Finite Group Databasebeta

← #110913

Subgroups — #110913

Order 512.

564 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
1C2×#1554C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#17327C211yesmaximal, characteristic
5#8096C211yesmaximal, characteristic
6#8073C211yesmaximal, characteristic
7#8010C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1554—22no—
16#1554—21no—
17#1554—21no—
18C2×#333C412yescomplemented
19C2×#332C412yescomplemented
20#1531C414yescomplemented
21#1528—21no—
22#1528—21no—
23#1526—22no—
24#1518—22no—
25#1515C414yes—
26#1508—22no—
27#1453—22no—
28#1451—21no—
29#1451—21no—
30C8×M16—21no—
31C8×M16—21no—
32#1303—22no—
33#1300—22no—
34#1293—22no—
35C22×M16—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432C2×C412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#431D411yescharacteristic
68#431D411yescharacteristic
69#431D412yes—
70#423—21no—
71#423—21no—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8C2×C412yes—
74#407—21no—
75#407D412yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16D412yes—
83C4×M16D411yescharacteristic
84C4×M16D411yescharacteristic
85C2×#97—21no—
86C2×#97C2×C412yes—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#383C2×C412yes—
90#382—21no—
91#381—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C22⋊C4—21no—
95#376—21no—
96#333—22no—
97#333—22no—
98#332—22no—
99#332—22no—
100#330—24no—
101#330—24no—
102#328—28no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—21no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—21no—
116C2×M16—44no—
117C2×M16—41no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8C22⋊C412yes—
184C4⋊C8C22⋊C412yes—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8C22⋊C414yes—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—41no—
198#97—41no—
199#97C22⋊C412yes—
200#97C22⋊C412yes—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—41no—
213C24#12411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C2×Q8—82no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
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×C4C4×D411yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12711yescharacteristic
244M16—24no—
245M16—44no—
246M16—44no—
247M16—42no—
248M16—82no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—82no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—82no—
275C42—42no—
276C42—42no—
277C42—82no—
278C42C2×C4211yescharacteristic
279C42C2×C22⋊C411yescharacteristic
280C42C2×C22⋊C411yescharacteristic
281C42#11711yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42#12011yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12311yescharacteristic
292C42#12411yescharacteristic
293C42#12411yescharacteristic
294C23—22no—
295C23—41no—
296C23—41no—
297C23#38811yescharacteristic
298C23#38811yescharacteristic
299C23#39011yescharacteristic
300Q8—82no—
301Q8—82no—
302Q8—82no—
303Q8—82no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4C4×C22⋊C411yescharacteristic
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4#39011yescharacteristic
341C8—24no—
342C8—42no—
343C8—44no—
344C8—42no—
345C8—42no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—82no—
357C22#143711yescharacteristic
358C22#193212yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—82no—
371C4—162no—
372C4#190212yes—
373C4#197612yes—
374C2—22no—
375C2—41no—
376C2—41no—
377C2#1094411yescharacteristic
378C2#1094611yescharacteristic
379C2#1095211yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
18. C2×#333C4semidirect
19. C2×#332C4semidirect
20. #1531C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15542
#176482
#175842
#173272
#80962
#80732
#80102