Finite Group Databasebeta

← #117388

Subgroups — #117388

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21833C211yesmaximal, 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#1988—22no—
17#1823—22no—
18#1574—22no—
19#1572—22no—
20#1565—22no—
21#1558—22no—
22#1556C414yescomplemented
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—82no—
100C22×Q8—82no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C22×C8—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—22no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8C4⋊C414yes—
126#117—82no—
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—22no—
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×C4⋊C4—41no—
142C2×C22⋊C4—22no—
143C2×C42—41no—
144C2×C42C22×C411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—41no—
165C4⋊C8C22⋊C414yes—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176#97—22no—
177#97—22no—
178#97—41no—
179#97—41no—
180C4×C8—22no—
181C4×C8—41no—
182C4×C8—41no—
183C4×C8—41no—
184C4×C8—41no—
185C4×C8—41no—
186#95—22no—
187#95—41no—
188C24#12311yescharacteristic
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4#11711yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12711yescharacteristic
229M16—42no—
230M16—42no—
231M16—82no—
232M16—82no—
233M16—82no—
234C2×C8—24no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C22⋊C4—82no—
252C42—82no—
253C42C2×C22⋊C411yescharacteristic
254C42C2×C4⋊C411yescharacteristic
255C42C4×D411yescharacteristic
256C42C4×D411yescharacteristic
257C42#12011yescharacteristic
258C42#12011yescharacteristic
259C42#12111yescharacteristic
260C42#12111yescharacteristic
261C42#12111yescharacteristic
262C42#12111yescharacteristic
263C42C22⋊Q811yescharacteristic
264C42C22⋊Q811yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42#12311yescharacteristic
267C42#12411yescharacteristic
268C42#12611yescharacteristic
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#38811yescharacteristic
274C23#39411yescharacteristic
275C23#39911yescharacteristic
276Q8—84no—
277Q8—84no—
278Q8—84no—
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#38011yescharacteristic
307C2×C4#38511yescharacteristic
308C2×C4#38511yescharacteristic
309C2×C4#38611yescharacteristic
310C2×C4#38811yescharacteristic
311C2×C4#39011yescharacteristic
312C2×C4#39211yescharacteristic
313C2×C4#39311yescharacteristic
314C2×C4#39411yescharacteristic
315C2×C4#39611yescharacteristic
316C2×C4#39611yescharacteristic
317C2×C4#39711yescharacteristic
318C2×C4#47112yes—
319C2×C4#47212yes—
320C8—44no—
321C8—82no—
322C8—82no—
323C8—82no—
324C8—82no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22#143911yescharacteristic
336C22#204511yescharacteristic
337C22#204611yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—84no—
347C4—162no—
348C4#190712yes—
349C4#200912yes—
350C4#204912yes—
351C4#206312yes—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2#1106111yescharacteristic
357C2#1113011yescharacteristic
358C2#1114311yescharacteristic
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
22. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#218332
#178082
#177462
#177422
#81052
#80942
#80662