Finite Group Databasebeta

← #99249

Subgroups — #99249

Order 512.

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

normal subgroups only

This group has 506 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#13947C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7036C212yesmaximal, complemented
5#7036C212yesmaximal, complemented
6C2×C4×C22⋊C4C2211yescharacteristic
7#1749C2212yes—
8#1749C2212yes—
9C2×#323—21no—
10C2×#323—21no—
11#1443—21no—
12#1440—21no—
13#1437—21no—
14#1434—21no—
15#1434C411yescharacteristic
16#1434C2211yescharacteristic
17#1434C2211yescharacteristic
18C22⋊#95C411yescharacteristic
19C22⋊#95C411yescharacteristic
20C4×#95C411yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C22⋊C4D411yescharacteristic
27C22×C22⋊C4D411yescharacteristic
28C22×C42—21no—
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—22no—
32C2×C22⋊C8—22no—
33C2×#97—22no—
34C2×#97—22no—
35C4×C22⋊C4—22no—
36C4×C22⋊C4—22no—
37C2×#95—21no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95C2×C411yescharacteristic
47C2×#95C2×C411yescharacteristic
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51C2×#95D411yescharacteristic
52#336—22no—
53#336—22no—
54#323—24no—
55#323—24no—
56C25C22⋊C411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—81no—
66C23×C4—81no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—42no—
74C22×C8—42no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—81no—
86C2×C22⋊C4—81no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42—81no—
92C2×C42—81no—
93C2×C42—81no—
94C2×C42#4112yes—
95C2×C42#4112yes—
96C22⋊C8—24no—
97C22⋊C8—24no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#97—82no—
101#97—82no—
102#95—22no—
103#95—22no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—42no—
109#95—42no—
110#95—42no—
111#95—42no—
112#95—81no—
113#95—81no—
114#95—81no—
115#95—81no—
116#95—81no—
117C24—21no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—41no—
124C24—41no—
125C24—41no—
126C24—41no—
127C24#10412yes—
128C24C2×C22⋊C411yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—42no—
152C22×C4—42no—
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—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4#10412yes—
192C22×C4#11711yescharacteristic
193C22×C4#11711yescharacteristic
194C22×C4#12312yes—
195C22×C4#12312yes—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C42—42no—
207C42—42no—
208C42—81no—
209C42—81no—
210C42—84no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23#38111yescharacteristic
239C23#41311yescharacteristic
240C23#41311yescharacteristic
241C23C2×#10411yescharacteristic
242C23C2×#10411yescharacteristic
243C23#42111yescharacteristic
244C23#42111yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
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—
308C8—84no—
309C8—84no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#152411yescharacteristic
334C22#152911yescharacteristic
335C22#153211yescharacteristic
336C22#153511yescharacteristic
337C22#182211yescharacteristic
338C22#182611yescharacteristic
339C22#182611yescharacteristic
340C22#182711yescharacteristic
341C22#182711yescharacteristic
342C22#182811yescharacteristic
343C22#182911yescharacteristic
344C4—41no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—82no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4—161no—
354C4—161no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2—82no—
360C2#872011yescharacteristic
361C2#874411yescharacteristic
362C2#875911yescharacteristic
363C2#877311yescharacteristic
364C2#877311yescharacteristic
365C2#877611yescharacteristic
366C2#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
#139472
#81522
#70362