Finite Group Databasebeta

← #107329

Subgroups — #107329

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34403C211yesmaximal, characteristic
2#17718C211yesmaximal, characteristic
3#17496C211yesmaximal, characteristic
4#8116C211yesmaximal, characteristic
5#8096C211yesmaximal, characteristic
6#7977C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2378C414yes—
14#2370—22no—
15#2356—22no—
16#2347—21no—
17#2346—21no—
18#2332—22no—
19#2331—21no—
20#2331—21no—
21#2320—22no—
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24#1554—22no—
25#1548—21no—
26#1546—21no—
27#1542C414yescomplemented
28#1497C414yescomplemented
29#1491—22no—
30#1481—22no—
31#1306—22no—
32#1303—22no—
33#1288—22no—
34#1282—21no—
35#1282—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42#533D412yes—
43#533Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8D412yes—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#431C2×C412yes—
69#423—21no—
70#423—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81#391—24no—
82#391—24no—
83#389—22no—
84#389—22no—
85#389—24no—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#383—22no—
93#383—21no—
94#383D412yes—
95#382—22no—
96#382—21no—
97#382—21no—
98#381C2×C412yes—
99#381D412yes—
100#380—21no—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—21no—
109C4×C22⋊C4D411yescharacteristic
110C4×C22⋊C4D411yescharacteristic
111#376—22no—
112#376—21no—
113#376—21no—
114#333—24no—
115#332—24no—
116Q8⋊C8—28no—
117C22×Q8—21no—
118C22×Q8C22⋊C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—41no—
123C2×M16—41no—
124C22×C8—22no—
125C22×C8—41no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8C22⋊C412yes—
131C4⋊Q8C22⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133C4⋊Q8C4⋊C412yes—
134#125—22no—
135#125—22no—
136#125—22no—
137#125—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8C4214yes—
145C22⋊Q8C4⋊C414yes—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8C22⋊C414yes—
153C4×Q8C4⋊C414yes—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C412yes—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q811yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—41no—
208#97—41no—
209#97—41no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—22no—
226#95C22⋊C414yes—
227C24C2×C22⋊C411yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8#9514yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4C2×C4211yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260M16—82no—
261M16—82no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—82no—
267C2×C8—82no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—82no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C4⋊C4#9514yes—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—82no—
289C42—24no—
290C42—24no—
291C42—42no—
292C42—42no—
293C42—82no—
294C42—84no—
295C42—84no—
296C42C2×C22⋊C411yescharacteristic
297C42C2×C22⋊C411yescharacteristic
298C42C2×C4⋊C411yescharacteristic
299C42C2×C4⋊C411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12311yescharacteristic
311C42#12311yescharacteristic
312C23—22no—
313C23—41no—
314C23—41no—
315C23C4×C22⋊C411yescharacteristic
316C23#37911yescharacteristic
317C23#38011yescharacteristic
318Q8—44no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
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—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4C2×#9511yescharacteristic
355C2×C4C4×C22⋊C411yescharacteristic
356C2×C4#38011yescharacteristic
357C2×C4#38111yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38511yescharacteristic
360C2×C4#38511yescharacteristic
361C2×C4#38511yescharacteristic
362C2×C4#38611yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38611yescharacteristic
366C8—82no—
367C8—82no—
368C8—82no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—82no—
379C22C22⋊#9511yescharacteristic
380C22#179412yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—44no—
393C4—44no—
394C4—82no—
395C4—82no—
396C4—84no—
397C4#190512yes—
398C2—22no—
399C2—41no—
400C2—41no—
401C2#1075611yescharacteristic
402C2#1077911yescharacteristic
403C2#1078611yescharacteristic
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
27. #1542C4semidirect
28. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#344032
#177182
#174962
#81162
#80962
#79772
#79282