Finite Group Databasebeta

← #98972

Subgroups — #98972

Order 512.

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

normal subgroups only

This group has 451 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#21487C211yesmaximal, characteristic
2#10822C211yesmaximal, characteristic, complemented
3#8596C211yesmaximal, characteristic, complemented
4#7616C212yesmaximal
5#7045C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×#435—21no—
8C22×C4×C8C2211yescharacteristic
9C4×#121C2211yescharacteristic
10#1954C411yescharacteristic
11#1939C411yescharacteristic
12#1923—21no—
13#1869—21no—
14#1797C2212yes—
15#1797C2211yescharacteristic
16#1770C411yescharacteristic
17#1753—21no—
18C2×#341—21no—
19C2×#341—21no—
20#1728C2212yes—
21C2×#335—21no—
22#1277—21no—
23#1271—22no—
24#1270—22no—
25C22×M16—21no—
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#121Q811yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33#436—42no—
34#435—44no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—22no—
44C2×#97—21no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#390D411yescharacteristic
51#389D411yescharacteristic
52#388—21no—
53#386—21no—
54#380Q811yescharacteristic
55C4×C4⋊C4C2×C411yescharacteristic
56C4×C22⋊C4—21no—
57#341—24no—
58#341—42no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—22no—
63#336—22no—
64#336—22no—
65#335—42no—
66C22×D4—21no—
67C22×D4C4⋊C411yescharacteristic
68C23×C4—21no—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×Q811yescharacteristic
71C23×C4#4111yescharacteristic
72C2×M16—24no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95#121—42no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—42no—
99C4×D4—42no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C4⋊C4C4⋊C411yescharacteristic
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42#4111yescharacteristic
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—82no—
126#97—42no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—41no—
130C4×C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134C24—21no—
135C24C22⋊Q811yescharacteristic
136C2×D4—22no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C2×C4⋊C411yescharacteristic
164C22×C4#11711yescharacteristic
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4C22⋊Q811yescharacteristic
168C22×C4#13512yes—
169C22×C4AGL(1,ℤ/8)12yes—
170M16—44no—
171M16—44no—
172M16—84no—
173M16—82no—
174M16—82no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—44no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8#9912yes—
200C2×C8#10412yes—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—41no—
218C42—41no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#38011yescharacteristic
233C23#47812yes—
234D4—82no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
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—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4C2×#9911yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4#41611yescharacteristic
271C2×C4#41811yescharacteristic
272C2×C4C2×#10411yescharacteristic
273C2×C4#42111yescharacteristic
274C2×C4#43012yes—
275C2×C4#46312yes—
276C2×C4#47211yescharacteristic
277C2×C4#47311yescharacteristic
278C8—22no—
279C8—22no—
280C8—41no—
281C8—41no—
282C8—84no—
283C8—81no—
284C8—81no—
285C8—162no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#179911yescharacteristic
301C22#180411yescharacteristic
302C22#180611yescharacteristic
303C22#181111yescharacteristic
304C22#181311yescharacteristic
305C22#181411yescharacteristic
306C22#181611yescharacteristic
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#150011yescharacteristic
317C4#150711yescharacteristic
318C4#151411yescharacteristic
319C4#151611yescharacteristic
320C2—42no—
321C2—161no—
322C2#857311yescharacteristic
323C2#861311yescharacteristic
324C2#862711yescharacteristic
325C2#866411yescharacteristic
326C2#867311yescharacteristic
327C2#868811yescharacteristic
328C2#869211yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10822C2semidirect
3. #8596C2semidirect
5. #7045C2semidirect

Maximal subgroups

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

subgroupindex
#214872
#108222
#85962
#76162
#70452