Finite Group Databasebeta

← #136145

Subgroups — #136145

Order 512.

469 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 98 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#22182C211yesmaximal, characteristic
2#12993C211yesmaximal, characteristic
3#12920C211yesmaximal, characteristic
4C2×#1348C211yesmaximal, characteristic
5#11328C211yesmaximal, characteristic
6#7258C211yesmaximal, characteristic
7#7183C211yesmaximal, characteristic
8C2×#467C2211yescharacteristic
9C22×#107C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×#389—21no—
12C2×#367—21no—
13C2×#358—21no—
14C2×C22⋊C16C2211yescharacteristic
15C2×C22⋊C16C2211yescharacteristic
16#1928—21no—
17#1921—21no—
18#1892—21no—
19#1878—21no—
20#1863—21no—
21#1808C2211yescharacteristic
22C2×#340—21no—
23C2×#340C2211yescharacteristic
24C2×#336C2211yescharacteristic
25#1380—21no—
26#1380—21no—
27#1348C414yescomplemented
28C22×Q16D411yescharacteristic
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×C22⋊Q8—21no—
32C2×C4×Q8—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—41no—
36C22×C42D411yescharacteristic
37C22×C16—21no—
38C22×C16—21no—
39#467—21no—
40#467—21no—
41#467C2×C412yes—
42#467Q812yes—
43C2×#107—21no—
44C2×#107—21no—
45C2×#107—42no—
46C2×#107D411yescharacteristic
47C2×#107D411yescharacteristic
48C2×#107D411yescharacteristic
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52#391—21no—
53#389—21no—
54#389—42no—
55#389—42no—
56#387—41no—
57#385—41no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—41no—
61C4×C22⋊C4—41no—
62C2×#95—41no—
63#367—42no—
64#358—42no—
65C22⋊C16—21no—
66C22⋊C16—21no—
67C22⋊C16—21no—
68C22⋊C16C2×C412yes—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—42no—
74#340C2×C412yes—
75#336—21no—
76#336—21no—
77C22×Q8—21no—
78C23×C4—41no—
79C23×C4—41no—
80C23×C4D811yescharacteristic
81C23×C4D811yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×Q16—21no—
84C2×Q16—41no—
85C2×Q16—41no—
86C2×Q16C4⋊C412yescomplemented
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×Q811yescharacteristic
98C22×C8#4111yescharacteristic
99C22×C8#4111yescharacteristic
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8—41no—
103C22⋊Q8—41no—
104C4×Q8—81no—
105C4×Q8—81no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C16—22no—
136C2×C16—41no—
137C2×C16—41no—
138C2×C16—41no—
139C2×C16—41no—
140C2×C16—41no—
141#107—21no—
142#107—22no—
143#107—22no—
144#107—41no—
145#107—41no—
146#107—41no—
147#107—41no—
148#107C4⋊C412yescomplemented
149#103—22no—
150#103—22no—
151#103—81no—
152#103—81no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—44no—
158#95—41no—
159C24C2×D811yescharacteristic
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4SD3212yes—
198C22×C4C4×D411yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#13511yescharacteristic
202C22×C4AGL(1,ℤ/8)11yescharacteristic
203C22×C4#13711yescharacteristic
204Q16—42no—
205Q16—81no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—81no—
219C2×C8C2×C4⋊C411yescharacteristic
220C2×C8C4×D411yescharacteristic
221C2×C8C22⋊Q811yescharacteristic
222C2×C8#12311yescharacteristic
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4—88no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C42—41no—
247C42—41no—
248C42—81no—
249C42—82no—
250C42—82no—
251C16—42no—
252C16—81no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C4×D811yescharacteristic
258C23#44711yescharacteristic
259C23#47411yescharacteristic
260C23C2×SD3211yescharacteristic
261C23#50811yescharacteristic
262C23#51011yescharacteristic
263C23#51011yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—81no—
267Q8—81no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—161no—
311C2×C4—161no—
312C2×C4—161no—
313C2×C4—161no—
314C2×C4#38511yescharacteristic
315C2×C4#44011yescharacteristic
316C2×C4#45211yescharacteristic
317C2×C4#45311yescharacteristic
318C2×C4#45611yescharacteristic
319C2×C4B(4,𝔽2)11yescharacteristic
320C2×C4#45811yescharacteristic
321C2×C4#47811yescharacteristic
322C8—21no—
323C8—21no—
324C8—21no—
325C8—21no—
326C8—81no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22#159511yescharacteristic
340C22#161011yescharacteristic
341C22#186211yescharacteristic
342C22C4×SD3211yescharacteristic
343C22#217611yescharacteristic
344C22#218511yescharacteristic
345C22#218911yescharacteristic
346C22#219011yescharacteristic
347C22#219111yescharacteristic
348C22#222511yescharacteristic
349C22#222911yescharacteristic
350C4—21no—
351C4—21no—
352C4—42no—
353C4—84no—
354C4—81no—
355C4—82no—
356C4—82no—
357C4—161no—
358C4—162no—
359C4—161no—
360C4#160311yescharacteristic
361C4#161811yescharacteristic
362C4#185911yescharacteristic
363C4#186711yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2#886611yescharacteristic
369C2#1210311yescharacteristic
370C2#1210711yescharacteristic
371C2#1228911yescharacteristic
372C2#1229311yescharacteristic
373C2#1293811yescharacteristic
374C2#1294311yescharacteristic
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 ◁ C2×#336 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1348C4semidirect
86. C2×Q16C4⋊C4semidirect
148. #107C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#221822
#129932
#129202
C2×#13482
#113282
#72582
#71832