Finite Group Databasebeta

← #72266341

Subgroups — #72266341

Order 4096.

Only the 1111 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 308 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72266293C211yesmaximal, characteristic, complemented
2#72266134C211yesmaximal, characteristic, complemented
3C2×#72233689C211yesmaximal, characteristic
4#72258375C211yesmaximal, characteristic, complemented
5C2×#6628900C414yescomplemented
6#72240423C218yesmaximal, complemented
7C2×#133278C2214yescomplemented
8C2×#6370693C411yescharacteristic, complemented
9C2×#6370693C411yescharacteristic, complemented
10C2×#6370693C411yescharacteristic, complemented
11C2×#6628896C2214yes—
12C23×H(7,𝔽2)C411yescharacteristic, complemented
13#72236899C211yesmaximal, characteristic, complemented
14#72236870C211yesmaximal, characteristic, complemented
15#72235937C2211yescharacteristic, complemented
16#72235054C2214yescomplemented
17#72231822C2214yescomplemented
18#72226161C211yesmaximal, characteristic, complemented
19#72226038C2214yescomplemented
20#72225881C2211yescharacteristic, complemented
21#72225873C2214yescomplemented
22#72225621C2211yescharacteristic, complemented
23#72219377C2211yescharacteristic, complemented
24C2×#7631286C2211yescharacteristic
25C2×#7631286C2211yescharacteristic
26C2×#7631286C2211yescharacteristic
27C2×#131967C2214yescomplemented
28C22×H(7,𝔽2)C2×C414yes—
29C24×H(5,𝔽2)C2×C411yescharacteristic
30C24×H(5,𝔽2)C2×C411yescharacteristic
31C24×H(5,𝔽2)C2×C411yescharacteristic
32C22×#3484D412yes—
33C23×D42C2311yescharacteristic
34C2×#35527D412yes—
35C2×#35512D412yes—
36C2×#35347C2×C412yes—
37C2×#35347C2×C412yes—
38C2×#35347C2×C412yes—
39C2×#35347D412yes—
40C2×#35346D412yes—
41C2×#35346C2312yes—
42C2×#35346C2312yes—
43C2×#35346C2312yes—
44C2×#35250D412yes—
45C2×#34980D412yes—
46C2×#34980D412yes—
47C2×#34736D412yes—
48C22×#2403D412yes—
49C22×#2403D412yes—
50C2×#34256D412yes—
51C2×#34171C2×C411yescharacteristic
52C2×#34171C2×C411yescharacteristic
53C2×#34171C2×C411yescharacteristic
54#6628896D4132yes—
55C2×#20585C2×C414yes—
56C2×#20585C2×C414yes—
57C2×#20585C2×C414yes—
58#382422C2312yescomplemented
59C2×#10965C2312yescomplemented
60C2×#8547C2312yescomplemented
61C2×#8547C2312yescomplemented
62C2×H(7,𝔽2)C22⋊C4116yescomplemented
63C23×H(5,𝔽2)C22⋊C412yes—
64C23×H(5,𝔽2)C22⋊C412yes—
65C23×H(5,𝔽2)C22⋊C412yes—
66C23×H(5,𝔽2)C22×C412yes—
67C23×H(5,𝔽2)C22×C412yes—
68C23×H(5,𝔽2)C22×C412yes—
69C24×#41C22⋊C412yes—
70C25×D4C22⋊C412yes—
71C25×D4C22⋊C412yes—
72C25×D4C22×C411yescharacteristic
73C25×D4C2×D411yescharacteristic
74C25×D4C2×D411yescharacteristic
75C25×D4C2×D411yescharacteristic
76C22×#534C22⋊C414yes—
77C23×#127C2×D411yescharacteristic
78C23×#121C22×C411yescharacteristic
79C23×#121C2×D411yescharacteristic
80C23×#121C2×D411yescharacteristic
81C23×#121C2×D411yescharacteristic
82C23×#120C2×D411yescharacteristic
83C23×#120C2×D411yescharacteristic
84C23×#120C2×D414yes—
85C23×#120C2×D411yescharacteristic
86C23×#120C2411yescharacteristic
87C22×#518C22⋊C412yes—
88C23×C4×D4C2×D411yescharacteristic
89C23×C4×D4C2×D411yescharacteristic
90#35346C2×D4116yes—
91#35346C2×D4116yes—
92#35346C2×D4116yes—
93C2×#2406C22⋊C414yes—
94C2×#2404C22⋊C414yes—
95C2×#2404C2×D414yes—
96C2×#2404C2×D414yes—
97C2×#2405C22⋊C414yes—
98C2×#2403C22⋊C414yes—
99C2×#2403C2×D414yes—
100C2×#2403C2×D414yes—
101C2×#2309C22⋊C412yes—
102C2×#2308C22⋊C412yes—
103C2×#2308C22×C412yes—
104C2×#2308C22×C412yes—
105C2×#2308C22×C412yes—
106C2×#2308C2×D414yes—
107C22×#386C22⋊C414yes—
108C2×#2280C22⋊C412yes—
109C22×#379C22⋊C412yes—
110C22×#379C22⋊C412yes—
111#20585C22⋊C4116yescomplemented
112#20585C22⋊C4116yescomplemented
113#20585C22⋊C4116yescomplemented
114C27#9912yes—
115C27#9912yescomplemented
116C27C2×C22⋊C411yescharacteristic
117C27C2×C22⋊C411yescharacteristic
118C27#12011yescharacteristic
119C27C22×D411yescharacteristic
120C22×H(5,𝔽2)#9918yes—
121C22×H(5,𝔽2)#9918yescomplemented
122C22×H(5,𝔽2)#9918yescomplemented
123C22×H(5,𝔽2)C2×C22⋊C418yes—
124C22×H(5,𝔽2)C2×C22⋊C418yes—
125C22×H(5,𝔽2)C2×C22⋊C418yes—
126C23×#41#9914yescomplemented
127C23×#41C2×C22⋊C414yes—
128C24×Q8#9912yescomplemented
129C24×D4#9914yes—
130C24×D4#9912yescomplemented
131C24×D4#9914yescomplemented
132C24×D4C2×C22⋊C411yescharacteristic
133C24×D4C2×C22⋊C411yescharacteristic
134C24×D4C2×C22⋊C411yescharacteristic
135C24×D4C2×C22⋊C411yescharacteristic
136C24×D4C2×C22⋊C411yescharacteristic
137C24×D4C2×C22⋊C414yes—
138C24×D4C2×C22⋊C414yes—
139C24×D4#12011yescharacteristic
140C24×D4#12011yescharacteristic
141C24×D4#12011yescharacteristic
142C24×D4#12011yescharacteristic
143C24×D4#12011yescharacteristic
144C24×D4#12011yescharacteristic
145C24×D4#12011yescharacteristic
146C24×D4#12011yescharacteristic
147C24×D4C23×C411yescharacteristic
148C24×D4C22×D411yescharacteristic
149C24×D4C22×D411yescharacteristic
150C25×C4#9912yes—
151C25×C4#9912yescomplemented
152C25×C4C2×C22⋊C411yescharacteristic
153C25×C4C2×C22⋊C411yescharacteristic
154C25×C4#12011yescharacteristic
155C25×C4#12011yescharacteristic
156C22×#120#120116yes—
157C22×#120C22×D418yes—
158C23×C4⋊C4C2×C22⋊C411yescharacteristic
159C23×C4⋊C4#12011yescharacteristic
160C23×C22⋊C4C2×C22⋊C414yes—
161C23×C22⋊C4C2×C22⋊C411yescharacteristic
162C23×C22⋊C4C2×C22⋊C411yescharacteristic
163C23×C22⋊C4#12011yescharacteristic
164C23×C22⋊C4#12011yescharacteristic
165C23×C22⋊C4#12011yescharacteristic
166C23×C22⋊C4C22×D411yescharacteristic
167C23×C22⋊C4C22×D411yescharacteristic
168C23×C22⋊C4C22×D411yescharacteristic
169C23×C42#12011yescharacteristic
170#2404#120116yes—
171#2404#120116yes—
172#2403#120116yes—
173#2403#120116yes—
174#2308C2×C22⋊C418yes—
175#2308C2×C22⋊C418yes—
176#2308C2×C22⋊C418yes—
177#2308#120116yes—
178C2×#381C2×C22⋊C414yes—
179C2×#379C2×C22⋊C414yes—
180C26#37911yescharacteristic
181C26#37911yescharacteristic
182C26#37911yescharacteristic
183C26#37911yescharacteristic
184C26C2×#9912yes—
185C26C2×#9912yes—
186C26C2×#9912yes—
187C26C2×#9914yes—
188C26C2×#9912yes—
189C26C2×#9914yes—
190C26C22×C22⋊C411yescharacteristic
191C26C22×C22⋊C411yescharacteristic
192C26C2×#12014yes—
193C26C2×#12011yescharacteristic
194C26C2×#12011yescharacteristic
195C2×H(5,𝔽2)AΣL(1,𝔽16)132yescomplemented
196C22×#41#37918yes—
197C22×#41C2×#9918yes—
198C23×Q8C2×#9914yes—
199C23×D4#37918yes—
200C23×D4#37918yes—
201C23×D4C2×#9914yes—
202C23×D4C2×#9914yes—
203C23×D4C2×#9914yes—
204C23×D4C2×#9918yes—
205C23×D4C2×#9914yes—
206C23×D4C2×#9918yes—
207C23×D4C22×C22⋊C414yes—
208C23×D4C2×#12014yes—
209C23×D4C2×#12014yes—
210C24×C4#37911yescharacteristic
211C24×C4#37911yescharacteristic
212C24×C4#37911yescharacteristic
213C24×C4#37911yescharacteristic
214C24×C4C2×#9912yes—
215C24×C4C2×#9912yes—
216C24×C4C2×#9914yes—
217C24×C4C2×#9914yes—
218C24×C4C22×C22⋊C411yescharacteristic
219C24×C4C2×#12011yescharacteristic
220C24×C4C2×#12011yescharacteristic
221C22×C22⋊C4#37918yes—
222C22×C22⋊C4C2×#12014yes—
223C22×C22⋊C4C2×#12014yes—
224C22×C22⋊C4C2×#12014yes—
225#381#37918yes—
226#379#37918yes—
227C25#178111yescharacteristic
228C25#178111yescharacteristic
229C25#178112yes—
230C25#178112yes—
231C25#178112yes—
232C25#178111yescharacteristic
233C25#178111yescharacteristic
234C25#178112yes—
235C25#178111yescharacteristic
236C25#178111yescharacteristic
237C25#178112yes—
238C25#178112yes—
239C25#178118yes—
240C25#178118yes—
241C25C2×#37912yes—
242C25C2×#37912yes—
243C25C2×#37911yescharacteristic
244C25#284612yes—
245C25#284612yes—
246C25C22×#9912yes—
247C25C22×#9912yes—
248C25C22×#9912yes—
249C25C22×#9912yes—
250C22×Q8#178118yes—
251C22×D4#178118yes—
252C22×D4C2×AΣL(1,𝔽16)116yes—
253C22×D4C2×AΣL(1,𝔽16)132yes—
254C23×C4#178112yes—
255C23×C4#178112yes—
256C23×C4#178112yes—
257C23×C4#178118yes—
258C23×C4#178118yes—
259C23×C4C2×#37912yes—
260C23×C4#284612yes—
261C23×C4#284612yes—
262C23×C4C22×#9912yes—
263C23×C4C22×#9912yes—
264C24#854111yescharacteristic
265C24#854111yescharacteristic
266C24#854111yescharacteristic
267C24#854114yes—
268C24#854114yes—
269C24#854111yescharacteristic
270C24#1268718yes—
271C24#12687116yescomplemented
272C24#1467911yescharacteristic
273C24C2×#178111yescharacteristic
274C24C2×#178114yes—
275C24C2×#178111yescharacteristic
276C24C2×#178111yescharacteristic
277C24C2×#178111yescharacteristic
278C24C2×#178111yescharacteristic
279C24C2×#178111yescharacteristic
280C24C2×#178114yes—
281C24C22×AΣL(1,𝔽16)18yes—
282C22×C4#854114yes—
283C22×C4#1268718yes—
284C22×C4#12687116yescomplemented
285C22×C4C2×#178114yes—
286C23#13382218yes—
287C23#13383912yes—
288C23C2×#854111yescharacteristic
289C23C2×#854111yescharacteristic
290C23C2×#854111yescharacteristic
291C23C2×#854111yescharacteristic
292C23#39210911yescharacteristic
293C23#39210911yescharacteristic
294C23#39210911yescharacteristic
295C23C2×#1268718yes—
296C23C2×#1268714yes—
297C2×C4#13382218yes—
298C22#7221517912yes—
299C22C2×#13382212yes—
300C22C2×#13382212yes—
301C22C2×#13383911yescharacteristic
302C22#7221851811yescharacteristic
303C22#7224041911yescharacteristic
304C2#7221883611yescharacteristic
305C2#7224042011yescharacteristic
306C2#7225805611yescharacteristic
subgroupG/H
center, socleC22#72218518
commutatorC24×D4C23×C4
Frattini, cosocleC23×#120C24
prime radicalC2×#72233689C2
LCSG ▷ C24×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×D42 ◁ G
LECSG ▷ C23×#120 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72266293C2semidirect
2. #72266134C2semidirect
4. #72258375C2semidirect
5. C2×#6628900C4semidirect
6. #72240423C2semidirect
7. C2×#133278C22semidirect
8. C2×#6370693C4semidirect
9. C2×#6370693C4semidirect
10. C2×#6370693C4semidirect
12. C23×H(7,𝔽2)C4semidirect
13. #72236899C2semidirect
14. #72236870C2semidirect
15. #72235937C22semidirect
16. #72235054C22semidirect
17. #72231822C22semidirect
18. #72226161C2semidirect
19. #72226038C22semidirect
20. #72225881C22semidirect
21. #72225873C22semidirect
22. #72225621C22semidirect
23. #72219377C22semidirect
27. C2×#131967C22semidirect
58. #382422C23semidirect
59. C2×#10965C23semidirect
60. C2×#8547C23semidirect
61. C2×#8547C23semidirect
62. C2×H(7,𝔽2)C22⋊C4semidirect
111. #20585C22⋊C4semidirect
112. #20585C22⋊C4semidirect
113. #20585C22⋊C4semidirect
115. C27#99semidirect
121. C22×H(5,𝔽2)#99semidirect
122. C22×H(5,𝔽2)#99semidirect
126. C23×#41#99semidirect
128. C24×Q8#99semidirect
130. C24×D4#99semidirect
131. C24×D4#99semidirect
151. C25×C4#99semidirect
195. C2×H(5,𝔽2)AΣL(1,𝔽16)semidirect
271. C24#12687semidirect
284. C22×C4#12687semidirect

Maximal subgroups

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

subgroupindex
#722662932
#722661342
C2×#722336892
#722583752
#722404232
#722368992
#722368702
#722261612