Finite Group Databasebeta

← #110888

Subgroups — #110888

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#20467C211yesmaximal, characteristic
3#17755C211yesmaximal, characteristic
4#17404C211yesmaximal, characteristic
5#8125C211yesmaximal, characteristic
6#7997C211yesmaximal, characteristic
7#7948C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—21no—
16#1994—21no—
17#1986—22no—
18#1826C414yes—
19#1768—22no—
20#1750—22no—
21#1745—21no—
22C4×#97—21no—
23#1556C412yescomplemented
24#1556C412yescomplemented
25#1552—22no—
26#1552—22no—
27#1472—22no—
28#1470C414yescomplemented
29#1467—22no—
30#1308—22no—
31#1291—22no—
32#1285—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125C2×C412yes—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117C2×C412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432C2×C412yes—
65#432D412yes—
66#432D412yes—
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C412yes—
71#423—21no—
72#423—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76#407—21no—
77#407C2×C412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8D412yes—
81C4×M16—21no—
82C4×M16—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—21no—
91C4×C22⋊C4C2×C412yes—
92#376—21no—
93C43D411yescharacteristic
94C43D411yescharacteristic
95#339—24no—
96#339—24no—
97#336—24no—
98#332—24no—
99#332—24no—
100#324—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119#125—22no—
120#125—22no—
121#125—22no—
122#125—22no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C4212yes—
150C2×C4⋊C4C4212yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8C22⋊C414yes—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8C22⋊C414yes—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—44no—
198#97—41no—
199#97—41no—
200#97—41no—
201C4×C8—22no—
202C4×C8—44no—
203C4×C8—41no—
204#95—22no—
205#95—22no—
206C24C4×D411yescharacteristic
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4C2×C4211yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4#11711yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238M16—44no—
239M16—82no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—24no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—81no—
273C42—81no—
274C42C2×C22⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12411yescharacteristic
288C42#12611yescharacteristic
289C42#12711yescharacteristic
290C23—22no—
291C23—41no—
292C23—41no—
293C23C4×C22⋊C411yescharacteristic
294C23#38611yescharacteristic
295C23#38811yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#39011yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39011yescharacteristic
340C8—44no—
341C8—44no—
342C8—82no—
343C8—82no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—82no—
354C22#143711yescharacteristic
355C22#191512yes—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4#190612yes—
372C2—22no—
373C2—41no—
374C2—41no—
375C2#1094011yescharacteristic
376C2#1094511yescharacteristic
377C2#1095011yescharacteristic
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
23. #1556C4semidirect
24. #1556C4semidirect
28. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#204672
#177552
#174042
#81252
#79972
#79482