Finite Group Databasebeta

← #110149

Subgroups — #110149

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17747C211yesmaximal, characteristic
2#17584C211yesmaximal, characteristic
3#17404C211yesmaximal, characteristic
4#17338C211yesmaximal, characteristic
5#8116C211yesmaximal, characteristic
6#7997C211yesmaximal, characteristic
7#7951C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14#2283C2211yescharacteristic
15#1559—22no—
16#1554—22no—
17#1549C412yescomplemented
18#1548C412yescomplemented
19#1518—22no—
20#1515—22no—
21#1508C414yes—
22#1472—22no—
23#1470—22no—
24#1467C414yescomplemented
25#1453—22no—
26#1453—22no—
27#1453—22no—
28#1306—22no—
29#1291—22no—
30#1285—22no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×#117—21no—
47C22×C42C2311yescharacteristic
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#432C2×C412yes—
58#432D412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431D412yes—
66#431D412yes—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72C2×C4⋊C8—21no—
73#407—21no—
74#407—21no—
75#407C2×C412yes—
76#407D412yes—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C2×#97—21no—
83#383—21no—
84#382—21no—
85#381—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4C2×C412yes—
89#376—21no—
90#333—24no—
91#332—24no—
92#328—28no—
93#324—28no—
94#322—26no—
95C82—22no—
96C22×Q8—21no—
97C22×Q8C4212yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C22×C8—41no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114#125—41no—
115#125—41no—
116#125—41no—
117#125—41no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C4212yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8C22⋊C414yes—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8C22⋊C414yes—
180C22⋊C8C22⋊C414yes—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—41no—
186#97—41no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191#95—22no—
192#95—41no—
193#95—41no—
194#95—41no—
195C24C4×D411yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4C2×C4211yescharacteristic
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4#11711yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12711yescharacteristic
226M16—82no—
227M16—82no—
228M16—82no—
229M16—82no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C42—42no—
253C42—42no—
254C42—82no—
255C42C2×C22⋊C411yescharacteristic
256C42#11711yescharacteristic
257C42C4×D411yescharacteristic
258C42C4×D411yescharacteristic
259C42C4×D411yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×D411yescharacteristic
263C42#12011yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42#12311yescharacteristic
268C42#12311yescharacteristic
269C42#12411yescharacteristic
270C42#12611yescharacteristic
271C23—41no—
272C23—41no—
273C23—41no—
274C23C4×C22⋊C411yescharacteristic
275C23#38811yescharacteristic
276C23#39011yescharacteristic
277Q8—82no—
278Q8—82no—
279Q8—82no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4C4×C22⋊C411yescharacteristic
307C2×C4C4×C22⋊C411yescharacteristic
308C2×C4#38611yescharacteristic
309C2×C4#38611yescharacteristic
310C2×C4#38611yescharacteristic
311C2×C4#38811yescharacteristic
312C2×C4#38811yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38811yescharacteristic
316C2×C4#39011yescharacteristic
317C2×C4#39011yescharacteristic
318C8—44no—
319C8—44no—
320C8—44no—
321C8—82no—
322C8—82no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—82no—
333C22#143711yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—82no—
346C4—82no—
347C4—162no—
348C4#190212yes—
349C2—41no—
350C2—41no—
351C2—41no—
352C2#1094111yescharacteristic
353C2#1095111yescharacteristic
354C2#1095211yescharacteristic
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. #1549C4semidirect
18. #1548C4semidirect
24. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#177472
#175842
#174042
#173382
#81162
#79972
#79512