Finite Group Databasebeta

← #125548

Subgroups — #125548

Order 512.

662 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 76 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
1#14914C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#8698C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C2×#385—21no—
8C22×#95C2211yescharacteristic
9#1779—21no—
10#1779—21no—
11#1446C411yescharacteristic
12#1446C2211yescharacteristic
13#1444C411yescharacteristic
14#1444C2211yescharacteristic
15#1441—21no—
16#1441—21no—
17C22⋊#95—21no—
18C22⋊#95C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C2×C22⋊Q8—44no—
28C2×C22⋊Q8—41no—
29C2×C22⋊Q8—41no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—41no—
40C2×C22⋊C8—41no—
41C2×C22⋊C8—41no—
42#385—44no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—41no—
51C2×#95—41no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95C2311yescharacteristic
58C25C22⋊C411yescharacteristic
59C22×Q8—22no—
60C22×Q8—42no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×Q8—81no—
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—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—41no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C22×C8—42no—
93C22×C8—42no—
94C22⋊Q8—44no—
95C22⋊Q8—44no—
96C22⋊Q8—44no—
97C22⋊Q8—44no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—82no—
121C22⋊C8—82no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#95—22no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95—81no—
136#95—81no—
137C24—22no—
138C24—22no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—41no—
146C24—41no—
147C24#9911yescharacteristic
148C24#9911yescharacteristic
149C24C2×C22⋊C411yescharacteristic
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—82no—
153C2×Q8—84no—
154C2×Q8—84no—
155C2×Q8—82no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4#13112yes—
236C22×C4#13512yes—
237C22×C4#13712yes—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C22⋊C4—48no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—24no—
257C23—24no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23AΣL(1,𝔽16)12yes—
285C23#35612yes—
286C23#40811yescharacteristic
287C23C2×#9911yescharacteristic
288C23#41311yescharacteristic
289C23#41611yescharacteristic
290C23#41911yescharacteristic
291C23#42111yescharacteristic
292C23#42111yescharacteristic
293Q8—84no—
294Q8—84no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C8—162no—
360C8—162no—
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—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22#146811yescharacteristic
392C22#148211yescharacteristic
393C22#149711yescharacteristic
394C22#151111yescharacteristic
395C22#151611yescharacteristic
396C22#153511yescharacteristic
397C22#155411yescharacteristic
398C22C2×AΣL(1,𝔽16)11yescharacteristic
399C22#212111yescharacteristic
400C22#212711yescharacteristic
401C22C2×#35611yescharacteristic
402C4—42no—
403C4—42no—
404C4—42no—
405C4—82no—
406C4—82no—
407C4—81no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—82no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—42no—
417C2—42no—
418C2#772111yescharacteristic
419C2#1154111yescharacteristic
420C2#1155511yescharacteristic
421C2#1181111yescharacteristic
422C2#1181511yescharacteristic
423C2#1183911yescharacteristic
424C2#1185411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7131C2semidirect
5. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#149142
#86982
#71312
#71232