Finite Group Databasebeta

← #202926

Subgroups — #202926

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#20533C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#9568C214yesmaximal, complemented
7#9561C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7877C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#107—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1990C2212yes—
19#1989—22no—
20#1988C2212yes—
21#1987C2212yes—
22#1956C2212yes—
23#1947—22no—
24#1947C2214yes—
25#1872—22no—
26#1803—22no—
27C8×C4⋊C4—22no—
28C8×C4⋊C4C2214yes—
29C4×C22⋊C8—22no—
30#1736—21no—
31#1736—21no—
32C2×#340—21no—
33#1542C418yescomplemented
34C2×#333C2212yes—
35C2×#332C2212yes—
36#1296C2214yes—
37#1277C2214yes—
38C22×M16—21no—
39C23×C8—21no—
40#533—22no—
41#533—22no—
42#533—22no—
43#533C2×C414yes—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4⋊Q8C2311yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59#431—22no—
60#431—22no—
61#431—22no—
62#431C2×C414yes—
63C2×#107—24no—
64C2×#107—44no—
65#423—22no—
66#423—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C414yes—
71C2×C4⋊C8D412yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—24no—
80C2×C4×C8C2312yes—
81#384—21no—
82#384—21no—
83#384—21no—
84#384C2312yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4C2312yes—
89C2×#95—21no—
90#340—24no—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—24no—
97#333C2×C418yescomplemented
98#332C2×C418yescomplemented
99C22×Q8—21no—
100C22×Q8—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—42no—
108C2×M16—42no—
109C2×M16—42no—
110C22×C8—24no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8C2×D414yes—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C22×C412yes—
124C4⋊Q8C22×C412yes—
125#125—24no—
126#125—42no—
127C22⋊Q8—24no—
128C22⋊Q8—24no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131C4×Q8—24no—
132C4×Q8—24no—
133C4×Q8—42no—
134C4×Q8—42no—
135#117—24no—
136#117—42no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4SD1614yes—
158C2×C4⋊C4Q1614yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42—82no—
166C2×C42C22×C412yes—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4112yes—
171C2×C42#4112yes—
172C2×C42#4112yes—
173C2×C42C2411yescharacteristic
174#107—48no—
175C4⋊C8—24no—
176C4⋊C8—24no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—42no—
182C4⋊C8C22×C414yes—
183C4⋊C8C22×C414yes—
184#103—84no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C4×C8—24no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—84no—
191#95—22no—
192#95—22no—
193C24#12111yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—81no—
201C2×Q8—81no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#11712yes—
221C22×C4C4×D412yes—
222C22×C4C4×D412yes—
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12312yes—
227C22×C4#12412yes—
228C22×C4#12612yes—
229C22×C4C2×SD1612yes—
230C22×C4C2×Q1612yes—
231C22×C4#13512yes—
232C22×C4#13512yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4111yescharacteristic
236M16—84no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—48no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4C2×SD1614yes—
272C4⋊C4C2×Q1614yes—
273C22⋊C4—44no—
274C42—44no—
275C42—44no—
276C42—82no—
277C42#11712yes—
278C42C4×D412yes—
279C42C4×D412yes—
280C42#12312yes—
281C42#12412yes—
282C42#12612yes—
283C42C23×C411yescharacteristic
284C42C2×#4111yescharacteristic
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#38812yes—
292C23#46512yes—
293C23#46712yes—
294C23C2×#12111yescharacteristic
295Q8—44no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—164no—
327C2×C4#38812yes—
328C2×C4#38812yes—
329C2×C4#38812yes—
330C2×C4C4×SD1614yes—
331C2×C4C4×Q1614yes—
332C2×C4#46512yes—
333C2×C4#46712yes—
334C2×C4#48714yes—
335C2×C4#48814yes—
336C2×C4C2×#11711yescharacteristic
337C2×C4C2×C4×D411yescharacteristic
338C2×C4C2×C4×D411yescharacteristic
339C2×C4C2×#12311yescharacteristic
340C2×C4C2×#12411yescharacteristic
341C2×C4C2×#12611yescharacteristic
342C2×C4C22×SD1611yescharacteristic
343C2×C4C22×Q1611yescharacteristic
344C2×C4C2×#13511yescharacteristic
345C2×C4C2×#13511yescharacteristic
346C2×C4#57612yes—
347C2×C4#57812yes—
348C8—44no—
349C8—44no—
350C8—44no—
351C8—82no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22#193314yes—
359C22C2×#38811yescharacteristic
360C22#304111yescharacteristic
361C22#304411yescharacteristic
362C22C2×#46511yescharacteristic
363C22C2×#46711yescharacteristic
364C22#306811yescharacteristic
365C22#306911yescharacteristic
366C4—22no—
367C4—21no—
368C4—21no—
369C4—22no—
370C4—44no—
371C4—44no—
372C4—82no—
373C4—82no—
374C4—162no—
375C4#193314yes—
376C4C2×C4×SD1611yescharacteristic
377C4C2×C4×Q1611yescharacteristic
378C4#295112yes—
379C4#309812yes—
380C4C2×#48711yescharacteristic
381C4C2×#48811yescharacteristic
382C2—24no—
383C2#1938411yescharacteristic
384C2#1938711yescharacteristic
385C2#1940211yescharacteristic
386C2#1940511yescharacteristic
387C2#2106711yescharacteristic
388C2#2107611yescharacteristic
389C2C2×#193311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ 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
6. #9568C2semidirect
7. #9561C2semidirect
33. #1542C4semidirect
97. #333C2×C4semidirect
98. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#218602
#213122
C2×#19472
#205332
C2×#15422
#95682
#95612
#80432
#78772