Finite Group Databasebeta

← #99253

Subgroups — #99253

Order 512.

546 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 82 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#13953C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7036C212yesmaximal, complemented
5#7036C212yesmaximal, complemented
6C2×#381C411yescharacteristic
7C2×C4×C22⋊C4C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9#1749C2212yes—
10#1749C2212yes—
11C2×#323—21no—
12C2×#323—21no—
13#1439—21no—
14#1439—21no—
15#1436—21no—
16#1436—21no—
17C4⋊#95C411yescharacteristic
18C4⋊#95C411yescharacteristic
19#1434C2211yescharacteristic
20#1434C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×#97—22no—
36C2×#97—22no—
37#381—42no—
38C4×C22⋊C4—22no—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—42no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53C2×#95D411yescharacteristic
54#336—22no—
55#336—22no—
56#323—24no—
57#323—24no—
58C25C22⋊C411yescharacteristic
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—81no—
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—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—81no—
82C2×C4⋊C4—81no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—81no—
94C2×C22⋊C4—81no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42—81no—
99C2×C42—81no—
100C2×C42—81no—
101C2×C42—81no—
102C2×C42#4112yes—
103C2×C42#4112yes—
104C22⋊C8—24no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#97—82no—
109#97—82no—
110#95—22no—
111#95—22no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
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#10112yes—
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—24no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
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—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—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—84no—
194C22×C4—84no—
195C22×C4#10012yes—
196C22×C4#10412yes—
197C22×C4#10412yes—
198C22×C4#11711yescharacteristic
199C22×C4#11711yescharacteristic
200C22×C4#12312yes—
201C22×C4#12312yes—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—84no—
209C4⋊C4—84no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—48no—
213C22⋊C4—84no—
214C42—42no—
215C42—42no—
216C42—81no—
217C42—81no—
218C42—84no—
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#38111yescharacteristic
247C23C2×#10011yescharacteristic
248C23C2×#10111yescharacteristic
249C23C2×#10411yescharacteristic
250C23C2×#10411yescharacteristic
251C23#42111yescharacteristic
252C23#42111yescharacteristic
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—84no—
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—81no—
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#152511yescharacteristic
342C22#153111yescharacteristic
343C22#153111yescharacteristic
344C22#153611yescharacteristic
345C22#182111yescharacteristic
346C22#182611yescharacteristic
347C22#182611yescharacteristic
348C22#182711yescharacteristic
349C22#182711yescharacteristic
350C22#182811yescharacteristic
351C22#182911yescharacteristic
352C4—41no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—82no—
360C4—82no—
361C4—161no—
362C4—161no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—82no—
368C2#872111yescharacteristic
369C2#874311yescharacteristic
370C2#875611yescharacteristic
371C2#877211yescharacteristic
372C2#877311yescharacteristic
373C2#877811yescharacteristic
374C2#877911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7036C2semidirect
5. #7036C2semidirect

Maximal subgroups

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

subgroupindex
#139532
#81522
#70362