Finite Group Databasebeta

← #136196

Subgroups — #136196

Order 512.

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

normal subgroups only

This group has 433 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#11256C211yesmaximal, characteristic
6#7258C211yesmaximal, characteristic
7#7183C211yesmaximal, characteristic
8C2×#467C2211yescharacteristic
9C22×#107C2211yescharacteristic
10C4×C22⋊Q8—21no—
11C2×#367—21no—
12C2×#358—21no—
13C2×C22⋊C16C2211yescharacteristic
14C2×C22⋊C16C2211yescharacteristic
15#1928—21no—
16#1921—21no—
17#1892—21no—
18#1878—21no—
19#1863—21no—
20#1808C2211yescharacteristic
21C2×#340—21no—
22C2×#340C2211yescharacteristic
23C2×#336C2211yescharacteristic
24C4⋊#95—21no—
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×C42D411yescharacteristic
36C22×C16—21no—
37C22×C16—21no—
38#467—21no—
39#467—21no—
40#467C2×C412yes—
41#467Q812yes—
42C2×#107—21no—
43C2×#107—21no—
44C2×#107—42no—
45C2×#107D411yescharacteristic
46C2×#107D411yescharacteristic
47C2×#107D411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51#391—21no—
52#389—21no—
53#387—41no—
54#385—41no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—41no—
58C4×C22⋊C4—41no—
59C2×#95—41no—
60C2×#95—41no—
61#367—42no—
62#358—42no—
63C22⋊C16—21no—
64C22⋊C16—21no—
65C22⋊C16—21no—
66C22⋊C16C2×C412yes—
67#340—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—42no—
72#340C2×C412yes—
73#336—21no—
74#336—21no—
75C22×Q8—21no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4D811yescharacteristic
79C23×C4D811yescharacteristic
80C23×C4C2×D411yescharacteristic
81C2×Q16—21no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16C4⋊C412yes—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C22×C8C2×Q811yescharacteristic
96C22×C8#4111yescharacteristic
97C22×C8#4111yescharacteristic
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102C4×Q8—81no—
103C4×Q8—81no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
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×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C16—22no—
131C2×C16—41no—
132C2×C16—41no—
133C2×C16—41no—
134C2×C16—41no—
135C2×C16—41no—
136#107—21no—
137#107—22no—
138#107—22no—
139#107—41no—
140#107—41no—
141#107—41no—
142#107—41no—
143#107C4⋊C412yes—
144#103—22no—
145#103—22no—
146#103—81no—
147#103—81no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—82no—
154#95—82no—
155C24C2×D811yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4SD3212yes—
194C22×C4C4×D411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4#13511yescharacteristic
198C22×C4AGL(1,ℤ/8)11yescharacteristic
199C22×C4#13711yescharacteristic
200Q16—42no—
201Q16—81no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—81no—
215C2×C8C2×C4⋊C411yescharacteristic
216C2×C8C4×D411yescharacteristic
217C2×C8C22⋊Q811yescharacteristic
218C2×C8#12311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—81no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—41no—
242C42—41no—
243C42—81no—
244C42—82no—
245C42—82no—
246C16—42no—
247C16—81no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23C4×D811yescharacteristic
253C23#44711yescharacteristic
254C23#47411yescharacteristic
255C23C2×SD3211yescharacteristic
256C23#50811yescharacteristic
257C23#51011yescharacteristic
258C23#51011yescharacteristic
259Q8—42no—
260Q8—42no—
261Q8—81no—
262Q8—81no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—161no—
302C2×C4—161no—
303C2×C4—161no—
304C2×C4—161no—
305C2×C4—161no—
306C2×C4—161no—
307C2×C4—161no—
308C2×C4—161no—
309C2×C4#38511yescharacteristic
310C2×C4#44011yescharacteristic
311C2×C4#45211yescharacteristic
312C2×C4#45411yescharacteristic
313C2×C4#45511yescharacteristic
314C2×C4B(4,𝔽2)11yescharacteristic
315C2×C4#45811yescharacteristic
316C2×C4#47811yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22#159811yescharacteristic
335C22#160711yescharacteristic
336C22#186211yescharacteristic
337C22C4×SD3211yescharacteristic
338C22#217611yescharacteristic
339C22#218511yescharacteristic
340C22#218911yescharacteristic
341C22#219011yescharacteristic
342C22#219111yescharacteristic
343C22#222511yescharacteristic
344C22#222911yescharacteristic
345C4—21no—
346C4—21no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—81no—
351C4—162no—
352C4—162no—
353C4—161no—
354C4—161no—
355C4#160611yescharacteristic
356C4#161511yescharacteristic
357C4#186011yescharacteristic
358C4#186711yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2#886911yescharacteristic
364C2#1215911yescharacteristic
365C2#1216311yescharacteristic
366C2#1227911yescharacteristic
367C2#1228311yescharacteristic
368C2#1293811yescharacteristic
369C2#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

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
#112562
#72582
#71832