Finite Group Databasebeta

← #97076

Subgroups — #97076

Order 512.

1236 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×#387C211yesmaximal, characteristic
2#8698C212yesmaximal
3#8698C212yesmaximal
4#8698C212yesmaximal
5C23×C4⋊C4C411yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C23×C42C411yescharacteristic
8C2×#387—24no—
9C2×#387—24no—
10C2×#387—24no—
11C22×#95C411yescharacteristic
12C22×#95C2211yescharacteristic
13C22×#95C2211yescharacteristic
14C22×#95C2211yescharacteristic
15#1779C2214yes—
16#1280—24no—
17#1280—24no—
18#1280—24no—
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C24×C4C2×C411yescharacteristic
22C24×C4C2×C411yescharacteristic
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×C4⋊C4—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—44no—
36C22×C4⋊C4C814yescomplemented
37C22×C42—22no—
38C22×C42—22no—
39C22×C42—22no—
40C22×C42—44no—
41C2×C22⋊C8—24no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—24no—
44#387—232no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95D414yes—
55C2×#95D414yes—
56C2×#95D414yes—
57C25C22×C411yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C2×C812yes—
101C23×C4C2×C812yes—
102C23×C4C2×C812yes—
103C23×C4M1612yes—
104C23×C4M1612yes—
105C23×C4M1612yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C22×C8—44no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C42—28no—
123C2×C42—48no—
124C2×C42—48no—
125C2×C42—44no—
126C2×C42—44no—
127C22⋊C8—48no—
128C22⋊C8—48no—
129C22⋊C8—48no—
130#95—28no—
131#95—28no—
132#95—28no—
133#95—28no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—22no—
138C24—22no—
139C24—22no—
140C24C2×C22⋊C411yescharacteristic
141C24C2×C22⋊C411yescharacteristic
142C24C2×C22⋊C411yescharacteristic
143C24C22×C811yescharacteristic
144C24C2×M1611yescharacteristic
145C24C2×M1611yescharacteristic
146C24C2×M1611yescharacteristic
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—416no—
188C22×C4—416no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
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—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—88no—
237C22×C4—88no—
238C22×C4C22⋊C814yes—
239C22×C4C22⋊C814yes—
240C22×C4C22⋊C814yes—
241C22×C4#10014yes—
242C22×C4#10114yes—
243C22×C4#12014yes—
244C2×C8—84no—
245C2×C8—84no—
246C2×C8—84no—
247C4⋊C4—232no—
248C4⋊C4—416no—
249C42—416no—
250C23—28no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#37911yescharacteristic
279C23C2×C22⋊C811yescharacteristic
280C23C2×C22⋊C811yescharacteristic
281C23C2×C22⋊C811yescharacteristic
282C23#40711yescharacteristic
283C23#40711yescharacteristic
284C23#40711yescharacteristic
285C23#41012yes—
286C23#41012yes—
287C23#41012yes—
288C23C2×#10012yes—
289C23C2×#10112yes—
290C23#41312yes—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C8—164no—
345C22—28no—
346C22—28no—
347C22—28no—
348C22—28no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22#145812yes—
374C22#146012yes—
375C22#146112yes—
376C22#146412yes—
377C22#146612yes—
378C22#146712yes—
379C22#177911yescharacteristic
380C22#178211yescharacteristic
381C22#178211yescharacteristic
382C22#178211yescharacteristic
383C22#178311yescharacteristic
384C22#178411yescharacteristic
385C22#178511yescharacteristic
386C4—44no—
387C4—44no—
388C4—44no—
389C4—44no—
390C4—44no—
391C4—44no—
392C4—82no—
393C2—22no—
394C2—22no—
395C2—22no—
396C2—22no—
397C2—22no—
398C2—22no—
399C2#830011yescharacteristic
400C2#830211yescharacteristic
401C2#830311yescharacteristic
402C2#854511yescharacteristic
403C2#855311yescharacteristic
404C2#855411yescharacteristic
405C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
36. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C22×#3872
#86982