Finite Group Databasebeta

← #72262607

Subgroups — #72262607

Order 2048.

At least 6801 conjugacy classes of subgroups; only the 1524 normal subgroups are shown.

The subgroup lattice has 418 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72262603C212yesmaximal, complemented
2#72259008C212yesmaximal, complemented
3#72240568C212yesmaximal, complemented
4#72238531C214yesmaximal, complemented
5#72235847C212yesmaximal, complemented
6C2×#188035C211yesmaximal, characteristic, complemented
7#72219307C211yesmaximal, characteristic, complemented
8C2×#7631286C211yesmaximal, characteristic
9C24×H(5,𝔽2)C2211yescharacteristic
10C22×#3484C412yescomplemented
11C23×D42C2211yescharacteristic
12C2×#35347C412yescomplemented
13C2×#35346C412yescomplemented
14C2×#34980C2212yes—
15C22×#2403C2212yes—
16C2×#34256C412yescomplemented
17C2×#34171C2211yescharacteristic
18C2×#21746C2212yescomplemented
19C2×#21746C2211yescharacteristic, complemented
20C2×#21746C2211yescharacteristic, complemented
21C22×#1781C2212yes—
22C22×#1781C2212yescomplemented
23C22×#1781C2212yescomplemented
24C22×#1781C2214yescomplemented
25#517411C2212yes—
26#517411C2214yescomplemented
27C2×#8101C2214yescomplemented
28C2×#8101C2214yescomplemented
29C23×H(5,𝔽2)C2×C412yes—
30C23×H(5,𝔽2)C2×C412yes—
31C23×H(5,𝔽2)D418yes—
32C25×D4C2×C411yescharacteristic
33C25×D4D411yescharacteristic
34C25×D4C2311yescharacteristic
35C25×D4C2312yes—
36C25×D4C2311yescharacteristic
37C22×#534C2×C414yes—
38C23×#127D411yescharacteristic
39C23×#121C2×C411yescharacteristic
40C23×#121D411yescharacteristic
41C23×#121D411yescharacteristic
42C23×#121D411yescharacteristic
43C23×#120C2×C414yes—
44C23×#120C2×C411yescharacteristic
45C23×#120D411yescharacteristic
46C23×#120D411yescharacteristic
47C23×#120C2311yescharacteristic
48C22×#518C2×C412yes—
49C23×C4×D4C2×C411yescharacteristic
50C23×C4×D4D411yescharacteristic
51C2×#2849D414yes—
52C2×#2846D414yes—
53C2×#2403D4116yes—
54C2×#2613D418yes—
55#34171D418yes—
56C2×#2308C2×C414yes—
57C2×#2308C2×C412yes—
58C22×#386C2×C414yes—
59C22×#379C2312yes—
60C22×#379C2314yes—
61C22×#379C2312yes—
62C22×#379C2312yes—
63C2×#1781D418yescomplemented
64#13867D418yescomplemented
65C27C22⋊C411yescharacteristic, complemented
66C27C22×C411yescharacteristic
67C27C2×D412yes—
68C27C2×D411yescharacteristic
69C27C2411yescharacteristic
70C22×H(5,𝔽2)C22⋊C418yes—
71C22×H(5,𝔽2)C22⋊C418yes—
72C24×D4C22⋊C411yescharacteristic, complemented
73C24×D4C22⋊C411yescharacteristic, complemented
74C24×D4C22⋊C411yescharacteristic, complemented
75C24×D4C22⋊C411yescharacteristic, complemented
76C24×D4C22⋊C411yescharacteristic, complemented
77C24×D4C22×C411yescharacteristic
78C24×D4C22×C414yes—
79C24×D4C22×C411yescharacteristic
80C24×D4C22×C411yescharacteristic
81C24×D4C2×D411yescharacteristic
82C24×D4C2×D414yes—
83C24×D4C2×D411yescharacteristic
84C24×D4C2×D411yescharacteristic
85C24×D4C2×D411yescharacteristic
86C24×D4C2×D411yescharacteristic
87C24×D4C2×D418yes—
88C24×D4C2×D411yescharacteristic
89C24×D4C2×D411yescharacteristic
90C24×D4C2×D411yescharacteristic
91C24×D4C2×D418yes—
92C24×D4C2×D414yes—
93C24×D4C2×D418yes—
94C24×D4C2×D414yes—
95C24×D4C2×D414yes—
96C24×D4C2×D414yes—
97C24×D4#4114yes—
98C24×D4#4114yes—
99C25×C4C22⋊C411yescharacteristic, complemented
100C25×C4C22⋊C411yescharacteristic, complemented
101C25×C4C22×C411yescharacteristic
102C25×C4C2×D412yes—
103C25×C4C2×D411yescharacteristic
104C22×#123C22⋊C418yescomplemented
105C22×#120C22⋊C418yescomplemented
106C22×#120C22⋊C4116yescomplemented
107C22×#120C2×D414yes—
108C22×#120C2×D414yes—
109C2×#518C22⋊C418yescomplemented
110C23×C4⋊C4C22⋊C411yescharacteristic, complemented
111C23×C4⋊C4C22⋊C411yescharacteristic, complemented
112C23×C22⋊C4C22⋊C411yescharacteristic, complemented
113C23×C22⋊C4C22⋊C411yescharacteristic, complemented
114C23×C22⋊C4C22⋊C411yescharacteristic, complemented
115C23×C22⋊C4C22⋊C411yescharacteristic, complemented
116C23×C22⋊C4C22⋊C411yescharacteristic, complemented
117C23×C22⋊C4C22×C411yescharacteristic
118C23×C22⋊C4C22×C414yes—
119C23×C22⋊C4C2×D411yescharacteristic
120C23×C22⋊C4C2×D411yescharacteristic
121C23×C22⋊C4C2×D414yescomplemented
122C23×C42C22⋊C411yescharacteristic, complemented
123#2308C22⋊C418yescomplemented
124C2×#386C22⋊C4116yescomplemented
125C2×#381C22⋊C418yescomplemented
126C2×#379C22⋊C418yescomplemented
127C2×#379C2×D414yes—
128C2×#379C2×D414yes—
129C2×#379C2×D418yes—
130C2×#379C2×D418yes—
131C2×#379C2×D418yescomplemented
132C2×#379C2×D414yescomplemented
133C26C2×C22⋊C411yescharacteristic
134C26C2×C22⋊C414yes—
135C26C2×C22⋊C411yescharacteristic
136C26C2×C22⋊C411yescharacteristic
137C26C2×C22⋊C411yescharacteristic
138C26C2×C22⋊C414yes—
139C26C2×C22⋊C411yescharacteristic
140C26#12018yes—
141C26#12014yes—
142C26#12012yes—
143C26#12114yes—
144C26#12312yes—
145C26C23×C411yescharacteristic
146C26C22×D412yes—
147C26C22×D411yescharacteristic
148C26C22×D412yes—
149C26C22×D412yes—
150C26C22×D411yescharacteristic
151C26C22×D412yes—
152C26C22×D412yes—
153C26C22×D412yes—
154C26C22×D414yes—
155C26C2×#4112yes—
156C26C2×#4112yes—
157C26H(5,𝔽2)12yes—
158C26H(5,𝔽2)14yes—
159C26H(5,𝔽2)12yes—
160C2×H(5,𝔽2)#99132yescomplemented
161C23×D4C2×C22⋊C412yes—
162C23×D4C2×C22⋊C412yes—
163C23×D4C2×C22⋊C418yes—
164C23×D4C2×C22⋊C418yes—
165C23×D4C2×C22⋊C412yes—
166C23×D4C2×C22⋊C414yes—
167C23×D4C2×C22⋊C412yes—
168C23×D4C4×D414yes—
169C23×D4C4×D414yes—
170C23×D4C4×D414yes—
171C23×D4C4×D414yes—
172C23×D4#120116yes—
173C23×D4#12014yes—
174C23×D4#12014yes—
175C23×D4#12014yes—
176C23×D4#12012yes—
177C23×D4#12012yes—
178C23×D4#12014yes—
179C23×D4#12014yes—
180C23×D4#12014yes—
181C23×D4#12014yes—
182C23×D4#12012yes—
183C23×D4#12012yes—
184C23×D4#12018yes—
185C23×D4#12114yes—
186C23×D4#12114yes—
187C23×D4#12118yes—
188C23×D4#12118yes—
189C23×D4#12118yes—
190C23×D4#12118yes—
191C23×D4#12312yes—
192C23×D4#12312yes—
193C23×D4#12318yes—
194C23×D4#12418yes—
195C23×D4#12718yes—
196C24×C4C2×C22⋊C411yescharacteristic
197C24×C4C2×C22⋊C414yes—
198C24×C4C2×C22⋊C411yescharacteristic
199C24×C4C2×C22⋊C411yescharacteristic
200C24×C4C2×C22⋊C411yescharacteristic
201C24×C4C2×C22⋊C411yescharacteristic
202C24×C4C2×C22⋊C414yes—
203C24×C4C2×C22⋊C411yescharacteristic
204C24×C4C2×C22⋊C411yescharacteristic
205C24×C4#12018yes—
206C24×C4#12014yes—
207C24×C4#12012yes—
208C24×C4#12114yes—
209C24×C4#12312yes—
210C22×C22⋊C4C2×C22⋊C414yes—
211C22×C22⋊C4C2×C22⋊C418yes—
212C22×C22⋊C4C2×C22⋊C418yes—
213C22×C22⋊C4C2×C22⋊C418yes—
214C22×C22⋊C4#12012yes—
215C22×C22⋊C4#12014yes—
216C22×C22⋊C4#12012yes—
217#379#120116yes—
218#379#120116yes—
219C25#37912yes—
220C25#37918yes—
221C25#37912yes—
222C25#37912yes—
223C25#37912yes—
224C25#37912yes—
225C25#38612yes—
226C25#38612yes—
227C25#38612yes—
228C25#38612yes—
229C25#39218yes—
230C25#39418yes—
231C25C22×C22⋊C411yescharacteristic
232C25C22×C22⋊C412yes—
233C25C22×C22⋊C411yescharacteristic
234C25C22×C22⋊C412yes—
235C25C22×C22⋊C411yescharacteristic
236C25C2×C4×D412yes—
237C25C2×C4×D412yes—
238C25#51812yes—
239C25#51812yes—
240C25C2×#12011yescharacteristic
241C25C2×#12012yes—
242C25C2×#12014yes—
243C25C2×#12011yescharacteristic
244C25C2×#12012yes—
245C25C2×#12012yes—
246C25C2×#12014yes—
247C25C2×#12011yescharacteristic
248C25C2×#12012yes—
249C25C2×#12012yes—
250C25C2×#12014yes—
251C25C2×#12011yescharacteristic
252C25C2×#12012yes—
253C25C2×#12014yes—
254C25C2×#12112yes—
255C25C2×#12112yes—
256C25C2×#12112yes—
257C25C2×#12112yes—
258C25C2×#12112yes—
259C25C2×#12112yes—
260C25C2×#12312yes—
261C25C2×#12311yescharacteristic
262C25C2×#12311yescharacteristic
263C25C2×#12412yes—
264C25C2×#12712yes—
265C25#53411yescharacteristic
266C25#53414yes—
267C25#53414yes—
268C25#53412yes—
269C25#53412yes—
270C25#53412yes—
271C25#53411yescharacteristic
272C25#53514yes—
273C25#53514yes—
274C25D4214yes—
275C25D4214yes—
276C25D4214yes—
277C25D4214yes—
278C25D4214yes—
279C25D4214yes—
280C25D4214yes—
281C25D4214yes—
282C25#54614yes—
283C25#54614yes—
284C25#54614yes—
285C25#54614yes—
286C25#54614yes—
287C25#54614yes—
288C25#54614yes—
289C25#54614yes—
290C22×D4#38618yes—
291C22×D4#38618yes—
292C22×D4#39018yes—
293C22×D4#39018yes—
294C22×D4#39218yes—
295C22×D4#39218yes—
296C22×D4#39418yes—
297C22×D4#39418yes—
298C22×D4C2×#99116yes—
299C22×D4C2×#99132yes—
300C23×C4#37912yes—
301C23×C4#37918yes—
302C23×C4#37912yes—
303C23×C4#37912yes—
304C23×C4#37912yes—
305C23×C4#37912yes—
306C23×C4#37912yes—
307C23×C4#37912yes—
308C23×C4#38612yes—
309C23×C4#38612yes—
310C23×C4#38612yes—
311C23×C4#38612yes—
312C23×C4#39218yes—
313C23×C4#39418yes—
314C2×C22⋊C4#379116yes—
315C24#143814yes—
316C24#143814yes—
317C24#178118yes—
318C24#1781116yescomplemented
319C24C2×#37911yescharacteristic
320C24C2×#37911yescharacteristic
321C24C2×#37912yes—
322C24C2×#37911yescharacteristic
323C24C2×#37912yes—
324C24C2×#38611yescharacteristic
325C24C2×#38611yescharacteristic
326C24C2×#38612yes—
327C24C2×#39012yes—
328C24#230811yescharacteristic
329C24#230811yescharacteristic
330C24#230812yes—
331C24#230912yes—
332C24D4×C22⋊C414yes—
333C24D4×C22⋊C414yes—
334C24D4×C22⋊C414yes—
335C24D4×C22⋊C414yes—
336C24C2×#39212yes—
337C24C2×#39212yes—
338C24C2×#39412yes—
339C24C2×#39412yes—
340C24#240314yes—
341C24#240311yescharacteristic
342C24#240312yes—
343C24#240312yes—
344C24#240314yes—
345C24#240312yes—
346C24#240311yescharacteristic
347C24#240511yescharacteristic
348C24#240511yescharacteristic
349C24#240512yes—
350C24#240514yes—
351C24#240514yes—
352C24#240512yes—
353C24#240512yes—
354C24#240814yes—
355C24#240814yes—
356C24#240814yes—
357C24#240814yes—
358C24#241012yes—
359C24#241014yes—
360C24#241012yes—
361C24#241014yes—
362C24#241012yes—
363C24#241014yes—
364C24#241014yes—
365C24#241014yes—
366C24#241012yes—
367C24#241014yes—
368C24#243314yes—
369C24#243314yes—
370C24#243514yes—
371C24#243514yes—
372C24#261312yes—
373C24#261312yes—
374C24#261314yes—
375C24#284612yes—
376C24#284612yes—
377C24#284614yes—
378C24C22×#9918yes—
379C2×D4#1898132yes—
380C22×C4#143814yes—
381C22×C4#143814yes—
382C22×C4#178118yes—
383C22×C4#1781116yescomplemented
384C23#1096518yes—
385C23C2×#143811yescharacteristic
386C23#1386711yescharacteristic
387C23#1386711yescharacteristic
388C23#1386714yes—
389C23#1386711yescharacteristic
390C23#1386711yescharacteristic
391C23#1390214yes—
392C23#1433511yescharacteristic
393C23#1467911yescharacteristic
394C23#1471512yes—
395C23#1471514yes—
396C23#1471512yes—
397C23#1472412yes—
398C23#1472412yes—
399C23#1472414yes—
400C23C2×#178118yes—
401C23C2×#178114yes—
402C23C2×#189818yes—
403C23#2174618yes—
404C23#2389514yes—
405C2×C4#1096518yes—
406C22#15409411yescharacteristic
407C22C2×#1096512yes—
408C22#35942818yes—
409C22#35952812yes—
410C22#35954912yes—
411C22#38242212yes—
412C22#39210912yes—
413C22#39406414yes—
414C2#7221837511yescharacteristic
415C2#7224054811yescharacteristic
416C2#7224059011yescharacteristic
subgroupG/H
center, socleC22#154094
commutatorC26C23×C4
Frattini, cosocleC27C24
prime radicalC2×#7631286C2
LCSG ▷ C26 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C27 ◁ G
LECSG ▷ C27 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72262603C2semidirect
2. #72259008C2semidirect
3. #72240568C2semidirect
4. #72238531C2semidirect
5. #72235847C2semidirect
6. C2×#188035C2semidirect
7. #72219307C2semidirect
10. C22×#3484C4semidirect
12. C2×#35347C4semidirect
13. C2×#35346C4semidirect
16. C2×#34256C4semidirect
18. C2×#21746C22semidirect
19. C2×#21746C22semidirect
20. C2×#21746C22semidirect
22. C22×#1781C22semidirect
23. C22×#1781C22semidirect
24. C22×#1781C22semidirect
26. #517411C22semidirect
27. C2×#8101C22semidirect
28. C2×#8101C22semidirect
63. C2×#1781D4semidirect
64. #13867D4semidirect
65. C27C22⋊C4semidirect
72. C24×D4C22⋊C4semidirect
73. C24×D4C22⋊C4semidirect
74. C24×D4C22⋊C4semidirect
75. C24×D4C22⋊C4semidirect
76. C24×D4C22⋊C4semidirect
99. C25×C4C22⋊C4semidirect
100. C25×C4C22⋊C4semidirect
104. C22×#123C22⋊C4semidirect
105. C22×#120C22⋊C4semidirect
106. C22×#120C22⋊C4semidirect
109. C2×#518C22⋊C4semidirect
110. C23×C4⋊C4C22⋊C4semidirect
111. C23×C4⋊C4C22⋊C4semidirect
112. C23×C22⋊C4C22⋊C4semidirect
113. C23×C22⋊C4C22⋊C4semidirect
114. C23×C22⋊C4C22⋊C4semidirect
115. C23×C22⋊C4C22⋊C4semidirect
116. C23×C22⋊C4C22⋊C4semidirect
121. C23×C22⋊C4C2×D4semidirect
122. C23×C42C22⋊C4semidirect
123. #2308C22⋊C4semidirect
124. C2×#386C22⋊C4semidirect
125. C2×#381C22⋊C4semidirect
126. C2×#379C22⋊C4semidirect
131. C2×#379C2×D4semidirect
132. C2×#379C2×D4semidirect
160. C2×H(5,𝔽2)#99semidirect
318. C24#1781semidirect
383. C22×C4#1781semidirect

Maximal subgroups

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

subgroupindex
#722626032
#722590082
#722405682
#722385312
#722358472
C2×#1880352
#722193072
C2×#76312862