Finite Group Databasebeta

← #124170

Subgroups — #124170

Order 512.

562 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 116 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×#398C211yesmaximal, characteristic
2C4×#324C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7104C212yesmaximal
6C2×#431—21no—
7C2×#398C411yescharacteristic
8C2×C4×C4⋊C4—21no—
9C2×C4×C4⋊C4C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11C4×C22⋊C8—21no—
12C2×#324—21no—
13#1445C2211yescharacteristic
14#1444C2211yescharacteristic
15#1442—21no—
16#1441—21no—
17#1441C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34#431—24no—
35#431—42no—
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40C2×C4×C8—41no—
41#398—22no—
42#398C814yescomplemented
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C4×C4⋊C4—41no—
47C4×C4⋊C4—42no—
48C4×C4⋊C4—42no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95C2311yescharacteristic
61#336—41no—
62#324—24no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C2×C812yes—
94C2×C4⋊C4C2×C812yes—
95C2×C4⋊C4M1612yes—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—24no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42D812yes—
113C2×C42SD1612yes—
114C4⋊C8—44no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—44no—
120C4×C8—82no—
121#95—24no—
122#95—21no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—41no—
132#95—41no—
133#95—42no—
134#95C2×C812yes—
135C24C2×C22⋊C411yescharacteristic
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
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×C4C22⋊C812yes—
192C22×C4C22⋊C812yes—
193C22×C4#9911yescharacteristic
194C22×C4#9911yescharacteristic
195C22×C4#10212yes—
196C22×C4#10212yes—
197C22×C4#10412yes—
198C22×C4#10412yes—
199C22×C4C22×C811yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4C2×D811yescharacteristic
202C22×C4C2×SD1611yescharacteristic
203C22×C4#13512yes—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C4⋊C4—48no—
220C42—22no—
221C42—22no—
222C42—22no—
223C42—22no—
224C42—42no—
225C42—48no—
226C42—42no—
227C42—42no—
228C42—81no—
229C42—82no—
230C42—82no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23C2×C22⋊C811yescharacteristic
236C23C2×#9911yescharacteristic
237C23#41311yescharacteristic
238C23C2×#10211yescharacteristic
239C23#41611yescharacteristic
240C23C2×#10411yescharacteristic
241C23C2×#10411yescharacteristic
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
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—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
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×C4D4⋊C814yes—
310C2×C4#32712yes—
311C2×C4#32712yes—
312C2×C4#35212yes—
313C2×C4#35612yes—
314C8—84no—
315C8—84no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#145911yescharacteristic
329C22C2×D4⋊C811yescharacteristic
330C22#147611yescharacteristic
331C22C2×#32711yescharacteristic
332C22#150111yescharacteristic
333C22#152411yescharacteristic
334C22#154011yescharacteristic
335C22C2×#35211yescharacteristic
336C22#212011yescharacteristic
337C22#212611yescharacteristic
338C22C2×#35611yescharacteristic
339C4—24no—
340C4—22no—
341C4—44no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—81no—
352C4#134614yes—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2#756511yescharacteristic
358C2#1147211yescharacteristic
359C2#1147811yescharacteristic
360C2#1169111yescharacteristic
361C2C2×#134611yescharacteristic
362C2#1172711yescharacteristic
363C2#1173611yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
42. #398C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3982
C4×#3242
#79082
#71232
#71042