Finite Group Databasebeta

← #115654

Subgroups — #115654

Order 512.

1310 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1310 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2542C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic, complemented
3#21799C211yesmaximal, characteristic, complemented
4#17448C212yesmaximal
5#7950C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8—22no—
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2542—24no—
12#2542—24no—
13C2×#399—22no—
14C2×#390C412yes—
15C2×#384C412yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1997—24no—
18#1966—24no—
19#1821—22no—
20#1784—22no—
21#1490—24no—
22#1486—24no—
23#1478—24no—
24#1285—24no—
25C23×Q8D412yes—
26C23×D4C2×C411yescharacteristic
27C24×C4—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C22⋊Q8—216no—
35C2×C22⋊Q8—416no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4D414yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—21no—
49C22×C42C2×C412yes—
50C22×C42C2311yescharacteristic
51#432—22no—
52#432—22no—
53#432—22no—
54#432—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#423—22no—
60#423D414yes—
61C2×#101—28no—
62#407—22no—
63#407—22no—
64C4×M16—22no—
65C4×M16D414yes—
66#399—216no—
67#390—24no—
68#390—24no—
69#384—24no—
70#384—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4D414yes—
73C2×#95—21no—
74#343—28no—
75D4⋊C8—216no—
76C25—21no—
77C22×Q8—24no—
78C22×Q8—22no—
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—48no—
82C22×D4—24no—
83C22×D4—21no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—42no—
87C22×D4C22⋊C412yes—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—24no—
91C23×C4—48no—
92C23×C4—41no—
93C23×C4—42no—
94C23×C4—41no—
95C23×C4C22⋊C414yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×M16—24no—
100C2×M16—24no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C22⋊Q8—464no—
110C4×D4—24no—
111C4×D4—24no—
112C4×D4—24no—
113C4×D4—24no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—216no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—416no—
133C2×C4⋊C4C22⋊C412yes—
134C2×C22⋊C4—216no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—416no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—41no—
140C2×C42—22no—
141C2×C42—44no—
142C2×C42C22⋊C414yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153#101—88no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#97—24no—
159#97—42no—
160C4×C8—24no—
161C4×C8—42no—
162#95—24no—
163C24—24no—
164C24—22no—
165C24—44no—
166C24—42no—
167C24—44no—
168C24—41no—
169C24—41no—
170C24C2×C22⋊C411yescharacteristic
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—84no—
176C2×Q8—88no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—48no—
182C2×D4—48no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4#10114yescomplemented
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—42no—
196C22×C4—416no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—84no—
206C22×C4—88no—
207C22×C4—84no—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4#12011yescharacteristic
213C22×C4#12012yes—
214C22×C4#12012yes—
215C22×C4#12011yescharacteristic
216C22×C4#12112yes—
217C22×C4#12112yes—
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220M16—48no—
221M16—48no—
222C2×C8—44no—
223C2×C8—44no—
224C4⋊C4—216no—
225C4⋊C4—24no—
226C4⋊C4—432no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C22⋊C4—432no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—44no—
236C42—44no—
237C42#12012yes—
238C42#12112yes—
239C42#12112yes—
240C42#12112yes—
241C42#12112yes—
242C42#12312yes—
243C42#12412yes—
244C42#12712yes—
245C23—22no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—24no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—416no—
254C23—44no—
255C23—42no—
256C23—42no—
257C23—82no—
258C23—88no—
259C23—82no—
260C23—82no—
261C23—82no—
262C23#37911yescharacteristic
263C23#38611yescharacteristic
264C23#38611yescharacteristic
265Q8—88no—
266Q8—88no—
267D4—28no—
268D4—48no—
269D4—44no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—416no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—88no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—88no—
294C2×C4—84no—
295C2×C4#38612yes—
296C2×C4#39012yes—
297C2×C4#39212yes—
298C2×C4#39212yes—
299C2×C4#39412yes—
300C2×C4#39412yes—
301C2×C4C2×#10112yes—
302C2×C4#41312yes—
303C2×C4#45314yes—
304C8—84no—
305C8—84no—
306C22—24no—
307C22—24no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—48no—
315C22—44no—
316C22—84no—
317C22—88no—
318C22—88no—
319C22—82no—
320C22—82no—
321C22#143811yescharacteristic
322C22#178411yescharacteristic
323C22#178511yescharacteristic
324C22#188712yes—
325C22#191212yes—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—48no—
331C4—84no—
332C4—82no—
333C4#190114yes—
334C4#200214yes—
335C2—21no—
336C2—21no—
337C2—22no—
338C2—22no—
339C2—84no—
340C2#1096811yescharacteristic
341C2#1097511yescharacteristic
342C2#1099011yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21829C2semidirect
3. #21799C2semidirect
5. #7950C2semidirect
185. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#218292
#217992
#174482
#79502