Finite Group Databasebeta

← #134227

Subgroups — #134227

Order 512.

794 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#381C211yesmaximal, characteristic
2C2×#1317C212yesmaximal
3#8557C212yesmaximal
4#7187C212yesmaximal
5C23×C42C411yescharacteristic
6C2×#381C411yescharacteristic
7C2×#381C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—22no—
11C2×#323C2214yes—
12#1434—22no—
13C22⋊#95C411yescharacteristic
14C22⋊#95C2211yescharacteristic
15C4×#95—22no—
16#1317—24no—
17#1280—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4C2×C411yescharacteristic
20C24×C4C2×C411yescharacteristic
21C22×C22⋊C4C2×C411yescharacteristic
22C22×C22⋊C4C2×C411yescharacteristic
23C22×C22⋊C4C2×C411yescharacteristic
24C22×C22⋊C4C2311yescharacteristic
25C22×C42—21no—
26C22×C42—21no—
27C22×C42—22no—
28C22×C42—21no—
29C22×C42—41no—
30C22×C42—41no—
31C22×C42—42no—
32C22×C42C814yescomplemented
33C22×C42D412yes—
34C2×C4⋊C8—42no—
35C2×C22⋊C8—24no—
36#381—22no—
37#381—22no—
38#381—22no—
39#381D412yes—
40#381Q812yes—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—42no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95D412yes—
50#336—42no—
51#323—24no—
52#323—24no—
53C25C22×C411yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—44no—
66C23×C4—44no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C4⋊C412yes—
77C23×C4C2×C812yes—
78C23×C4C2×C812yes—
79C23×C4C2×C812yes—
80C23×C4M1612yes—
81C23×C4M1612yes—
82C23×C4C2×D412yes—
83C22×C8—44no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4C4⋊C412yes—
97C2×C22⋊C4M1612yes—
98C2×C22⋊C4C2×D411yescharacteristic
99C2×C22⋊C4C2×Q811yescharacteristic
100C2×C22⋊C4#4112yes—
101C2×C42—24no—
102C2×C42—24no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—44no—
108C2×C42—44no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—42no—
112C2×C42—44no—
113C2×C42—44no—
114C2×C42—44no—
115C2×C42—44no—
116C2×C42—42no—
117C4⋊C8—84no—
118C22⋊C8—44no—
119C22⋊C8—44no—
120#95—28no—
121#95—24no—
122#95—24no—
123#95—42no—
124#95—44no—
125#95—44no—
126#95—42no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—42no—
132C24—42no—
133C24C2×C22⋊C411yescharacteristic
134C24C2×C4⋊C411yescharacteristic
135C24#11711yescharacteristic
136C24C22×C811yescharacteristic
137C24C2×M1611yescharacteristic
138C24C2×M1611yescharacteristic
139C24C2×M1611yescharacteristic
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
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—44no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—48no—
177C22×C4—44no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4C22⋊C812yes—
207C22×C4C22⋊C812yes—
208C22×C4#9911yescharacteristic
209C22×C4#9911yescharacteristic
210C22×C4#10012yes—
211C22×C4C4⋊C814yescomplemented
212C22×C4#12112yes—
213C22×C4C22⋊Q812yes—
214C2×C8—84no—
215C2×C8—84no—
216C2×C8—84no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—48no—
227C42—44no—
228C42—44no—
229C42—44no—
230C42—82no—
231C42—84no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—44no—
257C23—44no—
258C23—42no—
259C23—42no—
260C23#38011yescharacteristic
261C23C2×C22⋊C811yescharacteristic
262C23#40711yescharacteristic
263C23C2×#9911yescharacteristic
264C23#41011yescharacteristic
265C23C2×#10011yescharacteristic
266C23#41311yescharacteristic
267C23C2×C4⋊C811yescharacteristic
268C23#42311yescharacteristic
269C23#43111yescharacteristic
270C23#43211yescharacteristic
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#32312yes—
324C2×C4#32312yes—
325C2×C4#35214yes—
326C8—164no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—42no—
357C22C2×#32311yescharacteristic
358C22#145811yescharacteristic
359C22#146511yescharacteristic
360C22#146611yescharacteristic
361C22#179711yescharacteristic
362C22#179911yescharacteristic
363C22#180011yescharacteristic
364C22C2×#35212yes—
365C22#212012yes—
366C4—22no—
367C4—44no—
368C4—42no—
369C4—44no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—82no—
374C4—82no—
375C4#131714yescomplemented
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—41no—
381C2—41no—
382C2—41no—
383C2—41no—
384C2#855711yescharacteristic
385C2C2×#131711yescharacteristic
386C2#1142511yescharacteristic
387C2#1152612yes—
388C2#1279412yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. C22×C42C8semidirect
211. C22×C4C4⋊C8semidirect
375. C4#1317semidirect

Maximal subgroups

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

subgroupindex
C4×#3812
C2×#13172
#85572
#71872