Finite Group Databasebeta

← #117389

Subgroups — #117389

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2#17808C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8094C211yesmaximal, characteristic
7#8066C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2285C2211yescharacteristic
15#1990—22no—
16#1987—22no—
17#1824—22no—
18#1574—22no—
19#1572—22no—
20#1565—22no—
21#1558C414yescomplemented
22#1556—22no—
23#1549—22no—
24#1548—22no—
25#1548—22no—
26#1546—22no—
27#1305—22no—
28#1303—22no—
29#1299—22no—
30C23×Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
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⋊Q8C2×C412yes—
41C2×C4⋊Q8D412yes—
42C2×C4⋊Q8Q812yes—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432C2×C412yes—
54#432C2×C412yes—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#423—21no—
68#423D412yes—
69#423D412yes—
70#423D412yes—
71C2×C4⋊C8D412yes—
72#407—21no—
73#407—21no—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C2×C4×C8—21no—
82#384—21no—
83#382—21no—
84#380—21no—
85#380D412yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#340—28no—
89#335—24no—
90#334—24no—
91#333—28no—
92#332—28no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—84no—
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C22×C8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8C4⋊C414yes—
125#117—82no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C22⋊C4—22no—
142C2×C42—41no—
143C2×C42C22×C411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×Q811yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8C22⋊C414yes—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175#97—22no—
176#97—22no—
177#97—41no—
178#97—41no—
179C4×C8—22no—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—41no—
183C4×C8—41no—
184C4×C8—41no—
185#95—22no—
186#95—41no—
187C24#12311yescharacteristic
188C2×Q8—24no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×Q8—84no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—84no—
207C22×C4#11711yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12411yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12711yescharacteristic
225M16—42no—
226M16—42no—
227M16—82no—
228M16—82no—
229M16—82no—
230C2×C8—24no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—82no—
248C42—82no—
249C42C2×C22⋊C411yescharacteristic
250C42C2×C4⋊C411yescharacteristic
251C42C4×D411yescharacteristic
252C42C4×D411yescharacteristic
253C42#12011yescharacteristic
254C42#12011yescharacteristic
255C42#12111yescharacteristic
256C42#12111yescharacteristic
257C42#12111yescharacteristic
258C42#12111yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42#12311yescharacteristic
263C42#12411yescharacteristic
264C42#12611yescharacteristic
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23#38811yescharacteristic
270C23#39411yescharacteristic
271C23#39911yescharacteristic
272Q8—84no—
273Q8—84no—
274Q8—84no—
275Q8—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#38011yescharacteristic
302C2×C4#38511yescharacteristic
303C2×C4#38511yescharacteristic
304C2×C4#38611yescharacteristic
305C2×C4#38811yescharacteristic
306C2×C4#39011yescharacteristic
307C2×C4#39211yescharacteristic
308C2×C4#39311yescharacteristic
309C2×C4#39411yescharacteristic
310C2×C4#39611yescharacteristic
311C2×C4#39611yescharacteristic
312C2×C4#39711yescharacteristic
313C2×C4#47112yes—
314C2×C4#47312yes—
315C8—44no—
316C8—82no—
317C8—82no—
318C8—82no—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#143911yescharacteristic
331C22#204611yescharacteristic
332C22#204711yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—84no—
342C4—162no—
343C4#190812yes—
344C4#200912yes—
345C4#204812yes—
346C4#206412yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2#1106211yescharacteristic
352C2#1113011yescharacteristic
353C2#1114211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
21. #1558C4semidirect

Maximal subgroups

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

subgroupindex
#218342
#178082
#177462
#177422
#81052
#80942
#80662