Finite Group Databasebeta

← #107726

Subgroups — #107726

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#17590C211yesmaximal, characteristic
3#17501C211yesmaximal, characteristic
4#8110C211yesmaximal, characteristic
5#8062C211yesmaximal, characteristic
6#7973C211yesmaximal, characteristic
7#7920C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987C414yescomplemented
10#2986—22no—
11#2985—22no—
12#2985—22no—
13Q8×M16—22no—
14Q8×M16—22no—
15#2921—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2283C2211yescharacteristic
22#1518—22no—
23#1511C414yes—
24#1506—22no—
25#1493C414yescomplemented
26#1491—22no—
27#1485—22no—
28#1305—22no—
29#1299—22no—
30#1288—22no—
31#1281—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533C2×C412yes—
39#533D412yes—
40#533D412yes—
41C2×C4⋊Q8Q811yescharacteristic
42C2×C4⋊Q8Q811yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8D412yes—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#446—24no—
51#446—24no—
52#446—28no—
53#446—28no—
54C8×Q8—24no—
55C8×Q8—24no—
56#433—28no—
57#433—28no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432C2×C412yes—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431D412yes—
74#423—24no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423C2×C412yes—
79#407—21no—
80#407—21no—
81#407C2×C412yes—
82#407D412yes—
83C4×M16—24no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16—21no—
91C4×M16D412yes—
92C2×#97—21no—
93C2×C4×C8C2×C412yes—
94#383—21no—
95#382—21no—
96#381C2×C412yes—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C22⋊C4—21no—
100#376—21no—
101#328—28no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8C4⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—44no—
113C2×M16—41no—
114C2×M16—41no—
115C22×C8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C4⋊C412yes—
121C4⋊Q8C4⋊C412yes—
122C4⋊Q8C4⋊C412yes—
123C4⋊Q8C4⋊C412yes—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8C4214yes—
135C22⋊Q8C22⋊C414yes—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8C22⋊C414yes—
143C4×Q8C22⋊C414yes—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C4⋊C412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—28no—
172C4⋊C8—28no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8C22⋊C414yes—
189#97—28no—
190#97—22no—
191#97—41no—
192#97—41no—
193#97—41no—
194#97—41no—
195#97C22⋊C414yes—
196C4×C8—28no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203#95—22no—
204#95—22no—
205#95—41no—
206#95—41no—
207C24C4×D411yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8#9514yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4C2×C4211yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4#11711yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238M16—44no—
239M16—44no—
240M16—44no—
241M16—44no—
242M16—82no—
243M16—82no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—82no—
263C4⋊C4#9514yes—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—82no—
267C42—24no—
268C42—24no—
269C42—82no—
270C42C2×C22⋊C411yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42C2×C22⋊C411yescharacteristic
273C42C2×C4⋊C411yescharacteristic
274C42C2×C4⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12011yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42#12311yescharacteristic
285C42#12311yescharacteristic
286C23—41no—
287C23—41no—
288C23—41no—
289C23C4×C22⋊C411yescharacteristic
290C23#38511yescharacteristic
291C23#38611yescharacteristic
292Q8—44no—
293Q8—44no—
294Q8—44no—
295Q8—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4C2×#9511yescharacteristic
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4#37911yescharacteristic
324C2×C4#38011yescharacteristic
325C2×C4#38011yescharacteristic
326C2×C4#38111yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38511yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38611yescharacteristic
333C8—44no—
334C8—44no—
335C8—44no—
336C8—82no—
337C8—82no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—82no—
348C22C22⋊#9511yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—44no—
360C4—44no—
361C4—162no—
362C4#186112yes—
363C4#190512yes—
364C2—41no—
365C2—41no—
366C2—41no—
367C2#1075711yescharacteristic
368C2#1078011yescharacteristic
369C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2987C4semidirect
25. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#602492
#175902
#175012
#81102
#80622
#79732
#79202