Finite Group Databasebeta

← #134827

Subgroups — #134827

Order 512.

424 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 74 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#21682C211yesmaximal, characteristic
2#11529C212yesmaximal
3#8612C211yesmaximal, characteristic, complemented
4#8611C211yesmaximal, characteristic, complemented
5#7249C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×#435—21no—
8C2×#435—21no—
9C2×C8×D4C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#1955—21no—
12#1955—21no—
13#1954C411yescharacteristic
14#1954C2211yescharacteristic
15#1842C411yescharacteristic
16#1810—21no—
17#1810—21no—
18C4×#97C411yescharacteristic
19#1465C2212yes—
20#1465C2212yes—
21#1325—22no—
22#1318—22no—
23#1293—21no—
24#1293—21no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#127Q811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4—21no—
31C2×C4×D4C2×C411yescharacteristic
32C2×C4×D4C2311yescharacteristic
33#436—41no—
34#436—41no—
35#435—44no—
36#435—44no—
37C8×D4—41no—
38C8×D4—41no—
39C2×#108—41no—
40C2×#108—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8C2×C411yescharacteristic
45#407—22no—
46#407—22no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8D411yescharacteristic
53C2×C22⋊C8D411yescharacteristic
54C2×#97—21no—
55C2×#97—21no—
56C2×C4×C8C2×C411yescharacteristic
57C2×C4×C8C2×C411yescharacteristic
58#390—21no—
59#390—21no—
60#390D411yescharacteristic
61#390Q811yescharacteristic
62#384D411yescharacteristic
63C43C2×C411yescharacteristic
64#341—41no—
65#341—41no—
66#324—22no—
67#324—22no—
68#323—22no—
69#323—22no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4C4⋊C411yescharacteristic
73C22×D4C2×Q811yescharacteristic
74C23×C4—21no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95#127—42no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—41no—
103C4×D4—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C4⋊C411yescharacteristic
107C2×C4⋊C4C2×D411yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4#4111yescharacteristic
114C2×C22⋊C4#4111yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C4⋊C411yescharacteristic
118C2×C42C4⋊C411yescharacteristic
119C2×C42C22×C411yescharacteristic
120#108—82no—
121#108—82no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#97—24no—
138#97—42no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—41no—
142C24—21no—
143C24C22⋊Q811yescharacteristic
144C24C22⋊Q811yescharacteristic
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4—81no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179M16—82no—
180M16—82no—
181M16—82no—
182M16—82no—
183M16—82no—
184M16—82no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8#9911yescharacteristic
205C2×C8#9911yescharacteristic
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—22no—
218C42—22no—
219C42—21no—
220C42—21no—
221C42—22no—
222C42—41no—
223C42—81no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23#38011yescharacteristic
242D4—44no—
243D4—82no—
244D4—84no—
245D4—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
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—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4C2×#9911yescharacteristic
276C2×C4#41011yescharacteristic
277C2×C4#43012yes—
278C8—21no—
279C8—42no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—81no—
284C8—162no—
285C8—162no—
286C8#35312yes—
287C22—21no—
288C22—21no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#179911yescharacteristic
307C22#181111yescharacteristic
308C22#181311yescharacteristic
309C4—22no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—81no—
317C4C2×#35311yescharacteristic
318C4#212611yescharacteristic
319C2—21no—
320C2—21no—
321C2—81no—
322C2—81no—
323C2—161no—
324C2#861311yescharacteristic
325C2#1279711yescharacteristic
326C2#1281011yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#127Q8
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
3. #8612C2semidirect
4. #8611C2semidirect
5. #7249C2semidirect

Maximal subgroups

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

subgroupindex
#216822
#115292
#86122
#86112
#72492