Finite Group Databasebeta

← #414223

Subgroups — #414223

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23746C211yesmaximal, characteristic
2#23552C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic
6#11355C211yesmaximal, characteristic
7#11249C212yesmaximal
8#11242C212yesmaximal
9#10896C212yesmaximal
10#10865C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×#382C2211yescharacteristic
21#2101—22no—
22#2101—22no—
23#2058—22no—
24#2057—22no—
25#1993—22no—
26#1993—22no—
27#1771—22no—
28#1770—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#341C2212yes—
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1729C2211yescharacteristic
40#1728C2211yescharacteristic
41#1728C2211yescharacteristic
42#1728C2211yescharacteristic
43C23×Q8—21no—
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60C2×#108—24no—
61C2×#108—24no—
62C2×#108—42no—
63#428—24no—
64#428—42no—
65C2×#107—22no—
66C2×#106—22no—
67#424—22no—
68#424Q814yescomplemented
69C2×C4⋊C8—22no—
70#405—22no—
71#405—42no—
72#405Q814yescomplemented
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—42no—
76C2×#97D412yes—
77C2×#97D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#389D412yes—
81#389D412yes—
82#385—21no—
83#385—21no—
84#382—22no—
85#382—22no—
86C4×C4⋊C4—22no—
87C2×#95—21no—
88C2×#95—21no—
89#343—24no—
90#343—24no—
91#343—24no—
92#343—42no—
93#341—24no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—24no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105C22×Q8—44no—
106C22×Q8—42no—
107C22×Q8—84no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4C2411yescharacteristic
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16C2×Q812yes—
131C2×M16C2×Q812yes—
132C2×M16#4114yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×Q812yes—
145C22×C8C2×Q812yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C4⋊Q8—28no—
149C4⋊Q8—48no—
150C4⋊Q8—44no—
151#117—21no—
152#117—21no—
153#117—22no—
154#117—22no—
155#117C2×Q812yes—
156#117C2×Q812yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184#108—48no—
185#107—44no—
186#106—44no—
187C4⋊C8—28no—
188#97—24no—
189#97—42no—
190#97—42no—
191#97—42no—
192C4×C8—24no—
193C4×C8—42no—
194#95—22no—
195#95—22no—
196#95—42no—
197#95—42no—
198C24C2×#4111yescharacteristic
199C2×Q8—48no—
200C2×Q8—84no—
201C2×Q8—84no—
202C2×Q8—816no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—84no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4#12112yes—
220C22×C4#12112yes—
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4111yescharacteristic
226C22×C4C2×#4111yescharacteristic
227C22×C4C2×#4112yes—
228C22×C4C2×#4112yes—
229C22×C4H(5,𝔽2)12yes—
230C22×C42−1+412yes—
231M16—24no—
232M16—24no—
233M16—24no—
234M16—24no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243C2×C8—22no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—22no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—24no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8C22⋊Q812yes—
266C2×C8C22⋊Q812yes—
267C2×C8C22⋊Q814yes—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—48no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4C4⋊Q814yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—42no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—41no—
292C42—41no—
293C23—22no—
294C23—22no—
295C23C2×#12111yescharacteristic
296C23#53311yescharacteristic
297C23#53311yescharacteristic
298C23#55112yes—
299C23#55612yes—
300Q8—816no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—162no—
324C2×C4—162no—
325C2×C4C2×C22⋊Q811yescharacteristic
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4#52511yescharacteristic
328C2×C4#52912yes—
329C2×C4C2×C4⋊Q811yescharacteristic
330C2×C4#53211yescharacteristic
331C2×C4#53211yescharacteristic
332C2×C4#53312yes—
333C2×C4#53411yescharacteristic
334C2×C4#53611yescharacteristic
335C2×C4#53912yes—
336C2×C4#54412yes—
337C2×C4#54612yes—
338C2×C4#54812yes—
339C2×C4#55112yes—
340C2×C4#55412yes—
341C2×C4#55612yes—
342C2×C4#55712yes—
343C8—24no—
344C8—22no—
345C8—22no—
346C8—44no—
347C8—42no—
348C8—42no—
349C8—82no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22#253411yescharacteristic
357C22#254511yescharacteristic
358C22#258812yes—
359C22#260512yes—
360C22#285711yescharacteristic
361C22#306812yes—
362C22#306912yes—
363C4—44no—
364C4—42no—
365C4—42no—
366C4—81no—
367C4—84no—
368C4—81no—
369C4—162no—
370C4#239611yescharacteristic
371C4#255012yes—
372C4#256012yes—
373C4#256411yescharacteristic
374C4#258111yescharacteristic
375C4#262711yescharacteristic
376C2—22no—
377C2—22no—
378C2#1516411yescharacteristic
379C2#2813711yescharacteristic
380C2#2813811yescharacteristic
381C2#2867212yes—
382C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
68. #424Q8semidirect
72. #405Q8semidirect

Maximal subgroups

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

subgroupindex
#237462
#235522
C2×#19932
#204362
#113822
#113552
#112492
#112422
#108962
#108652
#108272