Finite Group Databasebeta

← #111913

Subgroups — #111913

Order 512.

1314 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 158 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#21790C211yesmaximal, characteristic
3C2×#1556C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5C2×#1285C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8—22no—
8C22×C4×D4C2211yescharacteristic
9C2×#432C2212yes—
10C2×#432C2212yes—
11#2542—24no—
12#2542—24no—
13C2×#399—22no—
14C2×#390C412yes—
15C2×#384C412yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1955—24no—
18#1779—22no—
19#1556—24no—
20#1556—24no—
21C2×#332—24no—
22#1466—24no—
23C2×#323—22no—
24#1285—24no—
25C23×Q8—21no—
26C23×D4C2×C411yescharacteristic
27C24×C4D412yes—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D414yes—
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×D4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4D412yes—
49C22×C42C2×C412yes—
50C22×C42C2311yescharacteristic
51#432—22no—
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432D414yes—
59C2×C4⋊C8—22no—
60C2×C4⋊C8D414yes—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—28no—
64C2×#97—22no—
65C2×#97—22no—
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#332—216no—
75#323—28no—
76C25C22⋊C412yes—
77C22×Q8—24no—
78C22×Q8—48no—
79C22×Q8—41no—
80C22×Q8—42no—
81C22×Q8—41no—
82C22×D4—24no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—42no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—48no—
95C23×C4C22⋊C412yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—24no—
100C22×C8—42no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C22⋊Q8—464no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—216no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—416no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4C22⋊C414yes—
134C2×C22⋊C4—216no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—416no—
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—24no—
152C4⋊C8—42no—
153C22⋊C8—24no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—88no—
158#97—24no—
159#97—42no—
160#97—42no—
161#97—42no—
162#95—24no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—44no—
170C24—44no—
171C24#10114yes—
172C24C2×C22⋊C411yescharacteristic
173C2×Q8—416no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—84no—
177C2×Q8—88no—
178C2×Q8—84no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—48no—
182C2×D4—48no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—416no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—84no—
208C22×C4—88no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C4×D412yes—
212C22×C4C4×D412yes—
213C22×C4#12012yes—
214C22×C4#12011yescharacteristic
215C22×C4#12012yes—
216C22×C4#12011yescharacteristic
217C22×C4#12112yes—
218C22×C4#12112yes—
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—48no—
224C2×C8—84no—
225C4⋊C4—216no—
226C4⋊C4—28no—
227C4⋊C4—432no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C22⋊C4—432no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C42—44no—
237C42—44no—
238C42#12012yes—
239C42#12112yes—
240C42#12112yes—
241C42#12112yes—
242C42#12112yes—
243C42#12312yes—
244C42#12412yes—
245C42#12712yes—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—48no—
252C23—44no—
253C23—44no—
254C23—48no—
255C23—44no—
256C23—42no—
257C23—42no—
258C23—44no—
259C23—42no—
260C23—42no—
261C23—84no—
262C23—82no—
263C23—82no—
264C23—84no—
265C23#37911yescharacteristic
266C23#38611yescharacteristic
267C23#38611yescharacteristic
268C23C2×#10112yes—
269C23#41312yes—
270Q8—88no—
271Q8—88no—
272D4—48no—
273D4—48no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—416no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—88no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—88no—
300C2×C4—82no—
301C2×C4#38612yes—
302C2×C4#39012yes—
303C2×C4#39212yes—
304C2×C4#39212yes—
305C2×C4#39412yes—
306C2×C4#39412yes—
307C2×C4#45314yes—
308C8—84no—
309C8—84no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—24no—
316C22—48no—
317C22—44no—
318C22—48no—
319C22—84no—
320C22—84no—
321C22—84no—
322C22—82no—
323C22—82no—
324C22—84no—
325C22#143811yescharacteristic
326C22#178411yescharacteristic
327C22#178511yescharacteristic
328C22#188712yes—
329C22#190112yes—
330C22#190112yes—
331C22#190912yes—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—48no—
338C4—82no—
339C4—84no—
340C4#200214yes—
341C2—21no—
342C2—21no—
343C2—22no—
344C2—84no—
345C2#945514yes—
346C2#1096811yescharacteristic
347C2#1097511yescharacteristic
348C2#1098811yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1556C2semidirect
5. C2×#1285C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#217902
C2×#15562
C2×#14662
C2×#12852