Finite Group Databasebeta

← #111920

Subgroups — #111920

Order 512.

1006 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2640C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic, complemented
3#21799C211yesmaximal, characteristic, complemented
4#17666C212yesmaximal
5#7855C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×#124C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2640—24no—
11#2640—24no—
12C2×#397—22no—
13C2×#393—22no—
14C2×#391C412yes—
15C2×#390C412yes—
16C2×#376C2211yescharacteristic
17#1997—24no—
18#1966—24no—
19#1821—22no—
20#1784—22no—
21#1533—24no—
22#1527—24no—
23#1527—24no—
24#1274—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8D412yes—
27C23×D4C2×C411yescharacteristic
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×#124—22no—
35C2×#124—22no—
36C2×#124—22no—
37C2×#124—22no—
38C2×#124—22no—
39C2×#124D414yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4D412yes—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—24no—
45C22×C42C2×C412yes—
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—22no—
51#431—22no—
52#431—22no—
53#431—22no—
54#431—22no—
55#423—22no—
56#423—22no—
57C2×#101—28no—
58#407—22no—
59#407—22no—
60C4×M16D414yes—
61C4×M16D414yes—
62#397—216no—
63#393—216no—
64#391—24no—
65#391—24no—
66#390—24no—
67#390—24no—
68#376—22no—
69#376D414yes—
70C2×#95—24no—
71C2×#95—21no—
72C2×#95—21no—
73#343—28no—
74#329—216no—
75C25—21no—
76C22×Q8—21no—
77C22×Q8—22no—
78C22×Q8—24no—
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—22no—
82C22×Q8—22no—
83C22×Q8—44no—
84C22×Q8—84no—
85C22×Q8C22⋊C412yes—
86C22×D4—21no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—44no—
90C22×D4C22⋊C412yes—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4C22⋊C414yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C2×M16—24no—
98C2×M16—24no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107#124—24no—
108#124—24no—
109#124—24no—
110#124—24no—
111#124—42no—
112#124—42no—
113#124—42no—
114#124—42no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—216no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—88no—
133C2×C42—22no—
134C2×C42—44no—
135C2×C42C22⋊C414yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42#4112yes—
141C2×C42#4112yes—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C4⋊C8—42no—
146#101—88no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8—42no—
151#97—24no—
152#97—24no—
153C4×C8—24no—
154C4×C8—24no—
155#95—216no—
156#95—24no—
157#95—24no—
158C24—22no—
159C24—44no—
160C24—44no—
161C24—44no—
162C24—41no—
163C24—41no—
164C24C2×C22⋊C411yescharacteristic
165C2×Q8—24no—
166C2×Q8—48no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—84no—
173C2×Q8—84no—
174C2×Q8—84no—
175C2×Q8—88no—
176C2×Q8—88no—
177C2×D4—24no—
178C2×D4—44no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—84no—
182C2×D4—84no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C4×D412yes—
201C22×C4C4×D412yes—
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12012yes—
205C22×C4#12012yes—
206C22×C4#12112yes—
207C22×C4#12112yes—
208C22×C4#12311yescharacteristic
209C22×C4#12311yescharacteristic
210M16—48no—
211M16—48no—
212C2×C8—44no—
213C2×C8—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—816no—
223C42—28no—
224C42—44no—
225C42#12012yes—
226C42#12112yes—
227C42#12112yes—
228C42#12112yes—
229C42#12112yes—
230C42#12312yes—
231C42#12412yes—
232C42#12712yes—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23—44no—
240C23—82no—
241C23—82no—
242C23—816no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23#37911yescharacteristic
248C23#38611yescharacteristic
249C23#38611yescharacteristic
250Q8—84no—
251Q8—88no—
252Q8—88no—
253Q8—84no—
254D4—84no—
255D4—84no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—88no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4#38612yes—
280C2×C4#39012yes—
281C2×C4#39212yes—
282C2×C4#39212yes—
283C2×C4#39412yes—
284C2×C4#39412yes—
285C2×C4#41314yes—
286C8—84no—
287C8—84no—
288C22—28no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—88no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—164no—
300C22—164no—
301C22#143811yescharacteristic
302C22#178512yes—
303C22#191112yes—
304C22#191212yes—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—84no—
311C4—82no—
312C4—84no—
313C4#196714yes—
314C2—22no—
315C2—22no—
316C2—22no—
317C2—162no—
318C2#1098211yescharacteristic
319C2#1098911yescharacteristic
320C2#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. #7855C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
#218292
#217992
#176662
#78552