Finite Group Databasebeta

← #141509

Subgroups — #141509

Order 512.

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

normal subgroups only

This group has 389 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#28706C211yesmaximal, characteristic
2#13299C211yesmaximal, characteristic
3#13224C211yesmaximal, characteristic
4#11900C211yesmaximal, characteristic
5#11878C211yesmaximal, characteristic
6#8609C211yesmaximal, characteristic
7#8160C211yesmaximal, characteristic
8C2×#498C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10#2563C2211yescharacteristic
11#2547C2211yescharacteristic
12C2×#354—22no—
13#2077—21no—
14#2077—21no—
15#1988C411yescharacteristic, complemented
16#1987C411yescharacteristic, complemented
17#1808—21no—
18#1807—21no—
19#1807—21no—
20#1799C2211yescharacteristic
21#1799C2211yescharacteristic
22C4×#106C411yescharacteristic, complemented
23C4×#99C2211yescharacteristic
24#1752—21no—
25#1459C2211yescharacteristic
26#1393—22no—
27#1351—22no—
28#1344—22no—
29#1289—21no—
30#1282—21no—
31C23×C8D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×#124—21no—
36C2×C4×D4C2311yescharacteristic
37#498—41no—
38#498—41no—
39C8×D4—41no—
40C8×D4—41no—
41C2×#107—41no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—21no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×#99—21no—
47C2×#99—21no—
48C2×#99D412yes—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C2×#97—41no—
53C2×C4×C8C2×C411yescharacteristic
54#397—21no—
55#397—21no—
56#393—21no—
57#393—21no—
58#391—21no—
59#390—21no—
60#382—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67C4×C4⋊C4C2×C411yescharacteristic
68C4×C4⋊C4C2×C411yescharacteristic
69C4×C22⋊C4—21no—
70#354—44no—
71#342—21no—
72#342—21no—
73#342—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—41no—
80#336—41no—
81#324—21no—
82#323D412yes—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×D4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—41no—
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103#124—42no—
104#124—42no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4#4111yescharacteristic
116C2×C22⋊C4—22no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4C2×D411yescharacteristic
124C2×C22⋊C4#4111yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42C22×C411yescharacteristic
129#107—82no—
130#106—42no—
131#106—42no—
132#106—41no—
133#106—41no—
134C4⋊C8—41no—
135#99—22no—
136#99—22no—
137#99—42no—
138#99—42no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143#97—82no—
144C4×C8—21no—
145C4×C8—21no—
146#95—21no—
147#95—21no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155C24#12111yescharacteristic
156C24#12711yescharacteristic
157C2×Q8—42no—
158C2×Q8—81no—
159C2×Q8—81no—
160C2×Q8—81no—
161C2×Q8—161no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4AGL(1,ℤ/8)12yes—
165C2×D4#13712yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12411yescharacteristic
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8#9911yescharacteristic
201C2×C8#9911yescharacteristic
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—82no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—161no—
229C22⋊C4—161no—
230C42—21no—
231C42—21no—
232C42—42no—
233C42—81no—
234C42—81no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#39011yescharacteristic
245C23#46812yes—
246C23#49611yescharacteristic
247C23#49711yescharacteristic
248Q8—162no—
249Q8—162no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4—161no—
287C2×C4—161no—
288C2×C4—161no—
289C2×C4C2×#9911yescharacteristic
290C2×C4#41011yescharacteristic
291C2×C4#44012yes—
292C8—21no—
293C8—41no—
294C8—41no—
295C8—41no—
296C8—82no—
297C8#35412yescomplemented
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22—81no—
312C22#196411yescharacteristic
313C22#197311yescharacteristic
314C22#197611yescharacteristic
315C4—22no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—161no—
321C4—161no—
322C4—162no—
323C4—161no—
324C4—161no—
325C4—321no—
326C4—321no—
327C4C2×#35411yescharacteristic
328C4#212911yescharacteristic
329C2—21no—
330C2—21no—
331C2—42no—
332C2—41no—
333C2—41no—
334C2#978611yescharacteristic
335C2#1322411yescharacteristic
336C2#1322911yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#498C4semidirect
15. #1988C4semidirect
16. #1987C4semidirect
22. C4×#106C4semidirect
297. C8#354semidirect

Maximal subgroups

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

subgroupindex
#287062
#132992
#132242
#119002
#118782
#86092
#81602