Finite Group Databasebeta

← #124347

Subgroups — #124347

Order 512.

481 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 90 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#21379C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic, complemented
3C2×#1377C211yesmaximal, characteristic, complemented
4#7133C212yesmaximal, complemented
5#7105C212yesmaximal, complemented
6C2×C4×#41C411yescharacteristic
7C22×M32—21no—
8C2×#432C411yescharacteristic
9C22×#97C2211yescharacteristic
10#2112—21no—
11#1985C411yescharacteristic
12#1985C2211yescharacteristic
13#1955C411yescharacteristic
14#1955C2211yescharacteristic
15#1869—21no—
16#1869—21no—
17#1749—21no—
18#1749—21no—
19#1377—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C22×#41C811yescharacteristic
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C4×#41—41no—
27C4×#41—41no—
28C4×#41—41no—
29C4×#41—41no—
30C2×C4×Q8C811yescharacteristic
31C2×C4×Q8C2×C411yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34C2×C4×D4C811yescharacteristic
35C2×C4×D4C2×C411yescharacteristic
36C2×#117—21no—
37C2×#117C811yescharacteristic
38C22×C42—21no—
39C22×C42C2×C411yescharacteristic
40C2×M32—24no—
41C2×M32—22no—
42C2×M32—22no—
43C2×M32—41no—
44C2×M32—42no—
45C22×C16—41no—
46#432—41no—
47#432—41no—
48C2×C4⋊C8C2×C411yescharacteristic
49C2×C4⋊C8C2×C411yescharacteristic
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—41no—
57C2×#97—41no—
58C2×#97C2×C411yescharacteristic
59C2×#97D411yescharacteristic
60C2×#97D411yescharacteristic
61C2×#97C2311yescharacteristic
62C22⋊C16—41no—
63C22⋊C16—41no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—44no—
71C22×Q8C2×C811yescharacteristic
72C22×D4—21no—
73C22×D4C2×C811yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C2×C811yescharacteristic
79C23×C4M1611yescharacteristic
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—41no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—42no—
104C4×D4—42no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—41no—
108C4×D4—41no—
109#117—42no—
110#117—42no—
111#117—41no—
112#117—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2×C811yescharacteristic
116C2×C4⋊C4C2×C811yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42C22⋊C411yescharacteristic
130C2×C42C2×C811yescharacteristic
131C2×C42M1611yescharacteristic
132C2×C42C22×C411yescharacteristic
133M32—24no—
134M32—44no—
135M32—42no—
136M32—42no—
137M32—42no—
138C2×C16—44no—
139C2×C16—42no—
140C2×C16—42no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C22⋊C8—42no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—41no—
151#97—41no—
152#97—41no—
153#97—41no—
154#97—42no—
155C24—21no—
156C24C22⋊C811yescharacteristic
157#41—44no—
158#41—44no—
159#41—44no—
160#41—44no—
161C2×Q8—42no—
162C2×Q8—41no—
163C2×D4—24no—
164C2×D4—41no—
165C2×D4—42no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C22⋊C811yescharacteristic
195C22×C4C22⋊C811yescharacteristic
196C22×C4C22⋊C811yescharacteristic
197C22×C4#9911yescharacteristic
198C22×C4#9911yescharacteristic
199C22×C4#10011yescharacteristic
200C22×C4#10011yescharacteristic
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C22×C811yescharacteristic
203C22×C4C2×M1611yescharacteristic
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8#10412yes—
226C2×C8#13512yes—
227C2×C8AGL(1,ℤ/8)11yescharacteristic
228C2×C8#13711yescharacteristic
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C42—42no—
249C42—41no—
250C42—41no—
251C16—44no—
252C16—82no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#32311yescharacteristic
265C23#32311yescharacteristic
266C23C2×C22⋊C811yescharacteristic
267Q8—42no—
268Q8—42no—
269D4—44no—
270D4—42no—
271D4—42no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—84no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#32311yescharacteristic
311C2×C4#32311yescharacteristic
312C2×C4C2×#9911yescharacteristic
313C2×C4C2×#10011yescharacteristic
314C2×C4#41611yescharacteristic
315C2×C4#41711yescharacteristic
316C2×C4C2×#10411yescharacteristic
317C2×C4#42111yescharacteristic
318C8—21no—
319C8—22no—
320C8—21no—
321C8—22no—
322C8—41no—
323C8—81no—
324C8—81no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#132012yes—
342C22C2×#32311yescharacteristic
343C22#147611yescharacteristic
344C22#147711yescharacteristic
345C4—21no—
346C4—21no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4#150411yescharacteristic
355C4#151611yescharacteristic
356C4#153011yescharacteristic
357C4#155111yescharacteristic
358C2—21no—
359C2—21no—
360C2—22no—
361C2—41no—
362C2—82no—
363C2#757811yescharacteristic
364C2C2×#132011yescharacteristic
365C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#323
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12846C2semidirect
3. C2×#1377C2semidirect
4. #7133C2semidirect
5. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#213792
#128462
C2×#13772
#71332
#71052