Finite Group Databasebeta

← #110713

Subgroups — #110713

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17715C211yesmaximal, characteristic
2#17575C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17326C211yesmaximal, characteristic
5#8094C211yesmaximal, characteristic
6#8054C211yesmaximal, characteristic
7#8038C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2285C2211yescharacteristic
15#1552—22no—
16#1549C412yescomplemented
17#1546C412yescomplemented
18#1544—21no—
19#1544—21no—
20#1520—22no—
21#1518C414yes—
22#1503—21no—
23#1503—21no—
24#1497C414yescomplemented
25#1493—22no—
26#1479—22no—
27#1454—22no—
28#1452—21no—
29#1451—21no—
30#1450—21no—
31C8×M16—21no—
32#1303—22no—
33#1298—22no—
34#1296—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
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#431—21no—
67#431C2×C412yes—
68#431C2×C412yes—
69#431C2×C412yes—
70#431D411yescharacteristic
71#431D411yescharacteristic
72#431D412yes—
73#431D411yescharacteristic
74#431D411yescharacteristic
75#423C2×C412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80#407—21no—
81#407—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8D412yes—
85C4×M16—21no—
86C4×M16—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#384—21no—
91#382C2×C412yes—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#332—22no—
97#332—22no—
98#332—22no—
99#332—22no—
100#328—24no—
101#328—24no—
102Q8⋊C8—28no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C4212yes—
124C4⋊Q8C4212yes—
125#125—22no—
126#125—22no—
127#125—41no—
128#125—41no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145#117—22no—
146#117—22no—
147#117—41no—
148#117—41no—
149#117—82no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—41no—
184C4⋊C8C22⋊C414yes—
185C4⋊C8C22⋊C412yes—
186C4⋊C8C22⋊C412yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C412yes—
198C22⋊C8C22⋊C412yes—
199#97—22no—
200#97—41no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—41no—
210C4×C8—41no—
211#95—22no—
212#95—22no—
213C24#12411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—44no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4#11711yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12611yescharacteristic
244C22×C4#12711yescharacteristic
245M16—42no—
246M16—42no—
247M16—82no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—44no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—82no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—82no—
276C42—42no—
277C42—42no—
278C42—82no—
279C42C2×C4211yescharacteristic
280C42C2×C22⋊C411yescharacteristic
281C42C2×C22⋊C411yescharacteristic
282C42#11711yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42#12011yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12311yescharacteristic
293C42#12411yescharacteristic
294C42#12411yescharacteristic
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23#38811yescharacteristic
300C23#38811yescharacteristic
301C23#39011yescharacteristic
302Q8—44no—
303Q8—82no—
304Q8—82no—
305Q8—82no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4C4×C22⋊C411yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#39011yescharacteristic
342C2×C4#39011yescharacteristic
343C2×C4#47112yes—
344C2×C4#47312yes—
345C8—42no—
346C8—42no—
347C8—42no—
348C8—42no—
349C8—44no—
350C8—82no—
351C8—82no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22#143711yescharacteristic
363C22#197711yescharacteristic
364C22#197911yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4—162no—
376C4#188112yes—
377C4#190812yes—
378C4#193012yes—
379C4#197612yes—
380C2—21no—
381C2—21no—
382C2—41no—
383C2—41no—
384C2#1094211yescharacteristic
385C2#1094411yescharacteristic
386C2#1095611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#124
LCSG ▷ C42 ▷ 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
16. #1549C4semidirect
17. #1546C4semidirect
24. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#177152
#175752
#174792
#173262
#80942
#80542
#80382