Finite Group Databasebeta

← #129704

Subgroups — #129704

Order 512.

650 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×#2991C211yesmaximal, characteristic
2#11639C212yesmaximal, complemented
3#11634C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic, complemented
5#11459C211yesmaximal, characteristic, complemented
6C2×C4×#41C411yescharacteristic
7#2991—24no—
8#2991—24no—
9C2×C8×Q8C411yescharacteristic
10C2×C8×Q8C2211yescharacteristic
11C2×#436—21no—
12C2×#436—21no—
13C2×#436C411yescharacteristic
14C2×#436C2211yescharacteristic
15C2×#431—21no—
16C2×#431C2211yescharacteristic
17C2×#423—21no—
18C2×#423C411yescharacteristic
19#2163C2212yes—
20#2162C2212yes—
21Q8⋊C16—24no—
22#1330—24no—
23#1322—21no—
24#1322—21no—
25#1314—21no—
26#1314—21no—
27C22×#41C811yescharacteristic
28C22×M16—21no—
29C22×M16—21no—
30C22×M16—21no—
31C4×#41—22no—
32C4×#41—22no—
33C4×#41—22no—
34C4×#41—22no—
35C2×C4×Q8C811yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C811yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×#117—21no—
42C2×#117C811yescharacteristic
43C22×C42—21no—
44C22×C42C2×C411yescharacteristic
45C8×Q8—22no—
46C8×Q8—22no—
47C8×Q8—22no—
48C8×Q8D414yes—
49#436—28no—
50#436—28no—
51#436—22no—
52#436—22no—
53#436—22no—
54#436D414yes—
55#431—28no—
56#431—22no—
57#431—22no—
58#423—28no—
59#423—22no—
60#423—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C4⋊C8C2×C411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8D411yescharacteristic
71C2×C22⋊C8D411yescharacteristic
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C4⋊C16—22no—
77C4⋊C16—22no—
78C22⋊C16—22no—
79C22⋊C16—22no—
80#346—22no—
81C4×C16—22no—
82C2×#41—22no—
83C2×#41—22no—
84C2×#41—44no—
85C22×Q8C2×C811yescharacteristic
86C22×D4—21no—
87C22×D4C2×C811yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C2×C811yescharacteristic
93C23×C4M1611yescharacteristic
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—82no—
98C2×M16—82no—
99C2×M16—82no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C4×Q8—22no—
107C4×Q8—22no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8C22⋊C412yes—
111C4×Q8M1612yes—
112C4×D4—24no—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4C22⋊C412yes—
121C4×D4M1612yes—
122#117—24no—
123#117—24no—
124#117—22no—
125#117—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2×C811yescharacteristic
129C2×C4⋊C4C2×C811yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42C22⋊C411yescharacteristic
143C2×C42C2×C811yescharacteristic
144C2×C42M1611yescharacteristic
145C2×C42C22×C411yescharacteristic
146C2×C16—42no—
147C2×C16—42no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8C22⋊C412yes—
156C4⋊C8C22⋊C412yes—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C4×C8—24no—
162C4×C8—24no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8C2×D412yes—
166C4×C8C2×D412yes—
167C24—21no—
168C24C22⋊C811yescharacteristic
169#41—48no—
170#41—44no—
171#41—44no—
172C2×Q8—24no—
173C2×Q8—21no—
174C2×Q8C22⋊C812yes—
175C2×D4—24no—
176C2×D4—21no—
177C2×D4—44no—
178C2×D4—42no—
179C2×D4C22⋊C812yes—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4C22⋊C811yescharacteristic
204C22×C4C22⋊C811yescharacteristic
205C22×C4C22⋊C811yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C22×C811yescharacteristic
208C22×C4C2×M1611yescharacteristic
209M16—84no—
210M16—84no—
211M16—84no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8#12012yes—
223C2×C8#12012yes—
224C4⋊C4—24no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4C22⋊C812yes—
230C4⋊C4C22⋊C812yes—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C42—24no—
236C42—24no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—21no—
244C42—21no—
245C42C2×C22⋊C411yescharacteristic
246C42C2×C22⋊C411yescharacteristic
247C42C2×M1611yescharacteristic
248C42C2×M1611yescharacteristic
249C16—82no—
250C16—82no—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—21no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C22⋊C811yescharacteristic
262Q8—24no—
263Q8—22no—
264Q8#34914yes—
265D4—42no—
266D4—48no—
267D4—42no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#35012yes—
299C2×C4#37911yescharacteristic
300C2×C4C2×C22⋊C811yescharacteristic
301C2×C4C2×C22⋊C811yescharacteristic
302C2×C4#40711yescharacteristic
303C2×C4#40711yescharacteristic
304C2×C4#40711yescharacteristic
305C8—22no—
306C8—22no—
307C8—42no—
308C8—42no—
309C8—41no—
310C8—41no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22#177911yescharacteristic
324C22C2×#35011yescharacteristic
325C22#211711yescharacteristic
326C4—21no—
327C4—24no—
328C4—21no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—42no—
334C4—41no—
335C4—42no—
336C4#179511yescharacteristic
337C4#179611yescharacteristic
338C4C2×#34912yes—
339C2—21no—
340C2—21no—
341C2—81no—
342C2—82no—
343C2#832411yescharacteristic
344C2#1268111yescharacteristic
345C2#1268311yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11639C2semidirect
3. #11634C2semidirect
4. #11460C2semidirect
5. #11459C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
#116392
#116342
#114602
#114592