Finite Group Databasebeta

← #107347

Subgroups — #107347

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34389C211yesmaximal, characteristic
2#17562C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#8096C211yesmaximal, characteristic
5#8025C211yesmaximal, characteristic
6#7975C211yesmaximal, characteristic
7#7945C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2381C412yescomplemented
13#2381C412yescomplemented
14#2368—22no—
15#2356—22no—
16#2343—22no—
17#2336—22no—
18#2329—22no—
19#2320—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C2×#383C2211yescharacteristic
23#1515—22no—
24#1511—22no—
25#1503C414yescomplemented
26#1497C414yescomplemented
27#1493—22no—
28#1479—22no—
29#1303—22no—
30#1294—22no—
31#1288—22no—
32#1284—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533C2×C412yes—
39#533D412yes—
40#533Q812yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8D412yes—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432C2×C412yes—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#431C2×C412yes—
62#423—21no—
63#423—21no—
64C2×C4⋊C8—21no—
65#407—21no—
66#407—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C2×C4×C8—21no—
71#391—24no—
72#391—24no—
73#389—24no—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#385—24no—
81#383—22no—
82#383—21no—
83#383—21no—
84#383—21no—
85#383C2×C412yes—
86#383C2×C412yes—
87#383D412yes—
88#382—22no—
89#382—21no—
90#382—21no—
91#381—21no—
92#381D412yes—
93#380—21no—
94#380—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4D412yes—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102#376—22no—
103#376—21no—
104#376—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95D412yes—
108#328—28no—
109Q8⋊C8—28no—
110C22×Q8—21no—
111C22×Q8C22⋊C412yes—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—41no—
117C2×M16—41no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C4212yes—
124C4⋊Q8C4212yes—
125C4⋊Q8C4⋊C412yes—
126C4⋊Q8C4⋊C412yes—
127#125—22no—
128#125—22no—
129#125—22no—
130#125—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8C22⋊C414yes—
138C22⋊Q8C4⋊C414yes—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8C22⋊C414yes—
146C4×Q8C4⋊C414yes—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
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—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193#97—22no—
194#97—41no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—41no—
198C4×C8—41no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95C22⋊C414yes—
216#95C22⋊C414yes—
217C24#12011yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—44no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8#9514yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254M16—82no—
255M16—82no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—82no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C4⋊C4—82no—
275C4⋊C4#9514yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—44no—
279C22⋊C4—82no—
280C42—24no—
281C42—24no—
282C42—82no—
283C42—82no—
284C42—84no—
285C42—84no—
286C42C2×C4211yescharacteristic
287C42C2×C22⋊C411yescharacteristic
288C42C2×C4⋊C411yescharacteristic
289C42C2×C4⋊C411yescharacteristic
290C42#11711yescharacteristic
291C42#11711yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C23—22no—
303C23—22no—
304C23—41no—
305C23#37911yescharacteristic
306C23#38511yescharacteristic
307C23#38611yescharacteristic
308Q8—44no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—41no—
319C2×C4—41no—
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—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—164no—
345C2×C4C2×#9511yescharacteristic
346C2×C4C4×C22⋊C411yescharacteristic
347C2×C4C4×C22⋊C411yescharacteristic
348C2×C4#38011yescharacteristic
349C2×C4#38011yescharacteristic
350C2×C4#38111yescharacteristic
351C2×C4#38511yescharacteristic
352C2×C4#38511yescharacteristic
353C2×C4#38511yescharacteristic
354C2×C4#38611yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38611yescharacteristic
357C8—82no—
358C8—82no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22C22⋊#9511yescharacteristic
369C22#179412yes—
370C22#190512yes—
371C4—21no—
372C4—21no—
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—84no—
386C4—84no—
387C4—162no—
388C2—22no—
389C2—22no—
390C2—41no—
391C2#1075211yescharacteristic
392C2#1078311yescharacteristic
393C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ 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. #2381C4semidirect
13. #2381C4semidirect
25. #1503C4semidirect
26. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#343892
#175622
#174792
#80962
#80252
#79752
#79452