Finite Group Databasebeta

← #108331

Subgroups — #108331

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2C2×#1559C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#7997C211yesmaximal, characteristic
5#7960C211yesmaximal, characteristic
6#7934C211yesmaximal, characteristic
7#7926C211yesmaximal, 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×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2283C2211yescharacteristic
23#1559—22no—
24#1559—22no—
25C2×#333C412yescomplemented
26C2×#332C412yescomplemented
27#1515—22no—
28#1511C414yes—
29#1503—22no—
30#1291—22no—
31#1286—22no—
32#1283—22no—
33#1282—22no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533D412yes—
40#533D412yes—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
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—21no—
61#432—21no—
62#432—24no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#431—21no—
70#431—21no—
71#431—24no—
72#431—21no—
73#431—21no—
74#431C2×C411yescharacteristic
75#431C2×C411yescharacteristic
76#431C2×C412yes—
77#431D412yes—
78#431D412yes—
79#424—24no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89C2×C4⋊C8C2×C412yes—
90#407—21no—
91#407C2×C412yes—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D412yes—
94#405—24no—
95C4×M16—21no—
96C4×M16D412yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#383—21no—
100#382—21no—
101#381C2×C412yes—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C22⋊C4—21no—
105#376—21no—
106#333—24no—
107#332—24no—
108#328—28no—
109C22×Q8—21no—
110C22×Q8C4212yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
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⋊Q8C22⋊C414yes—
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—41no—
160C2×C4⋊C4C4212yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—28no—
178C4⋊C8—28no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
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—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8C22⋊C412yes—
200C22⋊C8C22⋊C412yes—
201#97—28no—
202#97—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206C4×C8—28no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—22no—
210C4×C8—41no—
211C4×C8C22⋊C414yes—
212#95—22no—
213#95—22no—
214#95—41no—
215#95—41no—
216C24C4×D411yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×Q8#9514yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249M16—44no—
250M16—44no—
251M16—82no—
252M16—82no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4#9514yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—44no—
279C42—24no—
280C42—24no—
281C42—82no—
282C42C2×C22⋊C411yescharacteristic
283C42C2×C22⋊C411yescharacteristic
284C42C2×C22⋊C411yescharacteristic
285C42C2×C4⋊C411yescharacteristic
286C42C2×C4⋊C411yescharacteristic
287C42#11711yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42#12011yescharacteristic
291C42#12011yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C23—22no—
299C23—41no—
300C23—41no—
301C23C4×C22⋊C411yescharacteristic
302C23#38511yescharacteristic
303C23#38611yescharacteristic
304Q8—44no—
305Q8—44no—
306Q8—42no—
307Q8—42no—
308Q8—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
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—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4C2×#9511yescharacteristic
338C2×C4C4×C22⋊C411yescharacteristic
339C2×C4#37911yescharacteristic
340C2×C4#38011yescharacteristic
341C2×C4#38011yescharacteristic
342C2×C4#38111yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38511yescharacteristic
345C2×C4#38511yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#38611yescharacteristic
349C8—42no—
350C8—42no—
351C8—44no—
352C8—82no—
353C8—82no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—82no—
364C22C22⋊#9511yescharacteristic
365C22#175712yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—44no—
378C4—162no—
379C4#190211yescharacteristic
380C4#190311yescharacteristic
381C4#190512yes—
382C2—22no—
383C2—41no—
384C2—41no—
385C2#1076411yescharacteristic
386C2#1078411yescharacteristic
387C2#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
12. #2983C4semidirect
13. #2982C4semidirect
25. C2×#333C4semidirect
26. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#602502
C2×#15592
#175622
#79972
#79602
#79342
#79262