Finite Group Databasebeta

← #100461

Subgroups — #100461

Order 512.

455 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 102 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#33027C211yesmaximal, characteristic
2Q8×#97C211yesmaximal, characteristic
3#9890C211yesmaximal, characteristic
4#8175C211yesmaximal, characteristic
5#7682C211yesmaximal, characteristic
6#7049C211yesmaximal, characteristic
7#7030C211yesmaximal, characteristic
8C2×#477C411yescharacteristic
9C2×#462C411yescharacteristic
10C2×C8×Q8—21no—
11#2862C2211yescharacteristic
12C42×Q8C2211yescharacteristic
13#1987C411yescharacteristic
14#1981C411yescharacteristic
15#1928—21no—
16#1928—21no—
17#1878—21no—
18#1878—21no—
19#1831C2211yescharacteristic
20#1770—21no—
21#1770—22no—
22#1767—21no—
23#1767C2211yescharacteristic
24#1767C2211yescharacteristic
25#1744C2211yescharacteristic
26C4×#97—21no—
27C4×#97C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1295—21no—
30#1295—21no—
31#1279—21no—
32#1272—22no—
33C22×Q16—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39#477—42no—
40#462—42no—
41C8×Q8—24no—
42C8×Q8—24no—
43C2×#107—21no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C411yescharacteristic
48C2×C4⋊C8C2×C411yescharacteristic
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—21no—
53C2×#97—22no—
54C2×#97—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×#97D411yescharacteristic
58C2×#97D411yescharacteristic
59C2×C4×C8—22no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391—21no—
63#391—21no—
64#391—21no—
65#391—21no—
66#382—21no—
67#382—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4C2×C411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C4×C4⋊C4D411yescharacteristic
74C43—21no—
75C43C2311yescharacteristic
76#340—21no—
77#340—21no—
78#336—22no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336D412yes—
85Q8⋊C8—24no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8C22⋊C411yescharacteristic
89C2×Q16—42no—
90C2×Q16—42no—
91C2×Q16—42no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—22no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4×Q8—22no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×Q8—41no—
108C4×Q8—41no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#107—42no—
133C4⋊C8—28no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140#103—42no—
141#103—42no—
142#103—42no—
143#103—42no—
144#97—21no—
145#97—21no—
146#97—21no—
147#97—42no—
148#97—44no—
149#97—41no—
150#97—41no—
151#97—41no—
152#97—41no—
153#97Q1612yescomplemented
154C4×C8—28no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—41no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8#10012yes—
179C2×Q8#10412yes—
180C2×Q8#10412yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12311yescharacteristic
194Q16—84no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—42no—
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—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—24no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4#10412yes—
230C42—24no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—42no—
241C42—42no—
242C42—42no—
243C42—81no—
244C42—81no—
245C42C2×Q1611yescharacteristic
246C42#13511yescharacteristic
247C42#13711yescharacteristic
248C42#13711yescharacteristic
249C23#38611yescharacteristic
250Q8—22no—
251Q8—22no—
252Q8—22no—
253Q8—81no—
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—162no—
258Q8#32914yes—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
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—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×#10011yescharacteristic
293C2×C4#41311yescharacteristic
294C2×C4C2×#10411yescharacteristic
295C2×C4C2×#10411yescharacteristic
296C2×C4C2×#10411yescharacteristic
297C2×C4#42111yescharacteristic
298C2×C4C4×Q1611yescharacteristic
299C2×C4#44111yescharacteristic
300C2×C4#45111yescharacteristic
301C2×C4#45211yescharacteristic
302C2×C4#46211yescharacteristic
303C2×C4#46411yescharacteristic
304C8—24no—
305C8—42no—
306C8—44no—
307C8—82no—
308C8—82no—
309C8—81no—
310C8—81no—
311C22#187111yescharacteristic
312C22#187811yescharacteristic
313C22#188111yescharacteristic
314C22#189611yescharacteristic
315C22#190011yescharacteristic
316C22#190211yescharacteristic
317C22#190611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—24no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—161no—
333C4C2×#32911yescharacteristic
334C4#152411yescharacteristic
335C4#153011yescharacteristic
336C4#153211yescharacteristic
337C2#899011yescharacteristic
338C2#911911yescharacteristic
339C2#916011yescharacteristic
340C2#923211yescharacteristic
341C2#924811yescharacteristic
342C2#931211yescharacteristic
343C2#935011yescharacteristic
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
153. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#330272
Q8×#972
#98902
#81752
#76822
#70492
#70302