Finite Group Databasebeta

← #107245

Subgroups — #107245

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21332C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#8094C211yesmaximal, characteristic
5#8071C211yesmaximal, characteristic
6#7900C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2985—22no—
11#2984—22no—
12#2983C414yescomplemented
13#2962—22no—
14#2961—22no—
15#2924—22no—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2285C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1996—22no—
23#1996C414yes—
24#1740—22no—
25#1472—22no—
26#1470C414yescomplemented
27#1461—22no—
28#1303—22no—
29#1300—22no—
30#1279—22no—
31#1276—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8C2×C412yes—
44C2×C22⋊Q8D412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D412yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117D412yes—
50C22×C42C2311yescharacteristic
51#446—28no—
52#446—24no—
53#446—28no—
54#446—28no—
55C8×Q8—24no—
56#433—28no—
57#432—24no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432C2×C412yes—
65#432D412yes—
66#431—24no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C412yes—
74#431D412yes—
75#424—24no—
76#423—21no—
77#423D412yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80#407—21no—
81#407—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84#405—24no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89#384—21no—
90#382—21no—
91#380—21no—
92#380C2×C412yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4C2×C412yes—
99#376—21no—
100C43—21no—
101#342—28no—
102#324—28no—
103C22×Q8C4⋊C412yes—
104C22×Q8C4⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—22no—
109C2×M16—41no—
110C22×C8—22no—
111C22×C8—41no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—24no—
145#117—42no—
146#117—44no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C4214yes—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C4⋊C412yes—
154C2×C4⋊C4C4⋊C412yes—
155C2×C4⋊C4C4⋊C412yes—
156C2×C4⋊C4C4⋊C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C22⋊C414yes—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C22⋊C414yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C4⋊C8—28no—
178C4⋊C8—28no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8C22⋊C414yes—
186C4⋊C8C22⋊C414yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—28no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200C4×C8—28no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—41no—
204C4×C8—41no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209C24C2×C22⋊C411yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4#9514yes—
223C22×C4#9514yes—
224C22×C4C2×C4211yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242M16—42no—
243M16—42no—
244M16—82no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—82no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—42no—
272C22⋊C4—44no—
273C42—24no—
274C42—24no—
275C42—44no—
276C42—44no—
277C42—82no—
278C42—82no—
279C42C2×C22⋊C411yescharacteristic
280C42C2×C22⋊C411yescharacteristic
281C42#11711yescharacteristic
282C42#11711yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12011yescharacteristic
288C42#12011yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42#12311yescharacteristic
294C42#12311yescharacteristic
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23C2×#9511yescharacteristic
300C23#38611yescharacteristic
301C23#38611yescharacteristic
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—24no—
316C2×C4—24no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4C4×C22⋊C411yescharacteristic
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4#37911yescharacteristic
337C2×C4#38011yescharacteristic
338C2×C4#38011yescharacteristic
339C2×C4#38111yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#45412yes—
347C2×C4#45612yes—
348C8—44no—
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—
362C22C22⋊#9511yescharacteristic
363C22#188611yescharacteristic
364C22#188811yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—44no—
375C4—44no—
376C4—84no—
377C4—162no—
378C4#179312yes—
379C4#185712yes—
380C4#186812yes—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2#1074811yescharacteristic
386C2#1075011yescharacteristic
387C2#1076611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C22⋊C4
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
12. #2983C4semidirect
26. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#213322
#173852
#80942
#80712
#79002
#78702