Finite Group Databasebeta

← #107192

Subgroups — #107192

Order 512.

690 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21826C211yesmaximal, characteristic
3#17404C211yesmaximal, characteristic
4#8070C211yesmaximal, characteristic
5#8032C211yesmaximal, characteristic
6#7978C211yesmaximal, characteristic
7#7954C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2985—22no—
11#2984—22no—
12#2983C414yescomplemented
13#2962—22no—
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#1996—22no—
23#1996C414yes—
24#1820—22no—
25#1472—22no—
26#1470C414yescomplemented
27#1467—22no—
28#1300—22no—
29#1295—22no—
30#1288—22no—
31#1285—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8C2×C412yes—
44C2×C22⋊Q8D412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D412yes—
47C2×#117D412yes—
48C22×C42C2311yescharacteristic
49#446—28no—
50#446—24no—
51#446—28no—
52#446—28no—
53C8×Q8—24no—
54#433—28no—
55#432—24no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#432C2×C412yes—
65#432C2×C412yes—
66#432D412yes—
67#432D412yes—
68#431—24no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431C2×C412yes—
73#424—24no—
74#423—21no—
75#423—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82#405—24no—
83C4×M16—21no—
84C4×M16D412yes—
85C2×#97—21no—
86C2×#97D412yes—
87#383—21no—
88#383—21no—
89#382—21no—
90#382—21no—
91#381—21no—
92#381—21no—
93#380—21no—
94#380C2×C412yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4C2×C412yes—
99#376—21no—
100#376—21no—
101#342—28no—
102#324—28no—
103C22×Q8C4⋊C412yes—
104C22×Q8C4⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—22no—
109C2×M16—41no—
110C22×C8—22no—
111C22×C8—41no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C4214yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C4⋊C412yes—
149C2×C4⋊C4C4⋊C412yes—
150C2×C4⋊C4C4⋊C412yes—
151C2×C4⋊C4C4⋊C412yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C22⋊C414yes—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42C22⋊C414yes—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—28no—
171C4⋊C8—28no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188#97—28no—
189#97—22no—
190#97—22no—
191#97—41no—
192#97—41no—
193#97C22⋊C414yes—
194#97C22⋊C414yes—
195C4×C8—28no—
196C4×C8—22no—
197C4×C8—22no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206C24C2×C22⋊C411yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4#9514yes—
219C22×C4#9514yes—
220C22×C4C2×C4211yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4C2×C4⋊C411yescharacteristic
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×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238M16—44no—
239M16—82no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—82no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C42—24no—
265C42—24no—
266C42—82no—
267C42—82no—
268C42C2×C22⋊C411yescharacteristic
269C42C2×C22⋊C411yescharacteristic
270C42#11711yescharacteristic
271C42#11711yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12011yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23C2×#9511yescharacteristic
288C23#38611yescharacteristic
289C23#38611yescharacteristic
290Q8—44no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
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—44no—
315C2×C4—44no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4C4×C22⋊C411yescharacteristic
324C2×C4#37911yescharacteristic
325C2×C4#38011yescharacteristic
326C2×C4#38011yescharacteristic
327C2×C4#38111yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38511yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
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—82no—
347C22C22⋊#9511yescharacteristic
348C22#191212yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—44no—
362C4—44no—
363C4—162no—
364C4—162no—
365C2—22no—
366C2—41no—
367C2—41no—
368C2#1074111yescharacteristic
369C2#1075711yescharacteristic
370C2#1076611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
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
12. #2983C4semidirect
26. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#218262
#174042
#80702
#80322
#79782
#79542