Finite Group Databasebeta

← #135208

Subgroups — #135208

Order 512.

660 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4×#391C211yesmaximal, characteristic
2#8303C212yesmaximal
3#7189C212yesmaximal
4#7189C212yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#391C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C43C411yescharacteristic
10#1817—22no—
11C2×#324—22no—
12C2×#324C2214yes—
13#1436—22no—
14C4⋊#95C2211yescharacteristic
15C4⋊#95C2211yescharacteristic
16C4×#95—22no—
17C23×Q8C2×C411yescharacteristic
18C2×C4×Q8—22no—
19C2×C4×Q8—22no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8—42no—
23C22×C4⋊C4C2×C411yescharacteristic
24C22×C4⋊C4C2×C411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C4⋊C4C2311yescharacteristic
27C22×C42—21no—
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D412yes—
34C2×C22⋊C8—24no—
35#391—22no—
36#391—22no—
37#391—22no—
38#391C814yescomplemented
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C43—42no—
47#336—42no—
48#336—42no—
49#324—24no—
50#324—24no—
51#324—44no—
52C22×Q8—21no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8C2×C812yes—
61C22×Q8M1612yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C412yes—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—44no—
72C4×Q8—44no—
73C4×Q8—44no—
74C4×Q8—44no—
75C4×Q8—44no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2×C812yes—
88C2×C4⋊C4M1612yes—
89C2×C4⋊C4#4112yes—
90C2×C4⋊C4#4112yes—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—24no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—24no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—42no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—44no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C2×C812yes—
116C2×C42M1612yes—
117C22⋊C8—44no—
118C22⋊C8—44no—
119#95—28no—
120#95—28no—
121#95—44no—
122#95—44no—
123#95—42no—
124C24C2×C22⋊C411yescharacteristic
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C22⋊C814yes—
188C22×C4#9911yescharacteristic
189C22×C4#9911yescharacteristic
190C22×C4#9912yes—
191C22×C4#11711yescharacteristic
192C22×C4#11711yescharacteristic
193C22×C4#12314yes—
194C22×C4C22×C811yescharacteristic
195C22×C4C2×M1611yescharacteristic
196C22×C4C2×M1611yescharacteristic
197C22×C4C2×M1611yescharacteristic
198C2×C8—84no—
199C2×C8—84no—
200C2×C8—84no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—48no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C42—22no—
209C42—22no—
210C42—22no—
211C42—22no—
212C42—44no—
213C42—44no—
214C42—42no—
215C42—42no—
216C42—48no—
217C42—42no—
218C42—42no—
219C42—48no—
220C42—82no—
221C42—82no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23#38111yescharacteristic
227C23C2×C22⋊C811yescharacteristic
228C23#40711yescharacteristic
229C23C2×#9911yescharacteristic
230C23C2×#9911yescharacteristic
231C23#41311yescharacteristic
232C23#41311yescharacteristic
233Q8—44no—
234Q8—44no—
235Q8—44no—
236Q8—44no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
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—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4#35512yes—
297C2×C4#35612yes—
298C2×C4#43111yescharacteristic
299C2×C4#43111yescharacteristic
300C2×C4#43211yescharacteristic
301C2×C4#43211yescharacteristic
302C8—164no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22#145911yescharacteristic
316C22#145911yescharacteristic
317C22#146511yescharacteristic
318C22#146511yescharacteristic
319C22#181711yescharacteristic
320C22#181911yescharacteristic
321C22#182211yescharacteristic
322C22#212611yescharacteristic
323C22#212611yescharacteristic
324C22C2×#35511yescharacteristic
325C22C2×#35611yescharacteristic
326C4—22no—
327C4—22no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—44no—
334C4—44no—
335C4—42no—
336C4—44no—
337C4—82no—
338C4—82no—
339C2—21no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2#870011yescharacteristic
344C2#1147611yescharacteristic
345C2#1147811yescharacteristic
346C2#1153011yescharacteristic
347C2#1153211yescharacteristic
348C2#1285711yescharacteristic
349C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
38. #391C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3912
#83032
#71892