Finite Group Databasebeta

← #116167

Subgroups — #116167

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21889C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic
3#17751C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#8002C211yesmaximal, characteristic
7#7960C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1988—21no—
16#1981—21no—
17#1892—22no—
18#1832—22no—
19#1831—22no—
20#1805—22no—
21#1729—22no—
22#1561—22no—
23#1552—22no—
24#1549—21no—
25#1546—21no—
26#1508—22no—
27#1506—22no—
28#1503C414yescomplemented
29#1297—22no—
30#1292—22no—
31#1286—22no—
32C22×M16—21no—
33C23×C8D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423D412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74C2×#103—24no—
75#407—21no—
76#407—21no—
77C4×M16—21no—
78C4×M16—21no—
79#383—21no—
80#382D412yes—
81#381—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2×C412yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4D412yes—
89#376—21no—
90#376D412yes—
91C43—21no—
92#340—24no—
93#336—28no—
94#332—24no—
95#332—24no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—44no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—44no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4⋊C412yes—
119C4⋊Q8C4⋊C412yes—
120#125—22no—
121#125—22no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—22no—
141#117—22no—
142#117—42no—
143#117—42no—
144#117—41no—
145#117—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42—41no—
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
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186#103—84no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#97—41no—
192#97—41no—
193C4×C8—22no—
194C4×C8—41no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95—41no—
200#95C22⋊C414yes—
201C24#12111yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12611yescharacteristic
235M16—82no—
236M16—82no—
237M16—82no—
238M16—82no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—82no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—82no—
266C42—44no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—82no—
272C42—82no—
273C42C2×C4⋊C411yescharacteristic
274C42#11711yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42#12011yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12411yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C42#12711yescharacteristic
289C23—22no—
290C23—22no—
291C23—41no—
292C23#38611yescharacteristic
293C23#39411yescharacteristic
294C23#39611yescharacteristic
295Q8—44no—
296Q8—82no—
297Q8—82no—
298Q8—81no—
299Q8—81no—
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—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#38011yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#39011yescharacteristic
335C2×C4#39211yescharacteristic
336C2×C4#39311yescharacteristic
337C2×C4#39411yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39711yescharacteristic
340C2×C4#39911yescharacteristic
341C2×C4#45412yes—
342C2×C4#45512yes—
343C8—82no—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22#143911yescharacteristic
356C22#190212yes—
357C22#204012yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—84no—
368C4—82no—
369C4—82no—
370C4—162no—
371C4#186012yes—
372C4#200311yescharacteristic
373C4#200411yescharacteristic
374C4#205312yes—
375C2—22no—
376C2—22no—
377C2—41no—
378C2#1105911yescharacteristic
379C2#1107911yescharacteristic
380C2#1109211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×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
28. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#218892
#207552
#177512
#175452
#80522
#80022
#79602