Finite Group Databasebeta

← #119727

Subgroups — #119727

Order 512.

846 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 126 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#21825C211yesmaximal, characteristic
2#17397C211yesmaximal, characteristic
3#14159C211yesmaximal, characteristic
4#8051C212yesmaximal
5#8020C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8#2280C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11#1996—24no—
12#1819—22no—
13#1472—24no—
14#1463—22no—
15#1446C412yescomplemented
16#1445C412yescomplemented
17#1436—22no—
18C22⋊#95—22no—
19#1297—24no—
20#1294—24no—
21C23×Q8C2×C411yescharacteristic
22C24×C4—21no—
23C24×C4C2311yescharacteristic
24C22×M16—22no—
25C2×C22⋊Q8—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×#117—22no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4D412yes—
36C22×C42—21no—
37#407—22no—
38#407—22no—
39#407—22no—
40#407D414yes—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43#381—22no—
44#380—22no—
45#380D414yes—
46#379—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—24no—
54C2×#95—21no—
55C2×#95—24no—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58#342—28no—
59#324—28no—
60C25C2×D411yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—82no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—44no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—44no—
72C23×C4—82no—
73C23×C4—82no—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×D412yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4112yes—
83C23×C4#4112yes—
84C2×M16—24no—
85C2×M16—42no—
86C2×M16—84no—
87C22×C8—42no—
88C22⋊Q8—42no—
89C22⋊Q8—42no—
90C22⋊Q8—42no—
91C22⋊Q8—42no—
92C22⋊Q8—42no—
93C22⋊Q8—42no—
94C22⋊Q8—42no—
95C22⋊Q8—42no—
96#117—84no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C42—41no—
117C2×C42—42no—
118C2×C42—41no—
119C2×C42—82no—
120C22⋊C8—24no—
121C22⋊C8—24no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—48no—
128#95—42no—
129#95—42no—
130#95—84no—
131C24—22no—
132C24—22no—
133C24—42no—
134C24—42no—
135C24C2×C22⋊C411yescharacteristic
136C24#12011yescharacteristic
137C24#12112yes—
138C24#12312yes—
139C24#12311yescharacteristic
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8—84no—
145C2×Q8—84no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—816no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4#11711yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4#12011yescharacteristic
185C22×C4#12112yes—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4#12312yes—
189C22×C4#12312yes—
190C22×C4#12312yes—
191C22×C4#12311yescharacteristic
192C22×C4#12412yes—
193C22×C4#12412yes—
194C22×C4#12612yes—
195C22×C4#12612yes—
196M16—84no—
197M16—84no—
198C2×C8—44no—
199C2×C8—84no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—84no—
205C22⋊C4—48no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—84no—
211C22⋊C4—84no—
212C42—82no—
213C42—82no—
214C42—84no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—24no—
220C23—24no—
221C23—21no—
222C23—21no—
223C23—24no—
224C23—24no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—42no—
230C23—42no—
231C23—48no—
232C23—48no—
233C23#38111yescharacteristic
234C23#38611yescharacteristic
235C23#38611yescharacteristic
236C23#39412yes—
237C23#39712yes—
238C23#45412yes—
239C23#45612yes—
240Q8—84no—
241Q8—84no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—48no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—82no—
271C2×C4—88no—
272C2×C4—88no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4#38812yes—
282C2×C4#38812yes—
283C2×C4#39412yes—
284C2×C4#39712yes—
285C8—84no—
286C22—24no—
287C22—24no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—48no—
302C22—48no—
303C22—42no—
304C22—42no—
305C22#144011yescharacteristic
306C22#182812yes—
307C22#188412yes—
308C22#188611yescharacteristic
309C22#188811yescharacteristic
310C22#203814yes—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—84no—
316C4—84no—
317C4—82no—
318C4—82no—
319C4—164no—
320C2—21no—
321C2—22no—
322C2—24no—
323C2—22no—
324C2—21no—
325C2—42no—
326C2#1116011yescharacteristic
327C2#1116211yescharacteristic
328C2#1116611yescharacteristic
subgroupG/H
center, socleC22#1440
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. #1446C4semidirect
16. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#218252
#173972
#141592
#80512
#80202