Finite Group Databasebeta

← #136040

Subgroups — #136040

Order 512.

416 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 102 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#21960C211yesmaximal, characteristic
2#12920C211yesmaximal, characteristic
3#12919C211yesmaximal, characteristic
4C2×#1322C211yesmaximal, characteristic
5#11232C211yesmaximal, characteristic
6#7258C211yesmaximal, characteristic
7#7258C211yesmaximal, characteristic
8C22×#107C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2283—21no—
11C2×#367—21no—
12C2×#366—21no—
13C2×C22⋊C16C2211yescharacteristic
14C2×C22⋊C16C2211yescharacteristic
15#1949—21no—
16#1949—21no—
17#1949—21no—
18#1949—21no—
19#1808C2211yescharacteristic
20#1808C2211yescharacteristic
21#1807—21no—
22C2×#340—21no—
23C2×#340C2211yescharacteristic
24C4⋊#95—21no—
25#1380—21no—
26#1380—21no—
27#1322C414yescomplemented
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C42D411yescharacteristic
33C22×C16—21no—
34C22×C16—21no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—42no—
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×#107D411yescharacteristic
42C2×#106—21no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8C2×C412yes—
49C2×C4⋊C8Q812yes—
50#383—41no—
51#382—21no—
52#382—21no—
53#381—41no—
54#380—21no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—21no—
58C4×C22⋊C4—41no—
59#376—41no—
60C2×#95—41no—
61C2×#95—41no—
62#367—42no—
63#366—42no—
64C22⋊C16—21no—
65C22⋊C16—21no—
66C22⋊C16C2×C412yes—
67C22⋊C16C2×C412yes—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—42no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4D811yescharacteristic
78C23×C4D811yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C4⋊C412yes—
87C22×C8C22×C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×Q811yescharacteristic
91C22×C8#4111yescharacteristic
92C22×C8#4111yescharacteristic
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42C4⋊C412yes—
118C2×C42Q1612yes—
119C2×C16—22no—
120C2×C16—22no—
121C2×C16—41no—
122C2×C16—41no—
123C2×C16—41no—
124C2×C16—41no—
125#107—22no—
126#107—22no—
127#107—41no—
128#107—41no—
129#107—41no—
130#107—41no—
131#107—41no—
132#107—41no—
133#106—81no—
134#106—81no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—82no—
148#95—82no—
149C24C2×D811yescharacteristic
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—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4#10712yes—
183C22×C4#10712yes—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4C2×Q1611yescharacteristic
188C22×C4AGL(1,ℤ/8)11yescharacteristic
189C22×C4#13711yescharacteristic
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—22no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—81no—
203C2×C8C4×D411yescharacteristic
204C2×C8C4×D411yescharacteristic
205C2×C8C22⋊Q811yescharacteristic
206C2×C8C22⋊Q811yescharacteristic
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—82no—
232C42—82no—
233C16—42no—
234C16—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23C2×#10711yescharacteristic
240C23#44711yescharacteristic
241C23#48011yescharacteristic
242C23#50811yescharacteristic
243C23#50811yescharacteristic
244C23#50911yescharacteristic
245C23#51011yescharacteristic
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—161no—
287C2×C4—161no—
288C2×C4—161no—
289C2×C4—161no—
290C2×C4—161no—
291C2×C4—161no—
292C2×C4—161no—
293C2×C4—161no—
294C2×C4#38511yescharacteristic
295C2×C4#42811yescharacteristic
296C2×C4#45111yescharacteristic
297C2×C4#45411yescharacteristic
298C2×C4#45611yescharacteristic
299C2×C4B(4,𝔽2)11yescharacteristic
300C2×C4#45811yescharacteristic
301C2×C4#48411yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8—21no—
306C8—42no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22#159811yescharacteristic
320C22#161011yescharacteristic
321C22#185211yescharacteristic
322C22#215811yescharacteristic
323C22#215911yescharacteristic
324C22#218611yescharacteristic
325C22#218911yescharacteristic
326C22#219011yescharacteristic
327C22#219111yescharacteristic
328C22#223411yescharacteristic
329C22#223711yescharacteristic
330C4—21no—
331C4—21no—
332C4—42no—
333C4—42no—
334C4—162no—
335C4—161no—
336C4—161no—
337C4—161no—
338C4—162no—
339C4—161no—
340C4#160511yescharacteristic
341C4#161711yescharacteristic
342C4#185711yescharacteristic
343C4#186711yescharacteristic
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#885611yescharacteristic
349C2#1216011yescharacteristic
350C2#1216311yescharacteristic
351C2#1229011yescharacteristic
352C2#1229311yescharacteristic
353C2#1292111yescharacteristic
354C2#1292311yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1322C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#129202
#129192
C2×#13222
#112322
#72582