Finite Group Databasebeta

← #97033

Subgroups — #97033

Order 512.

798 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2#8300C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7595C212yesmaximal, complemented
5#7593C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×#117C2211yescharacteristic
8#2625—24no—
9#2625—24no—
10C2×#400—21no—
11C2×#398—21no—
12C2×#397—21no—
13C2×#397—21no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#380C411yescharacteristic
19#1817C2212yes—
20#1817C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23C2×#323—21no—
24C2×#323—21no—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—42no—
32C2×C22⋊Q8—42no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—42no—
36C2×#117—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53#400—28no—
54#398—28no—
55#397—28no—
56#397—28no—
57#391—22no—
58#391—22no—
59#391—22no—
60#391D414yes—
61#389—22no—
62#389—22no—
63#389—22no—
64#389D414yes—
65#380—22no—
66#380—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74#336—22no—
75#336—22no—
76#336—22no—
77#336—22no—
78#324—42no—
79#324—42no—
80#323—42no—
81#323—42no—
82C25C22⋊C411yescharacteristic
83C22×Q8—21no—
84C22×Q8—24no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8C22⋊C412yes—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—24no—
98C23×C4—42no—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—42no—
106C22×C8—42no—
107C22⋊Q8—48no—
108C22⋊Q8—48no—
109#117—48no—
110#117—48no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—82no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164C24—21no—
165C24—21no—
166C24—22no—
167C24—22no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24#9912yes—
173C24C2×C22⋊C411yescharacteristic
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—84no—
181C2×Q8—84no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—48no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4#9912yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#12012yes—
237C22×C4#12012yes—
238C22×C4#13114yes—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—48no—
253C4⋊C4—42no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—48no—
259C22⋊C4—84no—
260C42—42no—
261C42—42no—
262C42—42no—
263C42—42no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—24no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23#37911yescharacteristic
286C23#40812yes—
287C23C2×#9911yescharacteristic
288C23C2×#9911yescharacteristic
289C23#41312yes—
290Q8—84no—
291Q8—84no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4#40812yes—
344C2×C4#40812yes—
345C8—162no—
346C8—162no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—84no—
359C22—84no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#146812yes—
365C22#146812yes—
366C22#178011yescharacteristic
367C22#178111yescharacteristic
368C22#178511yescharacteristic
369C22#179511yescharacteristic
370C22#179611yescharacteristic
371C22#179611yescharacteristic
372C22#179611yescharacteristic
373C4—22no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—81no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4—82no—
383C4—82no—
384C2—22no—
385C2—22no—
386C2—82no—
387C2#830411yescharacteristic
388C2#832511yescharacteristic
389C2#832711yescharacteristic
390C2#851711yescharacteristic
391C2#852011yescharacteristic
392C2#852311yescharacteristic
393C2#852711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7595C2semidirect
5. #7593C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
#83002
#75952
#75932