Finite Group Databasebeta

← #129343

Subgroups — #129343

Order 512.

892 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#2327C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3C2×#1319C212yesmaximal
4#8303C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2327—24no—
9#2327—24no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#387—22no—
13C2×#387—22no—
14C2×#382C2211yescharacteristic
15C2×#376C411yescharacteristic
16#1817—22no—
17C2×#324—22no—
18C2×#324C2214yes—
19#1324—24no—
20#1319—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2×C411yescharacteristic
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8C814yescomplemented
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—42no—
34C2×C4×Q8—42no—
35C2×C4×Q8—42no—
36C2×C4×Q8—42no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C42—21no—
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D412yes—
47C2×C22⋊C8—24no—
48#391—22no—
49#391—22no—
50#391—22no—
51#391D414yes—
52#387—216no—
53#387—216no—
54#382—22no—
55#382D414yes—
56#376—22no—
57#376—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63#336—44no—
64#324—24no—
65#324—24no—
66#324—44no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×Q8—21no—
72C22×Q8—21no—
73C22×Q8—21no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8C22⋊C412yes—
83C22×Q8C2×C812yes—
84C22×Q8C2×C812yes—
85C22×Q8M1612yes—
86C22×Q8M1612yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C22×C8—44no—
97C4⋊Q8—28no—
98C4⋊Q8—28no—
99C4⋊Q8—28no—
100C4⋊Q8—28no—
101C4×Q8—48no—
102C4×Q8—48no—
103C4×Q8—48no—
104C4×Q8—48no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—21no—
132C2×C42—24no—
133C2×C42—42no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C412yes—
140C2×C42C2×C812yes—
141C2×C42M1612yes—
142C22⋊C8—44no—
143C22⋊C8—44no—
144#95—24no—
145#95—24no—
146#95—28no—
147#95—28no—
148#95—28no—
149C24C2×C22⋊C411yescharacteristic
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8C22⋊C814yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C22⋊C814yes—
223C22×C4#9912yes—
224C22×C4#9912yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#12014yes—
228C22×C4C22×C811yescharacteristic
229C22×C4C2×M1611yescharacteristic
230C22×C4C2×M1611yescharacteristic
231C22×C4C2×M1611yescharacteristic
232C2×C8—84no—
233C2×C8—88no—
234C4⋊C4—232no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—48no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—48no—
241C42—24no—
242C42—24no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—48no—
247C42—42no—
248C42—42no—
249C42—48no—
250C42C22⋊C814yes—
251C23—22no—
252C23—22no—
253C23#37911yescharacteristic
254C23C2×C22⋊C811yescharacteristic
255C23#40711yescharacteristic
256C23C2×#9911yescharacteristic
257C23C2×#9911yescharacteristic
258C23#41311yescharacteristic
259C23#41311yescharacteristic
260Q8—44no—
261Q8—44no—
262Q8—44no—
263Q8—44no—
264Q8—44no—
265Q8—44no—
266Q8—44no—
267Q8—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4C2×C22⋊C811yescharacteristic
321C2×C4C2×C22⋊C811yescharacteristic
322C2×C4#40711yescharacteristic
323C2×C4#40711yescharacteristic
324C8—164no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22#145911yescharacteristic
334C22#145911yescharacteristic
335C22#146511yescharacteristic
336C22#146511yescharacteristic
337C22#177911yescharacteristic
338C22#178111yescharacteristic
339C22#178511yescharacteristic
340C22#212912yes—
341C22#213312yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—44no—
349C4—44no—
350C4—42no—
351C4—42no—
352C4—82no—
353C4—82no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2#830111yescharacteristic
359C2#1148212yes—
360C2#1153612yes—
361C2#1269811yescharacteristic
362C2#1270211yescharacteristic
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
28. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
C2×#13242
C2×#13192
#83032