Finite Group Databasebeta

← #99488

Subgroups — #99488

Order 512.

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

normal subgroups only

This group has 356 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#8787C211yesmaximal, 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×D4⋊C8—21no—
23#1279—21no—
24#1277—21no—
25#1277—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—
75D4⋊C8—42no—
76C22×D4C4⋊C411yescharacteristic
77C23×C4—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—41no—
87C22×C8—41no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92#117—41no—
93#117—41no—
94#117—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C4⋊C411yescharacteristic
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42C2×Q811yescharacteristic
112C2×C42#4111yescharacteristic
113C2×C42#4111yescharacteristic
114C4⋊C8—22no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8D812yes—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—81no—
135C22⋊C8—81no—
136#97—42no—
137#97—41no—
138#97—41no—
139C4×C8—41no—
140C4×C8—41no—
141C4×C8—42no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—41no—
146#95—21no—
147#95—21no—
148#95—21no—
149C24—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—41no—
153C2×D4—41no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4C2×C4⋊C411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4C4×D411yescharacteristic
166C22×C4#12011yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12311yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
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—81no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42C2×D811yescharacteristic
227C42#13511yescharacteristic
228C42#13511yescharacteristic
229C42#13511yescharacteristic
230C42AGL(1,ℤ/8)11yescharacteristic
231C42AGL(1,ℤ/8)11yescharacteristic
232C42#13711yescharacteristic
233C42#13711yescharacteristic
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23#38511yescharacteristic
242D4—42no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—41no—
258C2×C4—41no—
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—81no—
269C2×C4—81no—
270C2×C4#42711yescharacteristic
271C2×C4#42811yescharacteristic
272C2×C4C4×D811yescharacteristic
273C2×C4#44011yescharacteristic
274C2×C4#44311yescharacteristic
275C2×C4#44411yescharacteristic
276C2×C4#44711yescharacteristic
277C2×C4#44911yescharacteristic
278C2×C4#45211yescharacteristic
279C2×C4#45211yescharacteristic
280C2×C4#45411yescharacteristic
281C2×C4#45511yescharacteristic
282C2×C4#47411yescharacteristic
283C2×C4#47811yescharacteristic
284C2×C4#48211yescharacteristic
285C2×C4#48511yescharacteristic
286C8—42no—
287C8—42no—
288C8—81no—
289C8—81no—
290C8—81no—
291C8—82no—
292C8—81no—
293C8—81no—
294C8—81no—
295C8—161no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#185411yescharacteristic
304C22#185811yescharacteristic
305C22#186011yescharacteristic
306C22#186111yescharacteristic
307C22#186211yescharacteristic
308C22#186611yescharacteristic
309C22#186811yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4#162511yescharacteristic
325C4#163811yescharacteristic
326C4#164111yescharacteristic
327C4#164611yescharacteristic
328C2—161no—
329C2#886311yescharacteristic
330C2#887711yescharacteristic
331C2#887811yescharacteristic
332C2#889711yescharacteristic
333C2#890311yescharacteristic
334C2#891711yescharacteristic
335C2#893011yescharacteristic
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. #8787C2semidirect
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
#87872
#85672
#75602
#70312
#70272