Finite Group Databasebeta

← #125267

Subgroups — #125267

Order 512.

512 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 78 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#15063C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7132C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C2211yescharacteristic
8#1749—21no—
9C4×C22⋊C8—21no—
10C2×#323—21no—
11C2×#323—21no—
12#1445C411yescharacteristic
13#1445C2211yescharacteristic
14#1444C2211yescharacteristic
15#1442C411yescharacteristic
16#1440—21no—
17#1439—21no—
18#1436—21no—
19#1434—21no—
20C22⋊#95C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×#97—41no—
36C2×C4×C8—41no—
37#380—41no—
38#380—41no—
39C4×C22⋊C4—41no—
40C4×C22⋊C4—41no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2311yescharacteristic
54#336—41no—
55#336—41no—
56#323—24no—
57#323—24no—
58C25C22⋊C411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—41no—
67C23×C4—42no—
68C23×C4—81no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—42no—
76C22×C8—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—81no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—81no—
95C2×C22⋊C4—81no—
96C2×C22⋊C4—81no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—42no—
102C2×C42—81no—
103C22⋊C8—24no—
104C22⋊C8—24no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—82no—
108C4×C8—82no—
109#95—22no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—41no—
117#95—41no—
118#95—42no—
119#95—81no—
120#95—81no—
121#95—81no—
122#95—81no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—41no—
130C24—41no—
131C24—41no—
132C24—41no—
133C24#10412yes—
134C24C2×C22⋊C411yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
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—81no—
193C22×C4—81no—
194C22×C4—84no—
195C22×C4—84no—
196C22×C4#9911yescharacteristic
197C22×C4#9911yescharacteristic
198C22×C4#10412yes—
199C22×C4#13112yes—
200C22×C4#13512yes—
201C22×C4#13712yes—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—42no—
216C42—42no—
217C42—81no—
218C42—81no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23#40811yescharacteristic
247C23C2×#9911yescharacteristic
248C23#41311yescharacteristic
249C23#41611yescharacteristic
250C23#41911yescharacteristic
251C23C2×#10411yescharacteristic
252C23C2×#10411yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—84no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4#35212yes—
316C8—84no—
317C8—84no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#146811yescharacteristic
342C22#148511yescharacteristic
343C22#149411yescharacteristic
344C22#150111yescharacteristic
345C22#150811yescharacteristic
346C22#152411yescharacteristic
347C22#155411yescharacteristic
348C22C2×#35211yescharacteristic
349C22#212111yescharacteristic
350C22#212811yescharacteristic
351C22#213211yescharacteristic
352C4—22no—
353C4—42no—
354C4—41no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—82no—
359C4—81no—
360C4—161no—
361C4—161no—
362C4—161no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—82no—
368C2#767511yescharacteristic
369C2#1154311yescharacteristic
370C2#1155411yescharacteristic
371C2#1172911yescharacteristic
372C2#1173511yescharacteristic
373C2#1178611yescharacteristic
374C2#1179311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7132C2semidirect
5. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#150632
#81522
C4×#3232
#71322
#71232