Finite Group Databasebeta

← #110480

Subgroups — #110480

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21829C211yesmaximal, characteristic
2#21202C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#8109C211yesmaximal, characteristic
6#7953C211yesmaximal, characteristic
7#7862C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#419C414yes—
14C2×C4×M16C2211yescharacteristic
15#2285C2211yescharacteristic
16#1997—22no—
17#1997C414yescomplemented
18#1960—22no—
19#1956—21no—
20#1956—21no—
21#1821—22no—
22#1558—22no—
23#1548—22no—
24#1548C414yescomplemented
25#1455—22no—
26#1453—22no—
27#1452—21no—
28#1450—21no—
29#1305—22no—
30#1285—22no—
31#1275—22no—
32C23×Q8C2×C412yes—
33C22×M16C2×C412yes—
34C2×C4⋊Q8—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⋊Q8—21no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#117C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432D412yes—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423C2×C412yes—
70#419—48no—
71C2×#103—24no—
72C2×#103—24no—
73#407—21no—
74#407—21no—
75#407—21no—
76#407D412yes—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16C2×C412yes—
84C4×M16D412yes—
85C2×#97D412yes—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88#384—21no—
89#382—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94#343—28no—
95#333—28no—
96#322—22no—
97#322—24no—
98C82—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—44no—
105C22×Q8—44no—
106C22×Q8C4214yes—
107C23×C4—22no—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C2×M16—22no—
116C2×M16—42no—
117C2×M16—41no—
118C2×M16C22⋊C414yes—
119C22×C8C22⋊C412yes—
120C22×C8C22⋊C412yes—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137#117—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—22no—
154C2×C42—22no—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8C22⋊C414yes—
175#103—48no—
176#103—48no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199#95—41no—
200#95—41no—
201C24C4×D411yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—48no—
205C2×Q8—48no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—82no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4C2×C4211yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12711yescharacteristic
242M16—44no—
243M16—44no—
244M16—44no—
245M16—44no—
246M16—82no—
247M16—82no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8#10014yes—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—82no—
270C42—82no—
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42#12011yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12411yescharacteristic
286C42#12611yescharacteristic
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23C4×C22⋊C411yescharacteristic
292C23#38811yescharacteristic
293C23#39011yescharacteristic
294Q8—48no—
295Q8—84no—
296Q8—84no—
297Q8—84no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C4×C22⋊C411yescharacteristic
325C2×C4C4×C22⋊C411yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38611yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#39011yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4C2×#10012yes—
337C2×C4#41312yes—
338C8—24no—
339C8—42no—
340C8—44no—
341C8—44no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—42no—
353C22—42no—
354C22#143711yescharacteristic
355C22#196511yescharacteristic
356C22#196711yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—84no—
366C4—162no—
367C4C4×#10012yes—
368C4#190012yes—
369C4#190212yes—
370C4#190912yes—
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2#1094711yescharacteristic
376C2#1095211yescharacteristic
377C2#1096211yescharacteristic
subgroupG/H
center, socleC22#1437
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
17. #1997C4semidirect
24. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#218292
#212022
#177462
#173342
#81092
#79532
#78622