Finite Group Databasebeta

← #329668

Subgroups — #329668

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29878C211yesmaximal, characteristic, complemented
2#29692C212yesmaximal
3#21850C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10622C211yesmaximal, characteristic, complemented
6#10482C212yesmaximal, complemented
7#10480C212yesmaximal, complemented
8#10279C211yesmaximal, characteristic, complemented
9#10278C211yesmaximal, characteristic, complemented
10#9135C212yesmaximal, complemented
11#8598C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#477C2212yes—
15C2×#476C2212yes—
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2857C2212yes—
19#2087—22no—
20#2076—22no—
21#2059—22no—
22#2042—22no—
23#2035—22no—
24#2033—22no—
25#1990C2212yes—
26#1984C2211yescharacteristic
27#1984C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1870—22no—
31#1834C2211yescharacteristic
32#1803C2212yes—
33#1802C2212yes—
34#1798—22no—
35#1797C2212yes—
36C2×#334C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×M16—21no—
44C2×C4⋊Q8C2312yes—
45#529—42no—
46#529—42no—
47C2×#126—21no—
48C2×#125D411yescharacteristic
49C2×#124D411yescharacteristic
50C2×#123—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8D411yescharacteristic
53C2×#121D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2311yescharacteristic
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#500—42no—
60#477—42no—
61#477—42no—
62#476—42no—
63#476—42no—
64#431—42no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#106—22no—
68#423—42no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8D412yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—22no—
86#389—22no—
87#389—22no—
88#384C2312yes—
89#380—22no—
90#380—22no—
91#340—22no—
92#340—24no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#334—42no—
99#333—42no—
100#333—42no—
101C22×Q8—22no—
102C22×Q8C2×D411yescharacteristic
103C22×D4—21no—
104C22×D4C2×D411yescharacteristic
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—42no—
110C2×M16—82no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121#126—44no—
122#125—42no—
123#124—42no—
124#123—44no—
125C22⋊Q8—42no—
126C22⋊Q8—84no—
127#121—42no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—42no—
133C4×D4—42no—
134#117—42no—
135#117—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—42no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42C2411yescharacteristic
161#107—44no—
162#107—44no—
163#106—44no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172#103—44no—
173#103—82no—
174#102—44no—
175#102—82no—
176C22⋊C8—42no—
177C22⋊C8—82no—
178C22⋊C8—82no—
179C22⋊C8—82no—
180C4×C8—22no—
181C4×C8—22no—
182#95—42no—
183#95—42no—
184C24—21no—
185C24#12111yescharacteristic
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—82no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225M16—84no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—84no—
264C42—21no—
265C42—22no—
266C42—21no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42H(5,𝔽2)12yes—
271C42H(5,𝔽2)12yes—
272C23—21no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23C2×#12111yescharacteristic
283Q8—42no—
284Q8—42no—
285Q8—162no—
286Q8—162no—
287D4—42no—
288D4—42no—
289D4—84no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4#53411yescharacteristic
321C2×C4#53412yes—
322C2×C4#53511yescharacteristic
323C2×C4#53812yes—
324C2×C4#57712yes—
325C2×C4#57712yes—
326C2×C4#57811yescharacteristic
327C2×C4#57811yescharacteristic
328C2×C4#57812yes—
329C8—42no—
330C8—42no—
331C8—82no—
332C8—82no—
333C8—81no—
334C8—81no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—82no—
344C22—82no—
345C22#261711yescharacteristic
346C22#304612yes—
347C22#305911yescharacteristic
348C22#306011yescharacteristic
349C22#306812yes—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—162no—
358C4—162no—
359C4#320112yes—
360C4#320112yes—
361C4#320112yes—
362C4#322211yescharacteristic
363C4#322311yescharacteristic
364C2—81no—
365C2—161no—
366C2#1954712yes—
367C2#1954912yes—
368C2#2972611yescharacteristic
369C2#2973711yescharacteristic
370C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29878C2semidirect
4. #10624C2semidirect
5. #10622C2semidirect
6. #10482C2semidirect
7. #10480C2semidirect
8. #10279C2semidirect
9. #10278C2semidirect
10. #9135C2semidirect
11. #8598C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#296922
#218502
#106242
#106222
#104822
#104802
#102792
#102782
#91352
#85982