Finite Group Databasebeta

← #124313

Subgroups — #124313

Order 512.

616 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 116 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#14155C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7132C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#391—21no—
7C2×#391—21no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#324—21no—
13C2×#324—21no—
14#1445C412yes—
15#1445C2212yes—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C2×C4⋊Q8—22no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8C814yescomplemented
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39#391—44no—
40#391—44no—
41C4×C4⋊C4—41no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—42no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95C2311yescharacteristic
51#336—41no—
52#336—41no—
53#324—24no—
54#324—42no—
55C22×Q8—24no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—41no—
61C22×Q8—41no—
62C22×Q8—42no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C22×C8—42no—
74C4⋊Q8—44no—
75C4⋊Q8—44no—
76C4⋊Q8—44no—
77C4⋊Q8—44no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C2×C814yes—
93C2×C4⋊C4SD1612yes—
94C2×C4⋊C4Q1612yes—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42C2×C812yes—
108C2×C42M1612yes—
109C4⋊C8—82no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C4×C8—82no—
115#95—41no—
116#95—41no—
117#95—42no—
118#95—44no—
119#95—44no—
120#95—44no—
121#95—44no—
122C24C2×C22⋊C411yescharacteristic
123C2×Q8—48no—
124C2×Q8—48no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—82no—
128C2×Q8—82no—
129C2×Q8—84no—
130C2×Q8—84no—
131C2×Q8—82no—
132C2×Q8—82no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
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×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C22⋊C812yes—
175C22×C4C22⋊C812yes—
176C22×C4#9911yescharacteristic
177C22×C4#9911yescharacteristic
178C22×C4#10012yes—
179C22×C4#10312yes—
180C22×C4#10312yes—
181C22×C4#10412yes—
182C22×C4C22×C811yescharacteristic
183C22×C4C2×M1611yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185C22×C4C2×Q1611yescharacteristic
186C22×C4#13712yes—
187C2×C8—44no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—82no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—42no—
205C42—42no—
206C42—81no—
207C42—81no—
208C42—84no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23C2×C22⋊C811yescharacteristic
214C23C2×#9911yescharacteristic
215C23C2×#10011yescharacteristic
216C23C2×#10311yescharacteristic
217C23#41911yescharacteristic
218C23C2×#10411yescharacteristic
219C23#42111yescharacteristic
220Q8—88no—
221Q8—88no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4#32312yes—
276C2×C4#32312yes—
277C2×C4Q8⋊C814yes—
278C2×C4#35312yes—
279C2×C4#35512yes—
280C8—84no—
281C8—162no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22C2×#32311yescharacteristic
295C22C2×Q8⋊C811yescharacteristic
296C22#147911yescharacteristic
297C22#150811yescharacteristic
298C22#152011yescharacteristic
299C22#152711yescharacteristic
300C22#154811yescharacteristic
301C22#212311yescharacteristic
302C22#212311yescharacteristic
303C22C2×#35311yescharacteristic
304C22C2×#35511yescharacteristic
305C4—24no—
306C4—22no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—84no—
314C4—81no—
315C4—84no—
316C4#131814yes—
317C2—21no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2#757511yescharacteristic
322C2#1142811yescharacteristic
323C2C2×#131811yescharacteristic
324C2#1178811yescharacteristic
325C2#1179411yescharacteristic
326C2#1190111yescharacteristic
327C2#1190711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
#141552
#85582
C4×#3242
#71322