Finite Group Databasebeta

← #122016

Subgroups — #122016

Order 512.

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

normal subgroups only

This group has 748 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
1#34595C211yesmaximal, characteristic
2#10835C211yesmaximal, characteristic
3C4×#343C211yesmaximal, characteristic
4#8029C212yesmaximal
5#7895C212yesmaximal
6C22×C4⋊C8C2211yescharacteristic
7C2×C4×M16C2212yes—
8C22×#97C2211yescharacteristic
9#2666—24no—
10#2659C412yes—
11#2646C412yes—
12#2627—22no—
13#2402—22no—
14#2399—22no—
15C4×C22⋊Q8—22no—
16#2283C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#343—22no—
19C2×#336—22no—
20#1295—24no—
21#1279—22no—
22#1279—22no—
23C22×M16—22no—
24C23×C8—21no—
25C23×C8—21no—
26C2×C22⋊Q8—22no—
27C2×C4×Q8—22no—
28C22×C4⋊C4D412yes—
29C22×C42D412yes—
30C22×C42C2311yescharacteristic
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8Q812yescomplemented
35C4×M16—22no—
36C4×M16—22no—
37C4×M16—22no—
38C4×M16Q814yescomplemented
39C2×#97—22no—
40C2×#97—22no—
41C2×#97—22no—
42C2×#97—21no—
43C2×#97Q812yescomplemented
44C2×C4×C8—22no—
45#400—24no—
46#400—24no—
47#398—24no—
48#398—24no—
49#398—24no—
50#398—24no—
51#395—24no—
52#395—24no—
53#391—22no—
54#389—22no—
55#387—24no—
56#385—24no—
57#383—22no—
58#382D414yes—
59#381—22no—
60#380—22no—
61#380C2×C414yes—
62C4×C4⋊C4—24no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4D412yes—
67C4×C4⋊C4D412yes—
68C4×C22⋊C4—22no—
69#376—22no—
70C2×#95C2×C411yescharacteristic
71C2×#95C2×C411yescharacteristic
72#343—28no—
73#336—44no—
74C22×Q8—22no—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C412yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—28no—
81C2×M16—24no—
82C2×M16—44no—
83C2×M16—42no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—42no—
92C22×C8—41no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—41no—
96C22×C8—41no—
97C22⋊Q8—44no—
98C22⋊Q8—44no—
99C4×Q8—82no—
100C4×Q8—82no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C22⋊C414yes—
111C2×C4⋊C4C22⋊C414yes—
112C2×C22⋊C4—24no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—42no—
118C2×C42—42no—
119C2×C42C2×D412yes—
120C2×C42C2×D412yes—
121C2×C42C2×Q811yescharacteristic
122C2×C42C2×Q812yes—
123C2×C42C2×Q811yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4112yes—
126C2×C42#4111yescharacteristic
127C4⋊C8—24no—
128C4⋊C8—42no—
129#97—24no—
130#97—24no—
131#97—42no—
132#97—42no—
133C4×C8—24no—
134C4×C8—42no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—22no—
139#95—22no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—22no—
144#95—22no—
145C24C2×C22⋊C411yescharacteristic
146C2×Q8—44no—
147C2×Q8—82no—
148C2×Q8—82no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4#10414yes—
168C22×C4C2×C22⋊C412yes—
169C22×C4C4×Q811yescharacteristic
170C22×C4C4×Q811yescharacteristic
171C22×C4C4×Q812yes—
172C22×C4#12011yescharacteristic
173C22×C4#12012yes—
174C22×C4#12011yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q812yes—
178C22×C4#12411yescharacteristic
179C22×C4#12411yescharacteristic
180C22×C4C4⋊Q811yescharacteristic
181C22×C4C4⋊Q811yescharacteristic
182M16—28no—
183M16—48no—
184M16—44no—
185C2×C8—24no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—48no—
192C2×C8—44no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—82no—
197C2×C8—81no—
198C2×C8—81no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—48no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—82no—
207C4⋊C4—84no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C42—42no—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—82no—
217C42C22⋊Q812yes—
218C42C22⋊Q812yes—
219C42C22⋊Q812yes—
220C42C22⋊Q812yes—
221C42#12412yes—
222C42#12412yes—
223C42C4⋊Q812yes—
224C42C4⋊Q812yes—
225C23—21no—
226C23—21no—
227C23—22no—
228C23#37911yescharacteristic
229C23#39111yescharacteristic
230C23#39111yescharacteristic
231C23C2×#10412yes—
232C23#42112yes—
233Q8—88no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#39112yes—
270C2×C4#39112yes—
271C2×C4#39312yes—
272C2×C4#39311yescharacteristic
273C2×C4#39311yescharacteristic
274C2×C4#39512yes—
275C2×C4#39511yescharacteristic
276C2×C4#39511yescharacteristic
277C2×C4#45311yescharacteristic
278C2×C4#45411yescharacteristic
279C2×C4#45511yescharacteristic
280C2×C4#45611yescharacteristic
281C8—44no—
282C8—48no—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22#144611yescharacteristic
293C22#178912yes—
294C22#179112yes—
295C22#179211yescharacteristic
296C22#179311yescharacteristic
297C22#198612yes—
298C22#199411yescharacteristic
299C22#199511yescharacteristic
300C4—22no—
301C4—22no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—44no—
307C4—82no—
308C4—84no—
309C4—84no—
310C4#202711yescharacteristic
311C4#202811yescharacteristic
312C4#206011yescharacteristic
313C4#206111yescharacteristic
314C2—22no—
315C2—21no—
316C2—21no—
317C2#840112yes—
318C2#840212yes—
319C2#1138611yescharacteristic
320C2#1139111yescharacteristic
321C2#1139211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
socleC23#379
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
34. C2×C4⋊C8Q8semidirect
38. C4×M16Q8semidirect
43. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#345952
#108352
C4×#3432
#80292
#78952