Finite Group Databasebeta

← #134879

Subgroups — #134879

Order 512.

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

normal subgroups only

This group has 447 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#21683C211yesmaximal, characteristic
2#11821C212yesmaximal
3#8611C211yesmaximal, characteristic, complemented
4#8608C211yesmaximal, characteristic, complemented
5#7322C212yesmaximal, complemented
6C2×C8×D4—21no—
7C2×C8×D4—21no—
8C2×C8×D4C411yescharacteristic, complemented
9C2×C8×D4C2211yescharacteristic
10#2397C2211yescharacteristic
11#1955—21no—
12#1955—21no—
13#1955C411yescharacteristic, complemented
14#1955C2211yescharacteristic
15#1843C411yescharacteristic, complemented
16#1810—21no—
17#1807—21no—
18#1799C2212yes—
19#1726C411yescharacteristic, complemented
20#1465C2212yes—
21#1352—22no—
22#1343—22no—
23#1293—21no—
24#1289—21no—
25C23×C8—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C2×C4×D4C2311yescharacteristic
32C8×D4—41no—
33C8×D4—41no—
34C8×D4—41no—
35C8×D4—41no—
36C8×D4—44no—
37C8×D4—44no—
38C2×#108—41no—
39C2×#106—41no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×#99—22no—
45#407—22no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D411yescharacteristic
52C2×C22⋊C8D411yescharacteristic
53C2×#97—21no—
54C2×#97C2×C411yescharacteristic
55C2×C4×C8—21no—
56C2×C4×C8C2×C411yescharacteristic
57#399—21no—
58#399D411yescharacteristic
59#399Q811yescharacteristic
60#392—21no—
61#392D411yescharacteristic
62#392Q811yescharacteristic
63#380—22no—
64#376C2×C411yescharacteristic
65#342—22no—
66#341—41no—
67#340—41no—
68#324—22no—
69#323—22no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4C4⋊C411yescharacteristic
73C22×D4#4111yescharacteristic
74C23×C4—21no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—42no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—42no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C4⋊C4#4111yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4C2×D411yescharacteristic
114C2×C22⋊C4C2×Q811yescharacteristic
115C2×C42—21no—
116C2×C42C22×C411yescharacteristic
117#108—82no—
118#106—82no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123#99—82no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—41no—
134#97—42no—
135C4×C8—24no—
136C4×C8—21no—
137C4×C8—21no—
138#95—21no—
139#95—42no—
140#95C4⋊C411yescharacteristic
141#95C4⋊C411yescharacteristic
142C24—21no—
143C24#12111yescharacteristic
144C24C22⋊Q811yescharacteristic
145C2×D4—22no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
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—82no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4#12111yescharacteristic
178C22×C4C22⋊Q811yescharacteristic
179M16—82no—
180M16—82no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8#9911yescharacteristic
202C2×C8#9911yescharacteristic
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23#38011yescharacteristic
239C23#47112yes—
240D4—44no—
241D4—42no—
242D4—48no—
243D4—48no—
244D4—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
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×C4—162no—
276C2×C4—162no—
277C2×C4—162no—
278C2×C4C2×#9911yescharacteristic
279C2×C4#41011yescharacteristic
280C8—21no—
281C8—42no—
282C8—41no—
283C8—41no—
284C8—41no—
285C8—81no—
286C8—162no—
287C8#35312yes—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22#179911yescharacteristic
308C22#181411yescharacteristic
309C22#181511yescharacteristic
310C4—21no—
311C4—42no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—322no—
319C4C2×#35311yescharacteristic
320C4#212611yescharacteristic
321C2—21no—
322C2—21no—
323C2—42no—
324C2—81no—
325C2—82no—
326C2#862611yescharacteristic
327C2#1279711yescharacteristic
328C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radical#2397C22
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. #8611C2semidirect
4. #8608C2semidirect
5. #7322C2semidirect
8. C2×C8×D4C4semidirect
13. #1955C4semidirect
15. #1843C4semidirect
19. #1726C4semidirect

Maximal subgroups

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

subgroupindex
#216832
#118212
#86112
#86082
#73222