Finite Group Databasebeta

← #124342

Subgroups — #124342

Order 512.

552 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 96 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#13961C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#7906C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7104C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C2×#431—21no—
8C2×#383C411yescharacteristic
9C2×#376C2211yescharacteristic
10#1817—21no—
11#1446C411yescharacteristic
12#1446C2211yescharacteristic
13#1445C411yescharacteristic
14#1445C2211yescharacteristic
15#1436—21no—
16#1436—21no—
17#1434—21no—
18#1434—21no—
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C2×C4×Q8—22no—
25C2×C4×Q8—41no—
26C2×C4×Q8—41no—
27C2×C4×Q8—42no—
28C2×C4×Q8C814yescomplemented
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35#431—24no—
36#431—42no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—41no—
41#383—42no—
42#376—41no—
43#376—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95C2311yescharacteristic
60#336—41no—
61#336—41no—
62C22×Q8—21no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8C2×C812yes—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—42no—
80C22×C8—42no—
81C4×Q8—44no—
82C4×Q8—44no—
83C4×Q8—44no—
84C4×Q8—44no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C2×C812yes—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—24no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—22no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42C2×C812yes—
109C2×C42M1612yes—
110C4⋊C8—44no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—44no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—41no—
132#95—41no—
133C24C2×C22⋊C411yescharacteristic
134C2×Q8—44no—
135C2×Q8—42no—
136C2×Q8—44no—
137C2×Q8—42no—
138C2×Q8—48no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
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—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C22⋊C812yes—
201C22×C4C22⋊C812yes—
202C22×C4#9911yescharacteristic
203C22×C4#9911yescharacteristic
204C22×C4#10012yes—
205C22×C4#10412yes—
206C22×C4C22×C811yescharacteristic
207C22×C4C2×M1611yescharacteristic
208C22×C4#13512yes—
209C22×C4AGL(1,ℤ/8)11yescharacteristic
210C22×C4#13711yescharacteristic
211C2×C8—44no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—22no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—42no—
230C42—42no—
231C42—48no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23C2×C22⋊C811yescharacteristic
237C23C2×#9911yescharacteristic
238C23C2×#10011yescharacteristic
239C23#41611yescharacteristic
240C23#41711yescharacteristic
241C23C2×#10411yescharacteristic
242C23#42111yescharacteristic
243Q8—44no—
244Q8—44no—
245Q8—44no—
246Q8—44no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#32312yes—
318C2×C4#32312yes—
319C8—84no—
320C8—162no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22C2×#32311yescharacteristic
334C22#147611yescharacteristic
335C22#147711yescharacteristic
336C22#150411yescharacteristic
337C22#151611yescharacteristic
338C22#153011yescharacteristic
339C22#155111yescharacteristic
340C22#212011yescharacteristic
341C22#212011yescharacteristic
342C22#212911yescharacteristic
343C22#213511yescharacteristic
344C4—22no—
345C4—22no—
346C4—42no—
347C4—44no—
348C4—41no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—81no—
355C4—82no—
356C4—161no—
357C4—161no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2#757811yescharacteristic
363C2#1142511yescharacteristic
364C2#1143411yescharacteristic
365C2#1175611yescharacteristic
366C2#1176311yescharacteristic
367C2#1184211yescharacteristic
368C2#1184911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ 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
#139612
#86992
#79062
#71312
#71042