Finite Group Databasebeta

← #220747

Subgroups — #220747

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic
2#22674C211yesmaximal, characteristic
3#22542C212yesmaximal
4C2×#1579C211yesmaximal, characteristic
5#9836C211yesmaximal, characteristic
6#9740C211yesmaximal, characteristic
7#9220C212yesmaximal
8#8983C212yesmaximal
9#8643C211yesmaximal, characteristic
10#8448C212yesmaximal
11#8203C211yesmaximal, characteristic
12C2×#557C2212yes—
13#3383—24no—
14#3383C418yescomplemented
15C2×#500C2211yescharacteristic
16C2×#499—21no—
17C2×#479—22no—
18C2×#475C2212yes—
19C2×C8×Q8C2212yes—
20C2×C8×Q8C2211yescharacteristic
21C2×#441—21no—
22C2×C4×Q16—21no—
23C2×C4×Q16C2211yescharacteristic
24Q8×C4⋊C4C2212yes—
25#1990C2212yes—
26#1981—22no—
27#1981C2212yes—
28#1956—22no—
29#1956C2212yes—
30#1943—21no—
31#1865—22no—
32#1844C2211yescharacteristic
33#1803—22no—
34#1803C2212yes—
35#1803C2212yes—
36#1770C2211yescharacteristic
37C4×#103—22no—
38C4×#103C2211yescharacteristic
39#1579—24no—
40C2×#334C2211yescharacteristic
41C2×#332C2212yes—
42C2×Q8⋊C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×#322C2211yescharacteristic
45#1296C2212yes—
46#1279C2211yescharacteristic
47#1277C2211yescharacteristic
48C22×Q16D411yescharacteristic
49#557—24no—
50#557—24no—
51#557—42no—
52#557C2×C418yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2312yes—
55C2×#125—22no—
56C2×#125—42no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2312yes—
61C2×C4×Q8C2311yescharacteristic
62#500—22no—
63#500C2×C414yes—
64#499—28no—
65#479—216no—
66#475—24no—
67#475C2×C418yes—
68C8×Q8—22no—
69C8×Q8—24no—
70C8×Q8—42no—
71C8×Q8C2×C414yes—
72#441—44no—
73C4×Q16—28no—
74C4×Q16C2×C414yes—
75C4×Q16D414yes—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107D411yescharacteristic
79C2×#106—22no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103D412yes—
95C2×#103D412yes—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8—22no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#389—22no—
103#389—42no—
104#384—22no—
105#384C2312yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2311yescharacteristic
108#340—22no—
109#340—22no—
110#340—42no—
111#340D412yes—
112#336—21no—
113#334—22no—
114#332—42no—
115Q8⋊C8—24no—
116Q8⋊C8—22no—
117#322—22no—
118C22×Q8—22no—
119C22×Q8C2×D411yescharacteristic
120C22×Q8C2×D412yes—
121C2×Q16C22⋊C414yes—
122C2×Q16SD1614yes—
123C2×Q16Q1614yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8C2×D412yes—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—28no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—44no—
136C4⋊Q8C22×C414yes—
137#125—28no—
138#125—44no—
139#125—48no—
140#125—48no—
141C4×Q8—21no—
142C4×Q8—22no—
143C4×Q8—21no—
144C4×Q8—22no—
145C4×Q8—28no—
146C4×Q8—28no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×Q8—44no—
150C4×Q8—82no—
151C4×Q8C22×C414yes—
152C4×Q8C22×C412yes—
153C4×Q8C2×D414yes—
154C4×Q8#4112yes—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C2411yescharacteristic
176#107—24no—
177#107—24no—
178#107C22⋊C414yes—
179#106—28no—
180#106—42no—
181#106—81no—
182C4⋊C8—24no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—22no—
188C4⋊C8—44no—
189C4⋊C8C22×C412yes—
190#103—28no—
191#103—42no—
192#103—44no—
193#103—44no—
194#103—44no—
195#103—82no—
196#103—82no—
197#103—82no—
198#103C22⋊C418yes—
199#97—42no—
200C4×C8—24no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—44no—
206C4×C8C22×C412yes—
207C4×C8C2×D412yes—
208#95—21no—
209#95—42no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8C2×C22⋊C414yes—
223C2×Q8C4×D412yes—
224C2×Q8C2×SD1614yes—
225C2×Q8C2×Q1614yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4H(5,𝔽2)11yescharacteristic
236Q16#103116yescomplemented
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8C2×C22⋊C412yes—
254C2×C8C2×C22⋊C412yes—
255C2×C8C2×SD1612yes—
256C2×C8C2×Q1612yes—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—48no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—84no—
277C4⋊C4—82no—
278C4⋊C4—162no—
279C4⋊C4C2×C22⋊C414yes—
280C4⋊C4C4×D412yes—
281C42—22no—
282C42—24no—
283C42—24no—
284C42—22no—
285C42—48no—
286C42—41no—
287C42—41no—
288C42—84no—
289C42C23×C411yescharacteristic
290C42C22×D411yescharacteristic
291C42C2×#4111yescharacteristic
292C42H(5,𝔽2)11yescharacteristic
293C23D4211yescharacteristic
294Q8—24no—
295Q8—24no—
296Q8—24no—
297Q8—22no—
298Q8—22no—
299Q8—42no—
300Q8—44no—
301Q8—44no—
302Q8—82no—
303Q8—82no—
304Q8C2×#10318yes—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C22×C22⋊C411yescharacteristic
326C2×C4C2×C4×D411yescharacteristic
327C2×C4#51811yescharacteristic
328C2×C4D4211yescharacteristic
329C2×C4#54611yescharacteristic
330C2×C4#54611yescharacteristic
331C2×C4C22×SD1611yescharacteristic
332C2×C4C22×Q1611yescharacteristic
333C2×C4#57512yes—
334C2×C4#57711yescharacteristic
335C2×C4#57712yes—
336C2×C4#57811yescharacteristic
337C8—22no—
338C8—42no—
339C8—42no—
340C8—81no—
341C8—82no—
342C8—81no—
343C8C2×#10314yes—
344C22D4×C22⋊C411yescharacteristic
345C22#327112yes—
346C22D4×SD1611yescharacteristic
347C22D4×Q1611yescharacteristic
348C22#328912yes—
349C4—28no—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—42no—
354C4—44no—
355C4—82no—
356C4—84no—
357C4—162no—
358C4C22×#10311yescharacteristic
359C4#289811yescharacteristic
360C4#294311yescharacteristic
361C4#295211yescharacteristic
362C4#327112yes—
363C4#329811yescharacteristic
364C4#329911yescharacteristic
365C2#1981012yes—
366C2#1981812yes—
367C2D4×#10311yescharacteristic
368C2#2244111yescharacteristic
369C2#2246511yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #3383C4semidirect
236. Q16#103semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
#226742
#225422
C2×#15792
#98362
#97402
#92202
#89832
#86432
#84482
#82032