Finite Group Databasebeta

← #134340

Subgroups — #134340

Order 512.

472 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 96 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#14050C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3#8560C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5#7189C212yesmaximal
6C2×#389C411yescharacteristic
7C2×#382C2211yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10#1797—21no—
11C2×#324C2212yes—
12C2×#324C2212yes—
13#1443—21no—
14#1443—21no—
15#1436—21no—
16#1436—21no—
17C4⋊#95C411yescharacteristic
18C4⋊#95C411yescharacteristic
19C4⋊#95C2211yescharacteristic
20C4×#95C411yescharacteristic
21#1324—24no—
22#1280—21no—
23#1280—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38#389—41no—
39#389—41no—
40#382—21no—
41#382—21no—
42#382D412yes—
43#382Q812yes—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59#336—41no—
60#336—41no—
61#324—22no—
62#324—22no—
63#324—22no—
64#324—22no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4C4⋊C412yes—
94C2×C4⋊C4C2×D411yescharacteristic
95C2×C4⋊C4C2×Q811yescharacteristic
96C2×C4⋊C4#4111yescharacteristic
97C2×C4⋊C4#4111yescharacteristic
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42—41no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—81no—
113C2×C42—81no—
114C2×C42C4⋊C412yes—
115C4⋊C8—82no—
116C4⋊C8—82no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#95—22no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—81no—
133#95—81no—
134#95—81no—
135#95—81no—
136C24C2×C22⋊C411yescharacteristic
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—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
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—41no—
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—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4#9911yescharacteristic
196C22×C4#9911yescharacteristic
197C22×C4#10112yes—
198C22×C4C2×C4⋊C411yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#13112yes—
205C22×C4#13112yes—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C42—24no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—81no—
229C42—81no—
230C42—82no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23#38011yescharacteristic
236C23#40811yescharacteristic
237C23#40811yescharacteristic
238C23C2×#9911yescharacteristic
239C23#41011yescharacteristic
240C23C2×#10111yescharacteristic
241C23#41311yescharacteristic
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
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—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
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#35512yes—
315C2×C4#42412yes—
316C2×C4#43312yes—
317C8—162no—
318C8—162no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22#146811yescharacteristic
332C22#147011yescharacteristic
333C22#147211yescharacteristic
334C22#147311yescharacteristic
335C22#179811yescharacteristic
336C22#179911yescharacteristic
337C22#180111yescharacteristic
338C22#212711yescharacteristic
339C22#212911yescharacteristic
340C22C2×#35511yescharacteristic
341C22#213211yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—42no—
351C4—82no—
352C4—82no—
353C4—82no—
354C4—81no—
355C4—81no—
356C4—161no—
357C2—21no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2#856211yescharacteristic
362C2#1155311yescharacteristic
363C2#1155611yescharacteristic
364C2#1157211yescharacteristic
365C2#1157311yescharacteristic
366C2#1280011yescharacteristic
367C2#1280111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#140502
C2×#13242
#85602
#85582
#71892