Finite Group Databasebeta

← #110527

Subgroups — #110527

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17575C211yesmaximal, characteristic
3#17464C211yesmaximal, characteristic
4#17325C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8084C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1997C412yescomplemented
16#1966C412yescomplemented
17#1901—22no—
18C4×#101—22no—
19#1520C414yes—
20#1518—22no—
21#1503—21no—
22#1503—21no—
23#1485—21no—
24#1485—21no—
25#1481—22no—
26#1479C414yescomplemented
27#1453—22no—
28#1452—21no—
29#1452—21no—
30#1450—21no—
31#1450—21no—
32#1305—22no—
33#1302—22no—
34#1279—22no—
35C22×M16—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#431C2×C412yes—
67#431D411yescharacteristic
68#431D411yescharacteristic
69#431D412yes—
70#431D411yescharacteristic
71#431D411yescharacteristic
72#423—21no—
73#423C2×C412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×#101—24no—
77#407—21no—
78#407D412yes—
79C2×C22⋊C8—21no—
80C2×C22⋊C8C2×C412yes—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16C2×C412yes—
86C4×M16D412yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#384—21no—
92#382—21no—
93#380—21no—
94#380C2×C412yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97#343—24no—
98#328—24no—
99#328—24no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102#322—22no—
103#322—22no—
104#322—22no—
105C82—22no—
106C22×Q8—21no—
107C22×Q8C4212yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C2×M16—22no—
112C2×M16—44no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16C22⋊C414yes—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128#125—41no—
129#125—41no—
130#125—41no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—82no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C4212yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—41no—
185C4⋊C8C22⋊C412yes—
186C4⋊C8C22⋊C412yes—
187#101—48no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8C22⋊C412yes—
197C22⋊C8C22⋊C412yes—
198#97—22no—
199#97—22no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8—41no—
212#95—22no—
213#95—41no—
214C24C4×D411yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—41no—
220C2×Q8—41no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C2×C4211yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12611yescharacteristic
245C22×C4#12711yescharacteristic
246M16—24no—
247M16—42no—
248M16—44no—
249M16—44no—
250M16—82no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—82no—
280C42—42no—
281C42—42no—
282C42—82no—
283C42C2×C22⋊C411yescharacteristic
284C42#11711yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12011yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12311yescharacteristic
296C42#12311yescharacteristic
297C42#12411yescharacteristic
298C42#12611yescharacteristic
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23C4×C22⋊C411yescharacteristic
304C23#38811yescharacteristic
305C23#39011yescharacteristic
306Q8—44no—
307Q8—82no—
308Q8—82no—
309Q8—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#39011yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#47112yes—
348C2×C4#47312yes—
349C8—42no—
350C8—42no—
351C8—44no—
352C8—44no—
353C8—42no—
354C8—42no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—81no—
365C22—81no—
366C22#143711yescharacteristic
367C22#197711yescharacteristic
368C22#197911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—82no—
379C4—162no—
380C4#175712yes—
381C4#188412yes—
382C4#190812yes—
383C4#193012yes—
384C2—21no—
385C2—21no—
386C2—41no—
387C2—41no—
388C2#1094211yescharacteristic
389C2#1095911yescharacteristic
390C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. #1997C4semidirect
16. #1966C4semidirect
26. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#214252
#175752
#174642
#173252
#81052
#80842
#78962