Finite Group Databasebeta

← #116865

Subgroups — #116865

Order 512.

740 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 132 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#21939C211yesmaximal, characteristic
2#20688C212yesmaximal
3#20685C211yesmaximal, characteristic
4#7982C211yesmaximal, characteristic
5#7946C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9#2280C2212yes—
10#2280C2212yes—
11#1854—24no—
12#1820—24no—
13#1819—24no—
14#1818—22no—
15#1817C414yescomplemented
16#1808—21no—
17#1807—21no—
18#1740—24no—
19#1729—21no—
20#1729—21no—
21#1289—22no—
22#1284—24no—
23C24×C4C2311yescharacteristic
24C22×M16—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—21no—
30C2×#117—22no—
31C2×#117—21no—
32C2×#117—21no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117D414yes—
37C22×C4⋊C4Q811yescharacteristic
38C22×C4⋊C4Q811yescharacteristic
39C22×C22⋊C4D412yes—
40C22×C42C2×C412yes—
41C2×#107—22no—
42C2×#106—22no—
43#407—22no—
44#407—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C412yes—
49C2×C22⋊C8C2×C412yes—
50C2×C22⋊C8D412yes—
51#381—22no—
52#381D414yes—
53#380—22no—
54#380—22no—
55#380—22no—
56#380—22no—
57#379—22no—
58#379—22no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63#342—216no—
64#340—22no—
65#340—22no—
66#336—24no—
67#336—24no—
68C25C2×D411yescharacteristic
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C4⋊C412yes—
72C23×C4C4⋊C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C23×C4C2×Q811yescharacteristic
79C23×C4#4111yescharacteristic
80C23×C4#4111yescharacteristic
81C23×C4#4112yes—
82C23×C4#4112yes—
83C2×M16—42no—
84C2×M16—42no—
85C2×M16—41no—
86C2×M16—41no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8C22⋊C412yes—
95C22×C8C22⋊C412yes—
96#117—41no—
97#117—42no—
98#117—42no—
99#117—42no—
100#117—41no—
101#117—42no—
102#117—42no—
103#117—41no—
104#117—42no—
105#117—41no—
106#117—44no—
107#117—44no—
108#117—44no—
109#117—44no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—42no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—42no—
136#107—82no—
137#106—82no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—42no—
142#95—24no—
143#95—24no—
144#95—42no—
145#95—42no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24C2×C4⋊C411yescharacteristic
150C24#12012yes—
151C24#12111yescharacteristic
152C24#12111yescharacteristic
153C24#12312yes—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—82no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4#11711yescharacteristic
179C22×C4C4×D412yes—
180C22×C4C4×D412yes—
181C22×C4#12011yescharacteristic
182C22×C4#12112yes—
183C22×C4#12112yes—
184C22×C4#12112yes—
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q812yes—
189C22×C4#12311yescharacteristic
190C22×C4#12312yes—
191C22×C4#12412yes—
192C22×C4#12411yescharacteristic
193C22×C4#12612yes—
194C22×C4#12711yescharacteristic
195M16—82no—
196M16—82no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—81no—
207C2×C8—81no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C22⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C22⋊C4—84no—
224C22⋊C4—84no—
225C22⋊C4—84no—
226C22⋊C4—84no—
227C42—42no—
228C42—44no—
229C42—42no—
230C42—84no—
231C42—84no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—88no—
248C23#38011yescharacteristic
249C23#38512yes—
250C23#39211yescharacteristic
251C23#39412yes—
252C23#39711yescharacteristic
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4#38611yescharacteristic
287C2×C4#38812yes—
288C2×C4#39011yescharacteristic
289C2×C4#39311yescharacteristic
290C2×C4#39612yes—
291C2×C4#39911yescharacteristic
292C2×C4#45812yes—
293C2×C4#45812yes—
294C8—42no—
295C8—42no—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—44no—
311C22—44no—
312C22—84no—
313C22—82no—
314C22—84no—
315C22#143911yescharacteristic
316C22#181212yes—
317C22#202212yes—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—21no—
322C4—82no—
323C4—82no—
324C4—164no—
325C4—164no—
326C4#189011yescharacteristic
327C4#189011yescharacteristic
328C4#203012yes—
329C2—22no—
330C2—22no—
331C2—42no—
332C2—42no—
333C2—41no—
334C2#1107011yescharacteristic
335C2#1109811yescharacteristic
336C2#1110211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1817C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#206882
#206852
#79822
#79462