Finite Group Databasebeta

← #124283

Subgroups — #124283

Order 512.

510 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 108 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#14150C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7122C212yesmaximal
6C2×#389—21no—
7C2×#382C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#324—21no—
13C2×#324—21no—
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1442C411yescharacteristic
17#1442C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20C4⋊#95—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C4×C8—41no—
38#389—42no—
39#389—42no—
40#382—22no—
41#382C814yescomplemented
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
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×#95C2311yescharacteristic
58#336—41no—
59#336—41no—
60#324—24no—
61#324—42no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C22×C8—42no—
74C22×C8—42no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
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—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4D812yes—
95C2×C4⋊C4SD1612yes—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—81no—
107C2×C42—81no—
108C2×C42C2×C812yes—
109C2×C42M1612yes—
110C4⋊C8—82no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—82no—
116#95—24no—
117#95—21no—
118#95—21no—
119#95—41no—
120#95—41no—
121#95—42no—
122#95—42no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—41no—
127#95—41no—
128#95—81no—
129#95—81no—
130#95C2×C812yes—
131#95C2×C812yes—
132C24C2×C22⋊C411yescharacteristic
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
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×C4C22⋊C812yes—
189C22×C4C22⋊C812yes—
190C22×C4#9911yescharacteristic
191C22×C4#9911yescharacteristic
192C22×C4#10112yes—
193C22×C4#10212yes—
194C22×C4#10212yes—
195C22×C4#10412yes—
196C22×C4C22×C811yescharacteristic
197C22×C4C2×M1611yescharacteristic
198C22×C4C2×D811yescharacteristic
199C22×C4C2×SD1611yescharacteristic
200C22×C4#13712yes—
201C2×C8—44no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—48no—
215C4⋊C4—84no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C42—22no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—42no—
223C42—42no—
224C42—81no—
225C42—81no—
226C42—84no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23C2×C22⋊C811yescharacteristic
232C23C2×#9911yescharacteristic
233C23C2×#10111yescharacteristic
234C23C2×#10211yescharacteristic
235C23#41911yescharacteristic
236C23C2×#10411yescharacteristic
237C23#42111yescharacteristic
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
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#32412yes—
308C2×C4#32412yes—
309C2×C4D4⋊C814yes—
310C8—84no—
311C8—162no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C2×#32411yescharacteristic
325C22C2×D4⋊C811yescharacteristic
326C22#147911yescharacteristic
327C22#150811yescharacteristic
328C22#151911yescharacteristic
329C22#152811yescharacteristic
330C22#154611yescharacteristic
331C22#212011yescharacteristic
332C22#212011yescharacteristic
333C22#212711yescharacteristic
334C22#213311yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—81no—
347C4—82no—
348C4—82no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2—21no—
353C2#757311yescharacteristic
354C2#1144811yescharacteristic
355C2#1145211yescharacteristic
356C2#1178411yescharacteristic
357C2#1179211yescharacteristic
358C2#1187511yescharacteristic
359C2#1187911yescharacteristic
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×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
41. #382C8semidirect

Maximal subgroups

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

subgroupindex
#141502
#85582
C4×#3242
#71232
#71222