Finite Group Databasebeta

← #100653

Subgroups — #100653

Order 512.

374 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 86 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#30042C211yesmaximal, characteristic
2#22315C211yesmaximal, characteristic
3#9895C211yesmaximal, characteristic
4#8235C211yesmaximal, characteristic
5#7662C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic
7#7023C211yesmaximal, characteristic
8#2862C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2072—21no—
11#2024C411yescharacteristic
12#2024C411yescharacteristic
13#2016—21no—
14#2016C411yescharacteristic
15#2016C411yescharacteristic
16#1985—21no—
17#1984—21no—
18#1940—21no—
19#1917—21no—
20#1916—21no—
21#1878—21no—
22#1878—21no—
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1771—21no—
26#1767C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1745C2211yescharacteristic
29C2×#333—21no—
30#1296—21no—
31#1295—21no—
32#1295—21no—
33#1272—21no—
34#1272—21no—
35C22×Q16—21no—
36C22×Q16—21no—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C4⋊Q8D411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×#103—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97D411yescharacteristic
54C2×#97D411yescharacteristic
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391—21no—
58#391—21no—
59#391—21no—
60#391—21no—
61#391C2×C411yescharacteristic
62#391C2×C411yescharacteristic
63#389—21no—
64#389—21no—
65#384D411yescharacteristic
66#382—21no—
67#382—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#333—24no—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8C22⋊C411yescharacteristic
92C22×Q8C22⋊C411yescharacteristic
93C2×Q16—42no—
94C2×Q16—42no—
95C2×Q16—81no—
96C2×Q16—81no—
97C2×Q16—81no—
98C2×Q16—81no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C22×C8—41no—
107C4⋊Q8—22no—
108C4⋊Q8—41no—
109C4⋊Q8—81no—
110C4⋊Q8—81no—
111C4×Q8—81no—
112C4×Q8—81no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—81no—
141C4⋊C8—81no—
142#103—81no—
143#103—81no—
144#103—81no—
145#103—81no—
146#97—21no—
147#97—21no—
148#97—21no—
149#97—41no—
150#97—41no—
151#97—81no—
152#97Q1612yescomplemented
153C4×C8—81no—
154C4×C8—81no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×Q8#10412yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12311yescharacteristic
193Q16—84no—
194Q16—84no—
195C2×C8—22no—
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—42no—
205C2×C8—42no—
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—41no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42C2×Q1611yescharacteristic
246C42#13511yescharacteristic
247C42#13711yescharacteristic
248C42#13711yescharacteristic
249C23#38611yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—81no—
253Q8—81no—
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—81no—
258Q8—162no—
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—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
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—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4C2×#10411yescharacteristic
294C2×C4#42111yescharacteristic
295C2×C4C4×Q1611yescharacteristic
296C2×C4#44111yescharacteristic
297C2×C4#45111yescharacteristic
298C2×C4#45211yescharacteristic
299C2×C4#46211yescharacteristic
300C2×C4#46411yescharacteristic
301C2×C4#47111yescharacteristic
302C2×C4#47311yescharacteristic
303C8—42no—
304C8—81no—
305C8—81no—
306C8—81no—
307C8—81no—
308C8—81no—
309C8—82no—
310C8—82no—
311C22#187111yescharacteristic
312C22#187811yescharacteristic
313C22#188111yescharacteristic
314C22#190511yescharacteristic
315C22#190811yescharacteristic
316C22#191111yescharacteristic
317C22#191411yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—42no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—161no—
335C4—161no—
336C2#899011yescharacteristic
337C2#916611yescharacteristic
338C2#918711yescharacteristic
339C2#926811yescharacteristic
340C2#929711yescharacteristic
341C2#933411yescharacteristic
342C2#937311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
152. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#300422
#223152
#98952
#82352
#76622
#70412
#70232