Finite Group Databasebeta

← #99405

Subgroups — #99405

Order 512.

403 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 90 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#21921C211yesmaximal, characteristic
2#11209C211yesmaximal, characteristic
3#8833C211yesmaximal, characteristic
4#8832C211yesmaximal, characteristic
5#7564C211yesmaximal, characteristic
6#7028C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#423C2211yescharacteristic
9#2296C2211yescharacteristic
10#1917—21no—
11#1917—21no—
12#1916—21no—
13#1916—21no—
14#1818—21no—
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17#1750—21no—
18C4×C22⋊C8C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#339—21no—
21C2×#339—21no—
22C2×#335—21no—
23C2×#334—21no—
24C4×#95C2211yescharacteristic
25#1277—21no—
26#1277—21no—
27#1271C411yescharacteristic, complemented
28#1271C411yescharacteristic, complemented
29#1271C411yescharacteristic, complemented
30#1271C411yescharacteristic, complemented
31C22×M16—21no—
32C2×#117D411yescharacteristic
33C2×#117D411yescharacteristic
34C22×C42—21no—
35C22×C42C2311yescharacteristic
36#423—42no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×#97—21no—
48C2×C4×C8—21no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51#389C2×C411yescharacteristic
52#389C2×C411yescharacteristic
53#389Q811yescharacteristic
54#389Q811yescharacteristic
55#385—21no—
56#385—21no—
57C4×C4⋊C4—21no—
58C4×C4⋊C4—21no—
59C2×#95—21no—
60C2×#95—21no—
61#339—42no—
62#339—42no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336C2×C411yescharacteristic
72#336C2×C411yescharacteristic
73#336C2×C411yescharacteristic
74#336C2×C411yescharacteristic
75#335—42no—
76#334—42no—
77C23×C4—41no—
78C23×C4C2×D411yescharacteristic
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—41no—
96#117—21no—
97#117—21no—
98#117—21no—
99#117—21no—
100#117—41no—
101#117—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C4⋊C411yescharacteristic
107C2×C4⋊C4C4⋊C411yescharacteristic
108C2×C4⋊C4C4⋊C411yescharacteristic
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42C22×C411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#97—81no—
141C4×C8—41no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—81no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—22no—
151#95—22no—
152#95—41no—
153#95—41no—
154C24#12311yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×C4⋊C411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4#12011yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#13511yescharacteristic
185C22×C4#13511yescharacteristic
186C22×C4AGL(1,ℤ/8)11yescharacteristic
187C22×C4#13711yescharacteristic
188M16—42no—
189M16—42no—
190M16—81no—
191M16—81no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—41no—
239C42—41no—
240C42—42no—
241C42—42no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—81no—
247C42—81no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23#38511yescharacteristic
255C23#48211yescharacteristic
256C23#48511yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#42711yescharacteristic
301C2×C4#42811yescharacteristic
302C2×C4#44311yescharacteristic
303C2×C4#44311yescharacteristic
304C2×C4#44411yescharacteristic
305C2×C4#44411yescharacteristic
306C2×C4#44911yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45411yescharacteristic
309C2×C4#45611yescharacteristic
310C2×C4B(4,𝔽2)11yescharacteristic
311C2×C4#45811yescharacteristic
312C8—42no—
313C8—42no—
314C8—81no—
315C8—81no—
316C8—81no—
317C8—82no—
318C8—81no—
319C8—81no—
320C8—81no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#185411yescharacteristic
332C22#185711yescharacteristic
333C22#186111yescharacteristic
334C22#186111yescharacteristic
335C22#186611yescharacteristic
336C22#186611yescharacteristic
337C22#186711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—84no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4#159711yescharacteristic
352C4#160311yescharacteristic
353C4#160911yescharacteristic
354C4#161511yescharacteristic
355C2—41no—
356C2—41no—
357C2#885511yescharacteristic
358C2#887411yescharacteristic
359C2#887411yescharacteristic
360C2#890311yescharacteristic
361C2#890511yescharacteristic
362C2#891911yescharacteristic
363C2#893611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1271C4semidirect
28. #1271C4semidirect
29. #1271C4semidirect
30. #1271C4semidirect

Maximal subgroups

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

subgroupindex
#219212
#112092
#88332
#88322
#75642
#70282