Finite Group Databasebeta

← #497603

Subgroups — #497603

Order 512.

865 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39616C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13302C214yesmaximal, complemented
4#13301C212yesmaximal, complemented
5#12803C212yesmaximal
6#12801C211yesmaximal, characteristic
7#12632C212yesmaximal
8C4×#355C211yesmaximal, characteristic
9C2×#557C2211yescharacteristic
10C2×D4×Q8C411yescharacteristic
11Q8×M16—22no—
12#2868C411yescharacteristic
13#2720—24no—
14#2660C412yes—
15#2568C2211yescharacteristic
16#2567C2212yes—
17#2566C412yes—
18#2542C411yescharacteristic
19#2434—22no—
20#2429—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#355C2212yes—
25C2×#355C2211yescharacteristic
26C2×#355C2211yescharacteristic
27#1998—24no—
28#1997—22no—
29#1801C2212yes—
30#1801C2214yes—
31C4×#101C2212yes—
32C4×#101C2214yes—
33#1394C2212yes—
34#1394C2214yes—
35#1394C2212yes—
36#1394C2214yes—
37C23×Q8—21no—
38#557—42no—
39#557—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8D412yes—
47C2×#125C2×C412yes—
48C2×#125D414yes—
49C2×#124C2311yescharacteristic
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8C2312yes—
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4C2×C412yes—
58C2×C4×D4C2×C411yescharacteristic
59#446—44no—
60#446—42no—
61C8×Q8—42no—
62#423—22no—
63#423—24no—
64C2×#101C2314yes—
65C2×#101C2312yes—
66C2×#101C2312yes—
67C4×M16—22no—
68C4×M16—24no—
69#400—22no—
70#399—22no—
71#398—24no—
72#396—24no—
73#395—24no—
74#395—24no—
75#391—21no—
76#391—22no—
77#391C2311yescharacteristic
78#391C2312yes—
79#390C2×C411yescharacteristic
80#390C2×C412yes—
81#389—21no—
82#389—24no—
83#388—22no—
84#388—24no—
85#387—22no—
86#385—24no—
87#385—24no—
88#384—22no—
89#384C2×C412yes—
90#384C2×C412yes—
91#384C2×C412yes—
92#382C2×C412yes—
93#382C2×C411yescharacteristic
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C22⋊C4—22no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#355—24no—
100#355—22no—
101#355Q814yescomplemented
102#343—24no—
103#343—22no—
104#343—24no—
105#343—22no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—84no—
111C22×Q8C22⋊C411yescharacteristic
112C22×Q8C22⋊C412yes—
113C22×Q8C2411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C23×C4—21no—
116C23×C4—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—24no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—44no—
127C4⋊Q8—84no—
128#125—44no—
129#125—44no—
130#125—44no—
131#124—22no—
132#124C2×Q812yes—
133#124#4112yes—
134C22⋊Q8—48no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C22⋊Q8—84no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—24no—
142C4×Q8—22no—
143C4×Q8—44no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—44no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C412yes—
179C2×C42C2×D414yes—
180C2×C42C2×D412yes—
181C4⋊C8—44no—
182C4⋊C8—42no—
183C4⋊C8—44no—
184C4⋊C8—42no—
185#101—22no—
186#101—24no—
187#101—22no—
188#101—24no—
189#101—22no—
190#101C2×Q814yescomplemented
191#97—44no—
192#97—42no—
193C4×C8—44no—
194C4×C8—42no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—24no—
201#95—22no—
202#95—24no—
203C24—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—84no—
217C2×Q8—88no—
218C2×Q8—88no—
219C2×Q8C22×Q811yescharacteristic
220C2×Q8C2×#4111yescharacteristic
221C2×Q82−1+412yes—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4C4×Q812yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—84no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244M16—24no—
245M16—42no—
246M16—42no—
247M16—44no—
248M16—42no—
249M16—44no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4#9912yes—
272C4⋊C4#9912yes—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—84no—
279C42—21no—
280C42—24no—
281C42—22no—
282C42—24no—
283C42—22no—
284C42—24no—
285C42—21no—
286C42—21no—
287C42—41no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42C4×Q812yes—
292C23—22no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C22×C22⋊C411yescharacteristic
298Q8—41no—
299Q8—41no—
300Q8—42no—
301Q8—42no—
302Q8—44no—
303Q8—42no—
304Q8—82no—
305Q8—82no—
306Q8—88no—
307Q8—82no—
308D4—42no—
309D4—42no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4C2×#9914yes—
341C2×C4C2×#9911yescharacteristic
342C2×C4C2×#9911yescharacteristic
343C2×C4C2×C4×Q811yescharacteristic
344C2×C4#51911yescharacteristic
345C2×C4#54812yes—
346C2×C4D4×Q812yes—
347C8—44no—
348C8—84no—
349C8—82no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—42no—
354C22—82no—
355C22—81no—
356C22—81no—
357C22Q8×C22⋊C411yescharacteristic
358C22C22×#9911yescharacteristic
359C22#288411yescharacteristic
360C4—21no—
361C4—24no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—42no—
369C4—44no—
370C4—41no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4—82no—
375C4#213412yes—
376C2—21no—
377C2—21no—
378C2—82no—
379C2Q8×#9911yescharacteristic
380C2C2×#213411yescharacteristic
381C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22517C2semidirect
3. #13302C2semidirect
4. #13301C2semidirect
101. #355Q8semidirect
190. #101C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#396162
#225172
#133022
#133012
#128032
#128012
#126322
C4×#3552