22⋊C4semidirect
124. #2697C22⋊C4semidirect
125. #2697C22⋊C4semidirect
126. #2684C22⋊C4semidirect
127. #2679C22⋊C4semidirect
128. #2679C22⋊C4semidirect
129. #2669C22⋊C4semidirect
130. #2561C22⋊C4semidirect
131. #2561C22⋊C4semidirect
132. #2539C22⋊C4semidirect
133. #2539C22⋊C4semidirect
150. #2376C22⋊C4semidirect
151. #2375C22⋊C4semidirect
152. #2374C22⋊C4semidirect
153. #2358C22⋊C4semidirect
154. #2312C22⋊C4semidirect
161. C2×#381C22⋊C4semidirect
162. C2×#381C22⋊C4semidirect
163. C2×#379C22⋊C4semidirect
164. C2×#379C22⋊C4semidirect
165. C2×#379C22⋊C4semidirect
166. C2×#379C22⋊C4semidirect
167. C2×#379C22⋊C4semidirect
169. #2273C22⋊C4semidirect
170. #2273C22⋊C4semidirect
311. C24#1790semidirect
327. C22×C4#1790semidirect
330. C23#12753semidirect

Maximal subgroups

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

subgroupindex
#722657662
#722590192
#722582342
#722411562
C2×#1332782
#722367542
#722351312
#722190342
#722190222