Finite Group Databasebeta

← #110678

Subgroups — #110678

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21311C211yesmaximal, characteristic
2#20544C211yesmaximal, characteristic
3#17562C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8071C211yesmaximal, characteristic
6#8002C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1989—22no—
16#1988C412yescomplemented
17#1987C412yescomplemented
18#1771—22no—
19#1768—22no—
20#1752—21no—
21C8×C22⋊C4—21no—
22#1735—21no—
23#1735—21no—
24#1515C414yescomplemented
25#1511—22no—
26#1503—22no—
27#1497C414yescomplemented
28#1491—22no—
29#1481—22no—
30#1300—22no—
31#1292—22no—
32#1284—22no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
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#432—21no—
57#432—21no—
58#432C2×C412yes—
59#432C2×C412yes—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#423—21no—
66#423—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8D412yes—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C2×#97—21no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#383C2×C412yes—
83#382—21no—
84#381D412yes—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92#376—21no—
93#376—21no—
94C43—21no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#336—22no—
100#336—22no—
101#328—28no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8C22⋊C412yes—
118C22×C8C22⋊C412yes—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4212yes—
126C4⋊Q8C4212yes—
127#125—22no—
128#125—22no—
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—24no—
148#117—22no—
149#117—22no—
150#117—42no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8C22⋊C414yes—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—41no—
201#97—82no—
202C4×C8—22no—
203C4×C8—41no—
204C4×C8—82no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95C22⋊C414yes—
211C24#12111yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—81no—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243C22×C4#12711yescharacteristic
244M16—82no—
245M16—82no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—81no—
258C2×C8—81no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—44no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—82no—
284C42—82no—
285C42C2×C4211yescharacteristic
286C42C2×C22⋊C411yescharacteristic
287C42#11711yescharacteristic
288C42#11711yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42#12111yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42#12311yescharacteristic
299C42#12411yescharacteristic
300C42#12611yescharacteristic
301C23—22no—
302C23—22no—
303C23—41no—
304C23#38611yescharacteristic
305C23#38811yescharacteristic
306C23#39011yescharacteristic
307Q8—44no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4C4×C22⋊C411yescharacteristic
341C2×C4C4×C22⋊C411yescharacteristic
342C2×C4C4×C22⋊C411yescharacteristic
343C2×C4#38611yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#39011yescharacteristic
351C2×C4#39011yescharacteristic
352C2×C4#45512yes—
353C2×C4#45612yes—
354C8—44no—
355C8—42no—
356C8—42no—
357C8—82no—
358C8—82no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—44no—
367C22#143711yescharacteristic
368C22#188212yes—
369C22#197612yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—84no—
381C4—84no—
382C4—82no—
383C4—82no—
384C4#188212yes—
385C4#188511yescharacteristic
386C4#188611yescharacteristic
387C2—22no—
388C2—22no—
389C2—41no—
390C2#1094411yescharacteristic
391C2#1095011yescharacteristic
392C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#121
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. #1988C4semidirect
17. #1987C4semidirect
24. #1515C4semidirect
27. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#213112
#205442
#175622
#174962
#80712
#80022
#79412