Finite Group Databasebeta

← #213979

Subgroups — #213979

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#17370C212yesmaximal
3#9518C214yesmaximal, complemented
4#9506C214yesmaximal, complemented
5#8757C212yesmaximal, complemented
6#8551C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C2×#520C2211yescharacteristic
9#2975C2214yes—
10#2579C412yes—
11#2579C2212yes—
12#2452—22no—
13#2451—22no—
14#2440—22no—
15#2439—22no—
16#2409C412yes—
17#2409C2212yes—
18#2317C411yescharacteristic
19#2316C2211yescharacteristic
20#2311C411yescharacteristic
21#2297C411yescharacteristic
22#2296C411yescharacteristic
23#1784C2214yes—
24#1465—22no—
25#1465C2214yes—
26#1463—22no—
27#1463C2214yes—
28#1463C2214yes—
29#1461—22no—
30#1458—22no—
31#1305C2218yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34C23×Q8C2311yescharacteristic
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—44no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40#520—44no—
41C2×C4×Q8D412yes—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4C2312yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117D412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4D412yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42C2×C411yescharacteristic
55#436—48no—
56#431—24no—
57C2×#101—44no—
58#407—24no—
59#407C2318yes—
60#398—22no—
61#398—22no—
62#396—22no—
63#396—22no—
64#395—22no—
65#395—22no—
66#393—22no—
67#393—22no—
68#391—21no—
69#391—21no—
70#391—22no—
71#391—22no—
72#390—21no—
73#390—21no—
74#389—21no—
75#389—21no—
76#389—22no—
77#389—22no—
78#389—22no—
79#387—22no—
80#387—22no—
81#386C2×C412yes—
82#386C2×C412yes—
83#386C2×C412yes—
84#386C2×C412yes—
85#385—22no—
86#385—22no—
87#385C2×C412yes—
88#385C2×C412yes—
89#385C2×C412yes—
90#385C2×C412yes—
91#384—21no—
92#382—21no—
93#380—22no—
94#380C2×C412yes—
95#380C2312yes—
96C4×C4⋊C4—22no—
97C4×C22⋊C4—22no—
98#324—24no—
99#324—24no—
100#324—24no—
101#324—44no—
102#323—24no—
103C2×#41—24no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—42no—
110C22×Q8—82no—
111C22×Q8—82no—
112C22×Q8C22⋊C411yescharacteristic
113C22×D4C22⋊C411yescharacteristic
114C22×D4C22×C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C2×M16—28no—
126C2×M16—44no—
127C4⋊Q8—48no—
128C4⋊Q8—48no—
129C22⋊Q8—84no—
130C22⋊Q8—84no—
131C4×Q8—44no—
132C4×D4—28no—
133C4×D4—44no—
134C4×D4—44no—
135#117—44no—
136#117—42no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22×C412yes—
168C2×C22⋊C4C22×C412yes—
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C2×D412yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42C22⋊C412yes—
177C2×C42C22⋊C412yes—
178C2×C42C2×D412yes—
179C4⋊C8—44no—
180#101—88no—
181C22⋊C8—28no—
182C22⋊C8—28no—
183C22⋊C8—44no—
184C4×C8—44no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195C24C2×C22⋊C412yes—
196C24C23×C411yescharacteristic
197#41—416no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—48no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—84no—
206C2×Q8—82no—
207C2×Q8—84no—
208C2×Q8—82no—
209C2×Q8—84no—
210C2×Q8#10114yescomplemented
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—42no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—82no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C22×D412yes—
251C22×C4H(5,𝔽2)14yes—
252M16—88no—
253M16—164no—
254C2×C8—48no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C42—24no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23C22×C22⋊C411yescharacteristic
299C23#51812yes—
300Q8—44no—
301Q8—88no—
302Q8—88no—
303D4—48no—
304D4—44no—
305C2×C4—28no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—162no—
346C2×C4C2×#10112yes—
347C2×C4C2×#10114yes—
348C2×C4#41312yes—
349C2×C4#53414yes—
350C8—88no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22#230811yescharacteristic
364C22C22×#10111yescharacteristic
365C22C2×#41311yescharacteristic
366C4—24no—
367C4—22no—
368C4—22no—
369C4—24no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—82no—
377C4—81no—
378C4—81no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—82no—
383C2#2175311yescharacteristic
384C2#2177611yescharacteristic
385C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9518C2semidirect
4. #9506C2semidirect
5. #8757C2semidirect
6. #8551C2semidirect
210. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
#355742
#173702
#95182
#95062
#87572
#85512