Finite Group Databasebeta

← #72262338

Subgroups — #72262338

Order 2048.

At least 8043 conjugacy classes of subgroups; only the 714 normal subgroups are shown.

The subgroup lattice has 323 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72262281C214yesmaximal, complemented
2#72262272C214yesmaximal, complemented
3#72251570C211yesmaximal, characteristic, complemented
4#72249444C211yesmaximal, characteristic
5#72236898C211yesmaximal, characteristic, complemented
6#72236837C211yesmaximal, characteristic, complemented
7#72235932C211yesmaximal, characteristic, complemented
8#72235848C211yesmaximal, characteristic, complemented
9#72226038C211yesmaximal, characteristic, complemented
10#9040959C412yescomplemented
11#9040958C412yescomplemented
12#8943604C412yescomplemented
13#8942673C412yescomplemented
14#8126631C2214yes—
15#7642273C2211yescharacteristic
16#7642273C2211yescharacteristic
17C2×#35346C2211yescharacteristic
18#555894C2211yescharacteristic, complemented
19#496231C2214yescomplemented
20#496225C2214yescomplemented
21#496194C2214yescomplemented
22#382422C2211yescharacteristic, complemented
23#379759C2212yescomplemented
24#379754C2212yescomplemented
25#201690C2212yescomplemented
26#201687C2212yescomplemented
27C2×#9497C2212yescomplemented
28#201322C2212yescomplemented
29C2×#8547C2211yescharacteristic, complemented
30C2×#8541C2211yescharacteristic, complemented
31C23×H(5,𝔽2)D411yescharacteristic
32C2×#3490D411yescharacteristic
33C2×#3490D411yescharacteristic
34C23×#120C2311yescharacteristic
35#59006D414yes—
36#58990D414yes—
37#52004D414yes—
38#51993D414yes—
39#51940D414yes—
40#51895D414yes—
41#50856C2×C414yes—
42#48113C2×C414yes—
43#48088C2×C414yes—
44#47827C2×C414yes—
45#45568D414yes—
46#45537D414yes—
47#36703C2×C412yes—
48#36703D412yes—
49C2×#2846D412yes—
50#35390C2×C411yescharacteristic
51#35390C2×C411yescharacteristic
52#35390D411yescharacteristic
53#35390D411yescharacteristic
54#35367C2×C411yescharacteristic
55#35367C2×C411yescharacteristic
56#35367D411yescharacteristic
57#35367D411yescharacteristic
58#35348C2×C412yes—
59#35348D412yes—
60#35346C2×C412yes—
61#35346C2×C412yes—
62#35346D412yes—
63#35346D412yes—
64#35222C2312yes—
65#35050D412yes—
66#34980C2312yes—
67C2×#2403D412yes—
68C2×#2403C2312yes—
69#34176D411yescharacteristic
70#34176D411yescharacteristic
71C2×#2309C2311yescharacteristic
72C2×#2308D411yescharacteristic
73C2×#2308C2311yescharacteristic
74C2×#1783C2312yes—
75C2×#1781C2312yes—
76C2×#1781C2312yes—
77C27C2×D411yescharacteristic
78C22×H(5,𝔽2)C22⋊C412yes—
79C22×H(5,𝔽2)C22⋊C412yes—
80C24×D4C2×D411yescharacteristic
81C24×D4C2×D412yes—
82C24×D4C2×D411yescharacteristic
83C24×D4C2411yescharacteristic
84C2×#551C22⋊C414yescomplemented
85C2×D42C22⋊C414yescomplemented
86C2×#543C22⋊C411yescharacteristic, complemented
87C2×#543C22⋊C411yescharacteristic, complemented
88C2×#538C22⋊C412yescomplemented
89C2×#535C2×D414yes—
90C2×#534C22⋊C411yescharacteristic, complemented
91C2×#534C22⋊C411yescharacteristic, complemented
92C2×#534C22×C412yes—
93C2×#534C2×D412yes—
94C2×#525C2×D412yes—
95C22×#120C22⋊C412yescomplemented
96C22×#120C22⋊C412yescomplemented
97C22×#120C22×C411yescharacteristic
98C22×#120C22×C411yescharacteristic
99C22×#120C2×D414yes—
100C22×#120C2×D411yescharacteristic
101C22×#120C2×D411yescharacteristic
102C22×#120C2×D412yes—
103C22×#120C2×D411yescharacteristic
104C22×#120C2×D411yescharacteristic
105C22×#120C2×D411yescharacteristic
106C22×#120C2×D411yescharacteristic
107C2×#518C22⋊C412yescomplemented
108C23×C22⋊C4C2×D411yescharacteristic
109C23×C22⋊C4C2×D411yescharacteristic
110C23×C22⋊C4C2×D411yescharacteristic
111#2804C22⋊C414yescomplemented
112#2748C22⋊C414yescomplemented
113#2709C22⋊C414yescomplemented
114#2697C22⋊C414yescomplemented
115#2684C22⋊C414yescomplemented
116#2684C22⋊C414yescomplemented
117#2679C22⋊C414yescomplemented
118#2669C22⋊C414yescomplemented
119#2561C22⋊C412yescomplemented
120#2561C22⋊C412yescomplemented
121#2539C22⋊C412yescomplemented
122#2539C22⋊C412yescomplemented
123#2435C22×C412yes—
124#2435C2×D412yes—
125#2435C2×D414yes—
126#2433C22×C412yes—
127#2433C2×D412yes—
128#2433C2×D414yes—
129#2405C2×D414yes—
130#2405C2×D414yes—
131#2403C22⋊C414yes—
132#2403C22⋊C414yes—
133#2403C22×C414yes—
134#2403C2×D414yes—
135#2403C2×D414yes—
136#2403C2×D414yes—
137#2403C2×D414yes—
138#2403C2×D414yes—
139#2380C22⋊C414yescomplemented
140#2379C22⋊C414yescomplemented
141#2376C22⋊C414yescomplemented
142#2375C22⋊C414yescomplemented
143#2358C22⋊C414yescomplemented
144#2358C22⋊C414yescomplemented
145#2315C22⋊C412yescomplemented
146#2312C22⋊C412yescomplemented
147#2309C2×D414yes—
148#2308C22⋊C412yes—
149#2308C22⋊C412yes—
150#2308C22×C412yes—
151#2308C2×D412yes—
152C2×#386C2×D414yes—
153#2280C2×D412yes—
154C2×#379C22⋊C412yescomplemented
155C2×#379C22⋊C412yescomplemented
156C2×#379C2×D412yes—
157C2×#379C2×D412yes—
158#2273C22⋊C411yescharacteristic, complemented
159#2273C22⋊C411yescharacteristic, complemented
160#2273C22⋊C411yescharacteristic, complemented
161#2273C22⋊C411yescharacteristic, complemented
162C26C2×C22⋊C411yescharacteristic
163C26C2×C22⋊C411yescharacteristic
164C26#12011yescharacteristic
165C26#12011yescharacteristic
166C26#12012yes—
167C26#12011yescharacteristic
168C26#12011yescharacteristic
169C26C22×D411yescharacteristic
170C26C22×D411yescharacteristic
171C22×#41#12012yes—
172C23×D4C2×C22⋊C411yescharacteristic
173C23×D4C2×C22⋊C411yescharacteristic
174C23×D4C2×C22⋊C411yescharacteristic
175C23×D4C2×C22⋊C414yes—
176C23×D4C2×C22⋊C412yes—
177C23×D4C2×C22⋊C411yescharacteristic
178C23×D4#12011yescharacteristic
179C23×D4#12011yescharacteristic
180C23×D4#12011yescharacteristic
181C23×D4#12014yes—
182C23×D4#12012yes—
183C23×D4#12011yescharacteristic
184C23×D4#12012yes—
185C23×D4C23×C411yescharacteristic
186C23×D4C22×D411yescharacteristic
187C23×D4C22×D414yes—
188C23×D4C22×D414yes—
189C23×D4C22×D411yescharacteristic
190C23×D4C22×D411yescharacteristic
191C24×C4#12011yescharacteristic
192C24×C4#12011yescharacteristic
193C24×C4#12012yes—
194C24×C4C22×D411yescharacteristic
195C2×#127C2×C22⋊C414yes—
196C2×#127#12014yes—
197C2×#124C2×C22⋊C414yes—
198C2×#124#12014yes—
199C2×#123C2×C22⋊C414yes—
200C2×#123#12014yes—
201C2×C22⋊Q8C2×C22⋊C412yes—
202C2×C22⋊Q8#12012yes—
203C2×#121C2×C22⋊C412yes—
204C2×#121#12012yes—
205C2×#120C2×C22⋊C414yes—
206C2×#120C2×C22⋊C414yes—
207C2×#120C2×C22⋊C414yes—
208C2×#120#12014yes—
209C2×#120#12014yes—
210C2×#120#12014yes—
211C2×#117#12012yes—
212C2×#117#12014yes—
213C22×C22⋊C4C2×C22⋊C411yescharacteristic
214C22×C22⋊C4C2×C22⋊C411yescharacteristic
215C22×C22⋊C4C2×C22⋊C411yescharacteristic
216C22×C22⋊C4C2×C22⋊C411yescharacteristic
217C22×C22⋊C4C2×C22⋊C411yescharacteristic
218C22×C22⋊C4C2×C22⋊C411yescharacteristic
219C22×C22⋊C4C2×C22⋊C412yes—
220C22×C22⋊C4C2×C22⋊C412yes—
221C22×C22⋊C4C2×C22⋊C412yes—
222C22×C22⋊C4#12011yescharacteristic
223C22×C22⋊C4#12011yescharacteristic
224C22×C22⋊C4#12011yescharacteristic
225C22×C22⋊C4#12011yescharacteristic
226C22×C22⋊C4#12011yescharacteristic
227C22×C22⋊C4#12011yescharacteristic
228C22×C22⋊C4#12012yes—
229C22×C22⋊C4#12012yes—
230C22×C22⋊C4#12014yes—
231#390C2×C22⋊C414yes—
232#390C2×C22⋊C414yes—
233#390#12014yes—
234#390#12014yes—
235#386C2×C22⋊C414yes—
236#386C2×C22⋊C414yes—
237#386C2×C22⋊C414yes—
238#386C2×C22⋊C414yes—
239#386#12014yes—
240#386#12014yes—
241#386#12014yes—
242#386#12014yes—
243#379C2×C22⋊C414yes—
244#379#12014yes—
245#379#12012yes—
246#379#12012yes—
247#379#12014yes—
248C4×C22⋊C4C2×C22⋊C412yes—
249C4×C22⋊C4C2×C22⋊C412yes—
250C4×C22⋊C4#12014yes—
251C25#37911yescharacteristic
252C25#37911yescharacteristic
253C25#37911yescharacteristic
254C25#37911yescharacteristic
255C25#37914yes—
256C25#37912yes—
257C25C22×C22⋊C411yescharacteristic
258C25C22×C22⋊C411yescharacteristic
259C25C2×#12012yes—
260C25C2×#12012yes—
261C25C2×#12012yes—
262C25C2×#12011yescharacteristic
263C25C2×#12012yes—
264C25C2×#12011yescharacteristic
265C25C2×#12011yescharacteristic
266C22×Q8#37912yes—
267C22×D4#37912yes—
268C22×D4C22×C22⋊C414yes—
269C22×D4C2×#12014yes—
270C22×D4C2×#12014yes—
271C22×D4C2×#12014yes—
272C23×C4#37911yescharacteristic
273C23×C4#37911yescharacteristic
274C23×C4#37911yescharacteristic
275C23×C4#37911yescharacteristic
276C23×C4#37912yes—
277C23×C4#37912yes—
278C23×C4#37912yes—
279C23×C4#37914yes—
280C23×C4B(4,𝔽2)12yes—
281C23×C4#45812yes—
282C23×C4C22×C22⋊C411yescharacteristic
283C23×C4C2×#12011yescharacteristic
284C23×C4C2×#12012yes—
285C23×C4C2×#12012yes—
286C2×C4⋊C4#37912yes—
287C2×C4⋊C4#37912yes—
288C2×C22⋊C4#37914yes—
289C2×C22⋊C4#37914yes—
290C2×C22⋊C4#37914yes—
291C2×C22⋊C4#37914yes—
292C2×C42#37914yes—
293C2×C42#37914yes—
294C24C2×#37912yes—
295C24C2×#37912yes—
296C24C2×#37911yescharacteristic
297C24#284611yescharacteristic
298C24#284612yes—
299C24#284612yes—
300C24#284611yescharacteristic
301C24C2×B(4,𝔽2)11yescharacteristic
302C24C2×#45811yescharacteristic
303C24#302611yescharacteristic
304C24#302611yescharacteristic
305C22×C4#179012yes—
306C22×C4#179012yes—
307C22×C4C2×#37912yes—
308C22×C4#284612yes—
309C23#1467911yescharacteristic
310C23C2×#179011yescharacteristic
311C23#2060911yescharacteristic
312C23#3289811yescharacteristic
313C23#3289811yescharacteristic
314C23#3290011yescharacteristic
315C23#3290011yescharacteristic
316C22#39212511yescharacteristic
317C22#39213512yes—
318C22#51179012yes—
319C2#7221847511yescharacteristic
320C2#7223834311yescharacteristic
321C2#7225166211yescharacteristic
subgroupG/H
center, socleC22#392125
commutatorC23×D4C23×C4
Frattini, cosocleC24×D4C24
prime radical#72249444C2
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. #72262281C2semidirect
2. #72262272C2semidirect
3. #72251570C2semidirect
5. #72236898C2semidirect
6. #72236837C2semidirect
7. #72235932C2semidirect
8. #72235848C2semidirect
9. #72226038C2semidirect
10. #9040959C4semidirect
11. #9040958C4semidirect
12. #8943604C4semidirect
13. #8942673C4semidirect
18. #555894C22semidirect
19. #496231C22semidirect
20. #496225C22semidirect
21. #496194C22semidirect
22. #382422C22semidirect
23. #379759C22semidirect
24. #379754C22semidirect
25. #201690C22semidirect
26. #201687C22semidirect
27. C2×#9497C22semidirect
28. #201322C22semidirect
29. C2×#8547C22semidirect
30. C2×#8541C22semidirect
84. C2×#551C22⋊C4semidirect
85. C2×D42C22⋊C4semidirect
86. C2×#543C22⋊C4semidirect
87. C2×#543C22⋊C4semidirect
88. C2×#538C22⋊C4semidirect
90. C2×#534C22⋊C4semidirect
91. C2×#534C22⋊C4semidirect
95. C22×#120C22⋊C4semidirect
96. C22×#120C22⋊C4semidirect
107. C2×#518C22⋊C4semidirect
111. #2804C22⋊C4semidirect
112. #2748C22⋊C4semidirect
113. #2709C22⋊C4semidirect
114. #2697C22⋊C4semidirect
115. #2684C22⋊C4semidirect
116. #2684C22⋊C4semidirect
117. #2679C22⋊C4semidirect
118. #2669C22⋊C4semidirect
119. #2561C22⋊C4semidirect
120. #2561C22⋊C4semidirect
121. #2539C22⋊C4semidirect
122. #2539C22⋊C4semidirect
139. #2380C22⋊C4semidirect
140. #2379C22⋊C4semidirect
141. #2376C22⋊C4semidirect
142. #2375C22⋊C4semidirect
143. #2358C22⋊C4semidirect
144. #2358C22⋊C4semidirect
145. #2315C22⋊C4semidirect
146. #2312C22⋊C4semidirect
154. C2×#379C22⋊C4semidirect
155. C2×#379C22⋊C4semidirect
158. #2273C22⋊C4semidirect
159. #2273C22⋊C4semidirect
160. #2273C22⋊C4semidirect
161. #2273C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#722622812
#722622722
#722515702
#722494442
#722368982
#722368372
#722359322
#722358482
#722260382