Finite Group Databasebeta

← #72321587

Subgroups — #72321587

Order 2048.

676 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72321574C212yesmaximal, complemented
2#72321570C212yesmaximal
3C2×#94506C211yesmaximal, characteristic, complemented
4C2×#94506C211yesmaximal, characteristic, complemented
5#72231854C211yesmaximal, characteristic
6#512545C411yescharacteristic
7#512545C2211yescharacteristic
8#479899C411yescharacteristic
9#153432C414yes—
10C2×#7511C2212yes—
11C2×#7511C2212yes—
12#149496—21no—
13#149496—21no—
14#149496C411yescharacteristic
15#149496C2211yescharacteristic
16#149495—21no—
17#149495C411yescharacteristic
18C2×#7330—21no—
19C2×#7330C2211yescharacteristic
20#94759C414yes—
21#94506—21no—
22#94506—21no—
23#94506—21no—
24#94506—21no—
25C2×#2258C812yes—
26C2×#2258C2×C412yes—
27C4×#504C811yescharacteristic
28C4×#504C2×C411yescharacteristic
29C8×#131C811yescharacteristic
30C2×M128C2×C414yes—
31C2×M128C2×C414yes—
32#13549—22no—
33#13549—22no—
34#13549C812yes—
35#13549C2×C412yes—
36C4×M64—21no—
37C4×M64—21no—
38C4×M64C2×C411yescharacteristic
39C4×M64C2×C411yescharacteristic
40C2×#1397—21no—
41C2×#1397—21no—
42C2×#1397C2×C411yescharacteristic
43C2×#1397C2311yescharacteristic
44#11399—21no—
45#11399—21no—
46#11399C811yescharacteristic
47#11399C2×C411yescharacteristic
48C8×M32—21no—
49C8×M32C811yescharacteristic
50C2×#1311—21no—
51C2×#1311C2×C411yescharacteristic
52#7514C8116yescomplemented
53#7511—22no—
54#7511—22no—
55#7511—22no—
56#7511—22no—
57#7330—21no—
58#7330—21no—
59#7330—21no—
60#7330—21no—
61#7330—21no—
62#7330—21no—
63#7330D411yescharacteristic
64#7330D411yescharacteristic
65#7330D411yescharacteristic
66#7330D411yescharacteristic
67C2×#504—21no—
68C2×#504C4211yescharacteristic
69C2×#504C2×C811yescharacteristic
70C8×#41—21no—
71C4×#131C2×C811yescharacteristic
72#2258—24no—
73#2258—24no—
74#2258—24no—
75#2258C2×C818yes—
76C2×M64—22no—
77C2×M64—22no—
78C2×M64C4212yes—
79C2×M64C2×C812yes—
80C22×C32—22no—
81C22×C32—22no—
82C22×C32C2×C812yes—
83C22×C32C22×C412yes—
84#2108—21no—
85#2108—22no—
86#2108—21no—
87#2108—21no—
88#2108C4211yescharacteristic
89#2108C2×C811yescharacteristic
90C4×M32—21no—
91C4×M32—21no—
92C4×M32C2×C811yescharacteristic
93C4×M32C2×C811yescharacteristic
94C2×C4×C16—21no—
95C2×C4×C16—21no—
96C2×C4×C16C2×C811yescharacteristic
97C2×C4×C16C22×C411yescharacteristic
98#1450—21no—
99#1450C2×C811yescharacteristic
100C8×M16—21no—
101C8×M16—22no—
102C2×C82—21no—
103C2×C82C2×C811yescharacteristic
104M128—24no—
105M128—24no—
106M128—24no—
107M128C2×C818yescomplemented
108C2×C64—24no—
109C2×C64C2×C818yescomplemented
110#1397—22no—
111#1397—22no—
112#1397—21no—
113#1397—21no—
114#1397C22⋊C411yescharacteristic
115#1397C22⋊C411yescharacteristic
116#1397M1612yes—
117#1397C2×D411yescharacteristic
118#1397C2×D411yescharacteristic
119#1397#4112yes—
120C4×C32—22no—
121C4×C32—22no—
122C4×C32—21no—
123C4×C32—21no—
124#1311—21no—
125#1311—21no—
126#1311—21no—
127#1311C22⋊C411yescharacteristic
128#1311C22⋊C411yescharacteristic
129#1311M1611yescharacteristic
130#1311M1611yescharacteristic
131C8×C16—21no—
132C8×C16—21no—
133C8×C16—21no—
134C2×#131C4×C811yescharacteristic
135C2×#131M3211yescharacteristic
136C4×#41M3211yescharacteristic
137#504—22no—
138#504—22no—
139#504—42no—
140#504—42no—
141#504C4×C814yes—
142#504M3214yes—
143C2×M32—22no—
144C2×M32—21no—
145C2×M32—21no—
146C2×M32—21no—
147C2×M32C4×C811yescharacteristic
148C2×M32C4×C811yescharacteristic
149C22×C16—21no—
150C22×C16—21no—
151C22×C16—22no—
152C22×C16—21no—
153C22×C16C2×C4211yescharacteristic
154C22×C16C22×C811yescharacteristic
155C8×Q8—41no—
156C8×D4—41no—
157C8×D4—42no—
158#431—22no—
159#431—21no—
160#405—21no—
161#405—21no—
162#405C4×C811yescharacteristic
163#405M3211yescharacteristic
164C4×M16—21no—
165C4×M16M3211yescharacteristic
166C2×C4×C8—21no—
167C2×C4×C8—22no—
168C2×C4×C8—21no—
169C2×C4×C8C22×C811yescharacteristic
170M64—28no—
171M64—22no—
172M64—22no—
173M64—28no—
174M64C4×C814yes—
175M64M3214yes—
176C2×C32—28no—
177C2×C32—22no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—28no—
182C2×C32C22⋊C812yes—
183C2×C32C22⋊C812yes—
184C2×C32C4×D412yes—
185C2×C32C4×D412yes—
186C2×C32C22×C814yes—
187C2×C32#13114yes—
188#346—22no—
189#346—21no—
190#346—21no—
191#346—21no—
192#346M3212yes—
193C4×C16—22no—
194C4×C16—21no—
195C4×C16—21no—
196C4×C16—21no—
197C4×C16—21no—
198C4×C16—21no—
199C4×C16C22⋊C811yescharacteristic
200C4×C16C22⋊C811yescharacteristic
201C4×C16C2×C22⋊C411yescharacteristic
202C4×C16#11711yescharacteristic
203C4×C16C2×M1611yescharacteristic
204C4×C16C2×M1611yescharacteristic
205#322—22no—
206#322M3212yes—
207C82—22no—
208C82—21no—
209C82—21no—
210C82C22⋊C811yescharacteristic
211C82C22⋊C811yescharacteristic
212C64—28no—
213C64—28no—
214C2×#41#34611yescharacteristic
215#131—22no—
216#131—22no—
217#131—21no—
218#131—21no—
219#131C8212yes—
220#131#34612yes—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16#34611yescharacteristic
224C2×M16#34611yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—42no—
228C22×C8—41no—
229C22×C8C2×C4×C811yescharacteristic
230C22×C8C2×M3211yescharacteristic
231C4×Q8—21no—
232C4×D4—21no—
233C4×D4—22no—
234#117—21no—
235#117#34611yescharacteristic
236C2×C42—21no—
237C2×C42C2×M3211yescharacteristic
238M32—21no—
239M32—21no—
240M32—24no—
241M32—24no—
242M32—44no—
243M32—41no—
244M32—41no—
245M32C8212yes—
246M32#34612yes—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—24no—
250C2×C16—24no—
251C2×C16—21no—
252C2×C16—22no—
253C2×C16—21no—
254C2×C16—44no—
255C2×C16—41no—
256C2×C16—41no—
257C2×C16C4×C22⋊C411yescharacteristic
258C2×C16C2×C4×C812yes—
259C2×C16C4×M1611yescharacteristic
260C2×C16#40512yes—
261C2×C16C2×C22⋊C811yescharacteristic
262C2×C16#43111yescharacteristic
263C2×C16C2×M3212yes—
264C2×C16C2×M3212yes—
265C4⋊C8—42no—
266C4⋊C8—41no—
267C22⋊C8—42no—
268C22⋊C8—41no—
269#97—21no—
270#97—21no—
271#97—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—22no—
277C4×C8—21no—
278C4×C8—42no—
279C4×C8—41no—
280C4×C8C2×C22⋊C811yescharacteristic
281C4×C8#43111yescharacteristic
282C4×C8C2×M3211yescharacteristic
283C4×C8C2×M3211yescharacteristic
284C32—22no—
285C32—24no—
286C32—22no—
287C32—24no—
288C32C8×D414yes—
289C32C8×D414yes—
290#41—21no—
291#41—21no—
292#41#131112yes—
293#41#139812yes—
294C2×Q8#139811yescharacteristic
295C2×D4—21no—
296C2×D4#139811yescharacteristic
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4C2×#34611yescharacteristic
301M16—24no—
302M16—21no—
303M16—21no—
304M16—22no—
305M16—22no—
306M16#131112yes—
307M16#139812yes—
308C2×C8—24no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—161no—
319C2×C8—162no—
320C2×C8C2×C8211yescharacteristic
321C2×C8#145011yescharacteristic
322C2×C8C4×C22⋊C811yescharacteristic
323C2×C8C2×#34611yescharacteristic
324C2×C8C4×M3211yescharacteristic
325C2×C8C4×M3211yescharacteristic
326C2×C8#211211yescharacteristic
327C2×C8#216311yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—21no—
330C22⋊C4—22no—
331C22⋊C4#139812yes—
332C42—22no—
333C42—21no—
334C42—21no—
335C42#211211yescharacteristic
336C42#216311yescharacteristic
337C16—22no—
338C16—22no—
339C16—41no—
340C16—42no—
341C16—41no—
342C16C8×M1612yes—
343C16C8×C22⋊C412yes—
344C16#211212yes—
345C16#216312yes—
346C23—21no—
347C23C2×#139811yescharacteristic
348Q8—21no—
349Q8#733012yes—
350D4—22no—
351D4—22no—
352D4—21no—
353D4#733012yes—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4C2×#131111yescharacteristic
363C2×C4#1141111yescharacteristic
364C2×C4#1259911yescharacteristic
365C2×C4C2×#139811yescharacteristic
366C2×C4C2×#139811yescharacteristic
367C2×C4C2×#139811yescharacteristic
368C8—21no—
369C8—22no—
370C8—21no—
371C8—22no—
372C8—82no—
373C8—81no—
374C8—81no—
375C8C8×C22⋊C811yescharacteristic
376C8C8×M3211yescharacteristic
377C8#1141111yescharacteristic
378C8#1259911yescharacteristic
379C22—22no—
380C22—41no—
381C22—42no—
382C22#14949511yescharacteristic
383C22#14951311yescharacteristic
384C22#15189111yescharacteristic
385C4—22no—
386C4—41no—
387C4—42no—
388C4—41no—
389C4#12824011yescharacteristic
390C4C2×#733011yescharacteristic
391C4#14951311yescharacteristic
392C4#15189111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2#7228635911yescharacteristic
subgroupG/H
centerC8C8×C22⋊C8
commutatorC2×C8C2×C82
Frattini, cosocleC2×#1397C23
socleC2#72286359
prime radicalC2×#41#346
LCSG ▷ C2×C8 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×#7330 ◁ G
LECSG ▷ C2×#1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72321574C2semidirect
3. C2×#94506C2semidirect
4. C2×#94506C2semidirect
52. #7514C8semidirect
107. M128C2×C8semidirect
109. C2×C64C2×C8semidirect

Maximal subgroups

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

subgroupindex
#723215742
#723215702
C2×#945062
#722318542