Finite Group Databasebeta

← #127335

Subgroups — #127335

Order 512.

877 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 877 conjugacy classes of subgroups — too many to draw. Only its 180 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#21786C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3#7237C214yesmaximal, complemented
4C22×C4×D4C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C2×#431C411yescharacteristic, complemented
7C22×#99—22no—
8C2×#380—22no—
9#1954—22no—
10#1954C412yescomplemented
11#1954C2212yes—
12#1799—22no—
13#1799—22no—
14#1799C414yescomplemented
15#1799C2214yes—
16#1779—22no—
17C2×#342C414yescomplemented
18C23×D4D411yescharacteristic
19C24×C4D412yes—
20C2×C4×D4—42no—
21C2×C4×D4—42no—
22C2×C4×D4C2×C412yes—
23C2×C4×D4C2×C411yescharacteristic
24C2×C4×D4D412yes—
25C2×C4×D4D411yescharacteristic
26C2×C4×D4Q811yescharacteristic
27C2×C4×D4C2311yescharacteristic
28C22×C4⋊C4—42no—
29C22×C4⋊C4Q811yescharacteristic
30C22×C22⋊C4—24no—
31C22×C22⋊C4C2×C412yes—
32C22×C42C2×C411yescharacteristic
33#431—42no—
34#431—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C412yescomplemented
37C2×#99—24no—
38C2×#99—24no—
39C2×#99—44no—
40C2×#99—44no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45C2×C4×C8—21no—
46C2×C4×C8C2×C412yescomplemented
47#380—24no—
48#380—24no—
49#380—42no—
50#380—42no—
51C2×#95—42no—
52#342—24no—
53#342C2×C418yescomplemented
54C25C22⋊C412yes—
55C22×D4—42no—
56C22×D4—42no—
57C22×D4C22⋊C412yes—
58C22×D4C4⋊C411yescharacteristic
59C22×D4D812yes—
60C22×D4SD1614yes—
61C22×D4Q1612yes—
62C22×D4C2×D411yescharacteristic
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—44no—
68C23×C4—84no—
69C23×C4C4211yescharacteristic
70C23×C4C22⋊C412yes—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C4⋊C411yescharacteristic
74C23×C4C4⋊C412yes—
75C23×C4C4⋊C412yes—
76C23×C4C2×D412yes—
77C22×C8—22no—
78C22×C8—22no—
79C4×D4—42no—
80C4×D4—42no—
81C4×D4—41no—
82C4×D4—41no—
83C4×D4—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—42no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—88no—
93C2×C4⋊C4C4⋊C412yes—
94C2×C4⋊C4C4⋊C411yescharacteristic
95C2×C4⋊C4C2×Q811yescharacteristic
96C2×C22⋊C4—28no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—48no—
100C2×C22⋊C4—48no—
101C2×C22⋊C4—44no—
102C2×C22⋊C4C4212yes—
103C2×C22⋊C4C22⋊C412yes—
104C2×C22⋊C4C4⋊C412yes—
105C2×C22⋊C4C22×C412yes—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42C4211yescharacteristic
109C2×C42C22⋊C411yescharacteristic
110C2×C42C4⋊C411yescharacteristic
111C2×C42C22×C411yescharacteristic
112C4⋊C8—42no—
113C4⋊C8—42no—
114#99—48no—
115#99—44no—
116#99—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—84no—
120C22⋊C8—84no—
121C4×C8—42no—
122C4×C8—42no—
123#95—44no—
124#95—44no—
125C24—22no—
126C24—24no—
127C24—22no—
128C24—82no—
129C24—82no—
130C24#9512yes—
131C24#9511yescharacteristic
132C24#10214yes—
133C24#10314yes—
134C24C2×C22⋊C412yes—
135C2×D4—42no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—42no—
139C2×D4—41no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—41no—
145C2×D4#10214yes—
146C2×D4#10314yes—
147C2×D4#10612yes—
148C2×D4#10712yes—
149C2×D4C2×D811yescharacteristic
150C2×D4C2×SD1612yes—
151C2×D4C2×Q1611yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—48no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—48no—
170C22×C4—84no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—81no—
177C22×C4—88no—
178C22×C4—88no—
179C22×C4—88no—
180C22×C4#9511yescharacteristic
181C22×C4#9511yescharacteristic
182C22×C4#9512yes—
183C22×C4#9511yescharacteristic
184C22×C4#9912yes—
185C22×C4#9912yes—
186C22×C4#10612yes—
187C22×C4#10712yes—
188C22×C4C2×C4211yescharacteristic
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C2×C4⋊C411yescharacteristic
191C22×C4C2×C4⋊C412yes—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—88no—
201C4⋊C4—88no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—48no—
207C22⋊C4—48no—
208C22⋊C4—88no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C23—24no—
216C23—24no—
217C23—22no—
218C23—22no—
219C23—24no—
220C23—24no—
221C23—22no—
222C23—22no—
223C23—41no—
224C23—41no—
225C23—44no—
226C23—44no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23—84no—
235C23—88no—
236C23H(3,ℤ/4)12yes—
237C23#34014yes—
238C23#34012yes—
239C23#34212yes—
240C23C2×#9511yescharacteristic
241C23C2×#10212yes—
242C23C2×#10312yes—
243D4—44no—
244D4—44no—
245D4—42no—
246D4—42no—
247D4—41no—
248D4—42no—
249D4—41no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—88no—
280C2×C4—88no—
281C2×C4—88no—
282C2×C4—1616no—
283C2×C4#34012yes—
284C2×C4#34212yes—
285C2×C4C2×#9911yescharacteristic
286C2×C4C2×#9911yescharacteristic
287C2×C4C2×#10611yescharacteristic
288C2×C4C2×#10711yescharacteristic
289C8—82no—
290C8—82no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—22no—
298C22—42no—
299C22—42no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—48no—
304C22—48no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22—84no—
312C22—84no—
313C22#128914yes—
314C22C2×H(3,ℤ/4)11yescharacteristic
315C22C2×#34011yescharacteristic
316C22C2×#34211yescharacteristic
317C4—21no—
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—168no—
327C4#212712yes—
328C2—21no—
329C2—24no—
330C2—21no—
331C2—42no—
332C2—44no—
333C2—42no—
334C2C2×#128911yescharacteristic
335C2#1257711yescharacteristic
336C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7237C2semidirect
5. C2×#431C4semidirect
6. C2×#431C4semidirect
10. #1954C4semidirect
14. #1799C4semidirect
17. C2×#342C4semidirect
36. C2×C4⋊C8C2×C4semidirect
46. C2×C4×C8C2×C4semidirect
53. #342C2×C4semidirect

Maximal subgroups

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

subgroupindex
#217862
C2×#17992
#72372