Finite Group Databasebeta

← #119540

Subgroups — #119540

Order 512.

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

normal subgroups only

This group has 856 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#21753C211yesmaximal, characteristic
3#14159C211yesmaximal, characteristic
4#8020C212yesmaximal
5#7942C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8#2280C2212yes—
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C2211yescharacteristic
11#1996—24no—
12#1901—24no—
13#1821—21no—
14#1819—22no—
15#1784—21no—
16#1446—21no—
17#1445—21no—
18#1436C414yescomplemented
19C22⋊#95—22no—
20#1294—24no—
21#1284—24no—
22C23×Q8D411yescharacteristic
23C24×C4—21no—
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×#117—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4D412yes—
37C22×C42C2×C412yes—
38C2×#101—24no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407D414yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45#381—22no—
46#380—22no—
47#380D414yes—
48#379—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—24no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—24no—
59C2×#95C2×C412yes—
60#343—24no—
61#342—28no—
62C25C2×D411yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—21no—
67C22×Q8—82no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—44no—
74C23×C4—82no—
75C23×C4—82no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4#4112yes—
83C23×C4#4112yes—
84C23×C4#4112yes—
85C23×C4#4112yes—
86C2×M16—24no—
87C2×M16—42no—
88C2×M16—84no—
89C22×C8—42no—
90C22⋊Q8—42no—
91C22⋊Q8—42no—
92C22⋊Q8—42no—
93C22⋊Q8—42no—
94C22⋊Q8—42no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98#117—84no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C42—24no—
119C2×C42—42no—
120C2×C42—82no—
121#101—84no—
122C22⋊C8—24no—
123C22⋊C8—24no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#95—24no—
127#95—24no—
128#95—48no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—41no—
133#95—84no—
134C24—22no—
135C24—22no—
136C24—42no—
137C24—42no—
138C24C2×C22⋊C411yescharacteristic
139C24#12011yescharacteristic
140C24#12112yes—
141C24#12312yes—
142C24#12311yescharacteristic
143C2×Q8—44no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—82no—
147C2×Q8—84no—
148C2×Q8—84no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—816no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4#11711yescharacteristic
184C22×C4#11711yescharacteristic
185C22×C4C4×D412yes—
186C22×C4C4×D412yes—
187C22×C4#12011yescharacteristic
188C22×C4#12112yes—
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C22×C4#12312yes—
193C22×C4#12312yes—
194C22×C4#12312yes—
195C22×C4#12412yes—
196C22×C4#12412yes—
197C22×C4#12612yes—
198C22×C4#12612yes—
199M16—48no—
200M16—84no—
201C2×C8—44no—
202C2×C8—84no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—84no—
208C22⋊C4—48no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—84no—
216C42—84no—
217C23—22no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—24no—
222C23—21no—
223C23—21no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—48no—
234C23—48no—
235C23#38111yescharacteristic
236C23#38611yescharacteristic
237C23#38611yescharacteristic
238C23#39412yes—
239C23#39712yes—
240C23#45412yes—
241C23#45612yes—
242Q8—84no—
243Q8—84no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—88no—
274C2×C4—88no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4#38812yes—
284C2×C4#38812yes—
285C2×C4#39412yes—
286C2×C4#39712yes—
287C8—84no—
288C22—24no—
289C22—24no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—48no—
304C22—48no—
305C22—42no—
306C22—42no—
307C22#144011yescharacteristic
308C22#182712yes—
309C22#188611yescharacteristic
310C22#188811yescharacteristic
311C22#191312yes—
312C22#203814yes—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—84no—
318C4—84no—
319C4—82no—
320C4—82no—
321C4—164no—
322C2—21no—
323C2—22no—
324C2—24no—
325C2—22no—
326C2—21no—
327C2—42no—
328C2#1114911yescharacteristic
329C2#1116211yescharacteristic
330C2#1117211yescharacteristic
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
18. #1436C4semidirect

Maximal subgroups

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

subgroupindex
#218252
#217532
#141592
#80202
#79422