Finite Group Databasebeta

← #107222

Subgroups — #107222

Order 512.

857 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34371C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8074C211yesmaximal, characteristic
6#7973C211yesmaximal, characteristic
7#7960C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2381—22no—
14#2365—22no—
15#2345—22no—
16Q8×C22⋊C4—22no—
17#2331—22no—
18#2327—22no—
19#2313C414yes—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#1997—22no—
23#1997C414yescomplemented
24#1966—22no—
25#1966C414yescomplemented
26#1784—22no—
27C2×#343—22no—
28#1303—22no—
29#1300—22no—
30#1288—22no—
31#1286—22no—
32C23×Q8D412yes—
33C22×M16—21no—
34C22×M16—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8D412yes—
46C2×C4⋊Q8Q812yes—
47C2×C4×Q8—24no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4D412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×#101—28no—
68#407—21no—
69#407—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16C2×C412yes—
75C4×M16C2×C412yes—
76C2×#97—21no—
77C2×C4×C8—21no—
78#391—24no—
79#391—28no—
80#389—24no—
81#387—28no—
82#387—28no—
83#387—28no—
84#383—21no—
85#383D412yes—
86#382—21no—
87#382D412yes—
88#381C2×C412yes—
89#381C2×C412yes—
90#380—24no—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99#376—21no—
100#376D412yes—
101#343—28no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—84no—
107C22×Q8C22⋊C414yes—
108C22×Q8C4⋊C414yes—
109C23×C4C22⋊C414yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—24no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—42no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—44no—
122C22×C8—41no—
123C22×C8—41no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4×Q8—88no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4C4214yes—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C4⋊C414yes—
155C2×C4⋊C4C4⋊C414yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—28no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181#101—88no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186#97—22no—
187#97—22no—
188#97—41no—
189#97—41no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194#95—28no—
195#95—28no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95C22⋊C414yes—
203#95C22⋊C414yes—
204C24C2×C22⋊C411yescharacteristic
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—84no—
224C22×C4#9518yes—
225C22×C4C2×C4211yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243M16—24no—
244M16—44no—
245M16—42no—
246M16—44no—
247M16—44no—
248M16—44no—
249M16—82no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—88no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—44no—
271C42—42no—
272C42—42no—
273C42—88no—
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C23—22no—
291C23—22no—
292C23—41no—
293C23C2×#9511yescharacteristic
294C23#37911yescharacteristic
295C23#38111yescharacteristic
296Q8—48no—
297Q8—48no—
298Q8—84no—
299Q8—84no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—28no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—84no—
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4#38011yescharacteristic
332C2×C4#38011yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38511yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#45412yes—
342C2×C4#45612yes—
343C8—44no—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22C22⋊#9511yescharacteristic
356C22#179312yes—
357C22#183012yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—48no—
368C4—84no—
369C4—84no—
370C4#185712yes—
371C4#188611yescharacteristic
372C4#188811yescharacteristic
373C4#191112yes—
374C2—22no—
375C2—22no—
376C2—41no—
377C2#1074011yescharacteristic
378C2#1075011yescharacteristic
379C2#1077011yescharacteristic
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
23. #1997C4semidirect
25. #1966C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#217992
C2×#19972
#80932
#80742
#79732
#79602