Finite Group Databasebeta

← #202866

Subgroups — #202866

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21833C211yesmaximal, characteristic
2C2×#1981C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20533C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#9566C214yesmaximal, complemented
7#9565C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7877C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103—21no—
17C22×#103—21no—
18#2285C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1990C2212yes—
21#1990C2212yes—
22#1988—22no—
23#1981—22no—
24#1981C2214yes—
25#1949—22no—
26#1948C2212yes—
27#1947C2212yes—
28#1823—22no—
29C8×C4⋊C4—22no—
30C8×C4⋊C4C2214yes—
31C4×C22⋊C8—22no—
32#1736—21no—
33#1546C418yescomplemented
34C2×#332C2214yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2214yes—
38C23×Q8—21no—
39C22×M16—21no—
40C23×C8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2×C414yes—
48C2×C4⋊Q8C2312yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—22no—
57#431—22no—
58#431—22no—
59#431C2×C414yes—
60#428—44no—
61C2×#107—22no—
62C2×#106—22no—
63#423—22no—
64#423C2×C414yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—44no—
76C2×#103—44no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C2×C4×C8—21no—
80C2×C4×C8—24no—
81C2×C4×C8C2312yes—
82#384C2311yescharacteristic
83#384C2311yescharacteristic
84#382—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4C2312yes—
93C2×#95—21no—
94#340—24no—
95#340—24no—
96#340—22no—
97#340—22no—
98#336—24no—
99#332C2×C4116yescomplemented
100C22×Q8—24no—
101C22×Q8—22no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—41no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—24no—
115C2×M16—42no—
116C2×M16—42no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8C2×D414yes—
124C4⋊Q8—24no—
125C4⋊Q8—24no—
126C4⋊Q8—24no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8C22×C414yes—
134#117—42no—
135#117—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4D814yes—
155C2×C4⋊C4SD1614yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—41no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—82no—
163C2×C42C22×C412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4112yes—
168C2×C42#4112yes—
169C2×C42#4112yes—
170C2×C42C2411yescharacteristic
171#107—44no—
172#106—44no—
173C4⋊C8—24no—
174C4⋊C8—24no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—42no—
180C4⋊C8C22×C414yes—
181C4⋊C8C22×C414yes—
182#103—48no—
183#103—48no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186C4×C8—24no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—84no—
190#95—22no—
191#95—22no—
192#95—41no—
193#95—41no—
194C24#12111yescharacteristic
195C2×Q8—24no—
196C2×Q8—42no—
197C2×Q8—44no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—48no—
203C2×Q8—48no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—84no—
207C2×Q8—84no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4#11712yes—
227C22×C4C4×D412yes—
228C22×C4C4×D412yes—
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12312yes—
233C22×C4#12412yes—
234C22×C4#12612yes—
235C22×C4C2×D812yes—
236C22×C4C2×SD1612yes—
237C22×C4#13512yes—
238C22×C4#13512yes—
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4111yescharacteristic
242M16—48no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—84no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4C2×D814yes—
275C4⋊C4C2×SD1614yes—
276C22⋊C4—82no—
277C42—44no—
278C42—82no—
279C42—82no—
280C42#11712yes—
281C42C4×D412yes—
282C42C4×D412yes—
283C42#12312yes—
284C42#12412yes—
285C42#12612yes—
286C42C23×C411yescharacteristic
287C42C2×#4111yescharacteristic
288C42C2×#4111yescharacteristic
289C42C2×#4111yescharacteristic
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#38812yes—
295C23#46512yes—
296C23#46612yes—
297C23C2×#12111yescharacteristic
298Q8—44no—
299Q8—44no—
300Q8—82no—
301Q8—82no—
302Q8—88no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—88no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#38812yes—
332C2×C4#38812yes—
333C2×C4#38812yes—
334C2×C4C4×D814yes—
335C2×C4C4×SD1614yes—
336C2×C4#46512yes—
337C2×C4#46612yes—
338C2×C4#48614yes—
339C2×C4#48814yes—
340C2×C4C2×#11711yescharacteristic
341C2×C4C2×C4×D411yescharacteristic
342C2×C4C2×C4×D411yescharacteristic
343C2×C4C2×#12311yescharacteristic
344C2×C4C2×#12411yescharacteristic
345C2×C4C2×#12611yescharacteristic
346C2×C4C22×D811yescharacteristic
347C2×C4C22×SD1611yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×#13511yescharacteristic
350C2×C4#57712yes—
351C2×C4#57812yes—
352C8—44no—
353C8—44no—
354C8—44no—
355C8—82no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22#193414yes—
363C22C2×#38811yescharacteristic
364C22#304211yescharacteristic
365C22#304411yescharacteristic
366C22C2×#46511yescharacteristic
367C22C2×#46611yescharacteristic
368C22#306911yescharacteristic
369C22#306911yescharacteristic
370C4—22no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—44no—
375C4—44no—
376C4—84no—
377C4—161no—
378C4—161no—
379C4#193414yes—
380C4C2×C4×D811yescharacteristic
381C4C2×C4×SD1611yescharacteristic
382C4#294712yes—
383C4#309712yes—
384C4C2×#48611yescharacteristic
385C4C2×#48811yescharacteristic
386C2—24no—
387C2#1939011yescharacteristic
388C2#1939211yescharacteristic
389C2#1940211yescharacteristic
390C2#1940411yescharacteristic
391C2#2106411yescharacteristic
392C2#2107711yescharacteristic
393C2C2×#193411yescharacteristic
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. #9566C2semidirect
7. #9565C2semidirect
33. #1546C4semidirect
99. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#218332
C2×#19812
#211702
#205332
C2×#15462
#95662
#95652
#80392
#78772