Finite Group Databasebeta

← #99496

Subgroups — #99496

Order 512.

366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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#22798C211yesmaximal, characteristic
2#9604C211yesmaximal, characteristic, complemented
3#8809C211yesmaximal, characteristic, complemented
4#8567C211yesmaximal, characteristic
5#7560C211yesmaximal, characteristic
6#7031C211yesmaximal, characteristic, complemented
7#7027C211yesmaximal, characteristic, complemented
8C2×#433—21no—
9#2293C2211yescharacteristic
10#1955—21no—
11#1924—21no—
12#1923—21no—
13#1842C2211yescharacteristic
14#1842C2211yescharacteristic
15#1798—21no—
16#1771—21no—
17#1767C2211yescharacteristic
18C4×C4⋊C8C2211yescharacteristic
19C4×C4⋊C8C2211yescharacteristic
20#1744C2211yescharacteristic
21C2×#335—21no—
22C2×#334—21no—
23C2×#334—21no—
24C2×D4⋊C8—21no—
25#1279—21no—
26#1277—21no—
27#1271C411yescharacteristic
28#1271C411yescharacteristic
29#1270C411yescharacteristic
30#1270C411yescharacteristic
31C2×C4×D4D411yescharacteristic
32C2×#117—21no—
33#433—42no—
34#433—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#390C2×C411yescharacteristic
57#390Q811yescharacteristic
58#389C2×C411yescharacteristic
59#389Q811yescharacteristic
60#388—21no—
61#388—21no—
62C4×C4⋊C4D411yescharacteristic
63C4×C22⋊C4—21no—
64C43C2311yescharacteristic
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336C2×C411yescharacteristic
71#336C2×C411yescharacteristic
72#336C2×C411yescharacteristic
73#336C2×C411yescharacteristic
74#335—42no—
75#334—42no—
76#334—42no—
77D4⋊C8—42no—
78C22×D4C4⋊C411yescharacteristic
79C23×C4—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94#117—41no—
95#117—41no—
96#117—41no—
97#117—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C4⋊C4C4⋊C411yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C42—21no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×Q811yescharacteristic
114C2×C42#4111yescharacteristic
115C2×C42#4111yescharacteristic
116C4⋊C8—22no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—42no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8SD1612yes—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—81no—
138C22⋊C8—81no—
139#97—42no—
140#97—41no—
141#97—41no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—42no—
145C4×C8—41no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152C24—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C2×C4⋊C411yescharacteristic
167C22×C4C4×D411yescharacteristic
168C22×C4C4×D411yescharacteristic
169C22×C4#12011yescharacteristic
170C22×C4C22⋊Q811yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4#12311yescharacteristic
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—81no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42C2×SD1611yescharacteristic
230C42#13511yescharacteristic
231C42#13511yescharacteristic
232C42#13511yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C42#13711yescharacteristic
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#38511yescharacteristic
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4#42711yescharacteristic
274C2×C4#42811yescharacteristic
275C2×C4C4×SD1611yescharacteristic
276C2×C4#44211yescharacteristic
277C2×C4#44311yescharacteristic
278C2×C4#44411yescharacteristic
279C2×C4#44911yescharacteristic
280C2×C4#44911yescharacteristic
281C2×C4#45011yescharacteristic
282C2×C4#45211yescharacteristic
283C2×C4#45311yescharacteristic
284C2×C4#45611yescharacteristic
285C2×C4#47611yescharacteristic
286C2×C4#47811yescharacteristic
287C2×C4#48211yescharacteristic
288C2×C4#48511yescharacteristic
289C8—42no—
290C8—42no—
291C8—42no—
292C8—42no—
293C8—81no—
294C8—82no—
295C8—81no—
296C8—81no—
297C8—81no—
298C8—161no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#185411yescharacteristic
307C22#185811yescharacteristic
308C22#185911yescharacteristic
309C22#186111yescharacteristic
310C22#186411yescharacteristic
311C22#186611yescharacteristic
312C22#186811yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#162411yescharacteristic
328C4#163111yescharacteristic
329C4#165211yescharacteristic
330C4#165411yescharacteristic
331C2—161no—
332C2#886311yescharacteristic
333C2#887611yescharacteristic
334C2#888011yescharacteristic
335C2#888911yescharacteristic
336C2#890311yescharacteristic
337C2#891611yescharacteristic
338C2#893211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
socleC23#385
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9604C2semidirect
3. #8809C2semidirect
6. #7031C2semidirect
7. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#227982
#96042
#88092
#85672
#75602
#70312
#70272