Finite Group Databasebeta

← #106095

Subgroups — #106095

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic
3#21826C211yesmaximal, characteristic
4#7976C211yesmaximal, characteristic
5#7962C211yesmaximal, characteristic
6#7899C211yesmaximal, characteristic
7#7857C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983C412yescomplemented
13#2982C412yescomplemented
14#2962—22no—
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1997C412yescomplemented
24#1996—22no—
25#1996C414yes—
26#1966C412yescomplemented
27#1901—22no—
28#1820—22no—
29#1801—22no—
30#1288—22no—
31#1286—22no—
32#1279—22no—
33#1274—22no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125Q812yes—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8D412yes—
50C2×#117D412yes—
51C22×C42C2311yescharacteristic
52#446—24no—
53#446—28no—
54#446—24no—
55#446—24no—
56C8×Q8—24no—
57C8×Q8—24no—
58C8×Q8—24no—
59#433—28no—
60#432—24no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#431—24no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C411yescharacteristic
71#431C2×C411yescharacteristic
72#431C2×C412yes—
73#431D412yes—
74#431D412yes—
75#424—24no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423—21no—
81#423C2×C412yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×#101—24no—
86#407—21no—
87#407—21no—
88C2×C22⋊C8—21no—
89#405—24no—
90C4×M16C2×C412yes—
91C4×M16D412yes—
92C2×C4×C8D412yes—
93#383—21no—
94#383—21no—
95#382—21no—
96#382—21no—
97#381—21no—
98#381—21no—
99#380—21no—
100#380C2×C412yes—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4C2×C412yes—
105#376—21no—
106#376—21no—
107#343—24no—
108#342—28no—
109C22×Q8C4212yes—
110C22×Q8C22⋊C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—41no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129#125—22no—
130#125—22no—
131#125—22no—
132#125—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C4212yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C4⋊C412yes—
158C2×C4⋊C4C4⋊C412yes—
159C2×C4⋊C4C4⋊C414yes—
160C2×C4⋊C4C4⋊C414yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4C22⋊C414yes—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42C22⋊C414yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×Q811yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C4⋊C8—28no—
180C4⋊C8—28no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192#101—84no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—28no—
201#97—22no—
202#97—22no—
203C4×C8—28no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8C22⋊C412yes—
207C4×C8C22⋊C412yes—
208C4×C8C22⋊C414yes—
209#95—22no—
210#95—22no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217C24C2×C22⋊C411yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4#9514yes—
232C22×C4#9514yes—
233C22×C4C2×C4211yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251M16—44no—
252M16—42no—
253M16—44no—
254M16—42no—
255M16—82no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—81no—
286C42—81no—
287C42C2×C22⋊C411yescharacteristic
288C42C2×C22⋊C411yescharacteristic
289C42#11711yescharacteristic
290C42#11711yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42#12011yescharacteristic
296C42#12011yescharacteristic
297C42C22⋊Q811yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C23—22no—
304C23—41no—
305C23—41no—
306C23C2×#9511yescharacteristic
307C23#38611yescharacteristic
308C23#38611yescharacteristic
309Q8—44no—
310Q8—42no—
311Q8—42no—
312Q8—44no—
313Q8—82no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4C4×C22⋊C411yescharacteristic
346C2×C4C4×C22⋊C411yescharacteristic
347C2×C4#37911yescharacteristic
348C2×C4#38011yescharacteristic
349C2×C4#38011yescharacteristic
350C2×C4#38111yescharacteristic
351C2×C4#38511yescharacteristic
352C2×C4#38511yescharacteristic
353C2×C4#38511yescharacteristic
354C2×C4#38511yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38611yescharacteristic
357C8—42no—
358C8—42no—
359C8—44no—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—82no—
371C22C22⋊#9511yescharacteristic
372C22#191212yes—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—44no—
384C4—44no—
385C4—162no—
386C4—162no—
387C4#175712yes—
388C4#190911yescharacteristic
389C4#191011yescharacteristic
390C2—22no—
391C2—41no—
392C2—41no—
393C2#1074111yescharacteristic
394C2#1074211yescharacteristic
395C2#1076411yescharacteristic
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
13. #2982C4semidirect
23. #1997C4semidirect
26. #1966C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#218792
#218262
#79762
#79622
#78992
#78572