Finite Group Databasebeta

← #108167

Subgroups — #108167

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34389C211yesmaximal, characteristic
2C2×#1554C211yesmaximal, characteristic
3C2×#1497C211yesmaximal, characteristic
4#8032C211yesmaximal, characteristic
5#7916C211yesmaximal, characteristic
6#7916C211yesmaximal, characteristic
7#7894C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2381—21no—
14#2381—21no—
15#2368—22no—
16#2356—22no—
17#2343—22no—
18#2336—22no—
19#2329C414yes—
20#2320—22no—
21#2283C2211yescharacteristic
22C2×#383C2211yescharacteristic
23#1554—22no—
24#1554C414yescomplemented
25C2×#333—21no—
26C2×#332—21no—
27#1497—22no—
28#1497C414yescomplemented
29C2×Q8⋊C8—22no—
30#1295—22no—
31#1280—22no—
32#1280—22no—
33#1279—22no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8D412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431C2×C412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#391—24no—
80#391—24no—
81#389—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#383—22no—
90#383—21no—
91#383—21no—
92#383—21no—
93#383—21no—
94#383—21no—
95#382—22no—
96#382D412yes—
97#381—21no—
98#380C2×C412yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—21no—
104#376—22no—
105#376—21no—
106C2×#95C2×C412yes—
107C2×#95D412yes—
108C2×#95D412yes—
109#333—24no—
110#332—24no—
111Q8⋊C8—28no—
112C22×Q8—21no—
113C22×Q8C22⋊C412yes—
114C23×C4C22⋊C414yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C22⋊C412yes—
127C4⋊Q8C22⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129C4⋊Q8C4⋊C412yes—
130#125—22no—
131#125—22no—
132#125—22no—
133#125—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8C4214yes—
141C22⋊Q8C4⋊C414yes—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8C22⋊C414yes—
149C4×Q8C4⋊C414yes—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—41no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—41no—
204C4×C8—41no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—82no—
226C2×Q8#9514yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4C2×C4211yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—44no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4#9514yes—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C42—24no—
282C42—24no—
283C42—82no—
284C42—84no—
285C42—84no—
286C42C2×C22⋊C411yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42C2×C4⋊C411yescharacteristic
289C42C2×C4⋊C411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42#12311yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C23—22no—
303C23—22no—
304C23—22no—
305C23C4×C22⋊C411yescharacteristic
306C23#37911yescharacteristic
307C23#38011yescharacteristic
308Q8—44no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—164no—
342C2×C4C2×#9511yescharacteristic
343C2×C4C4×C22⋊C411yescharacteristic
344C2×C4#38011yescharacteristic
345C2×C4#38111yescharacteristic
346C2×C4#38511yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38511yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38611yescharacteristic
354C8—82no—
355C8—82no—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22C22⋊#9511yescharacteristic
366C22#179112yes—
367C22#179412yes—
368C22#179612yes—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—44no—
382C4—44no—
383C4—84no—
384C4—84no—
385C2—41no—
386C2—41no—
387C2#849014yes—
388C2#1074711yescharacteristic
389C2#1077711yescharacteristic
390C2#1077911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1554C4semidirect
28. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#343892
C2×#15542
C2×#14972
#80322
#79162
#78942