Finite Group Databasebeta

← #118045

Subgroups — #118045

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21839C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4C2×#1497C211yesmaximal, characteristic
5#8031C211yesmaximal, characteristic
6C2×#1287C211yesmaximal, characteristic
7#7912C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1992—22no—
16#1988—21no—
17#1988—21no—
18#1987—21no—
19#1981—21no—
20#1947—22no—
21#1872—22no—
22#1852—22no—
23#1825—22no—
24C4×#107—22no—
25#1763—22no—
26#1497—22no—
27#1497C414yescomplemented
28C2×Q8⋊C8—22no—
29#1295—22no—
30#1287—22no—
31#1280—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533Q812yes—
40C2×C4⋊Q8D411yescharacteristic
41C2×C4⋊Q8D411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117—21no—
49C22×C4⋊C4D412yes—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431D412yes—
68C2×#107—24no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×#103—24no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97D412yes—
81C2×C4×C8D412yes—
82#384D412yes—
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C2×#95—21no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8C4⋊C412yes—
115C4⋊Q8C4⋊C412yes—
116#125—22no—
117#125—22no—
118#125—41no—
119#125—41no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136#117—22no—
137#117—22no—
138#117—41no—
139#117—41no—
140#117—82no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—82no—
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172#107—48no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181#103—48no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—41no—
194#97—41no—
195#97—41no—
196C4×C8—22no—
197C4×C8C22⋊C414yes—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202C24#12111yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—44no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—42no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#11711yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12711yescharacteristic
240C2×C8—24no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—82no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—42no—
271C42—42no—
272C42—82no—
273C42—82no—
274C42—82no—
275C42C2×C22⋊C411yescharacteristic
276C42C2×C4⋊C411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42#12311yescharacteristic
286C42#12311yescharacteristic
287C42#12311yescharacteristic
288C42#12311yescharacteristic
289C42#12411yescharacteristic
290C42#12411yescharacteristic
291C23—22no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23#38811yescharacteristic
296C23#39211yescharacteristic
297C23#39611yescharacteristic
298Q8—44no—
299Q8—44no—
300Q8—82no—
301Q8—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#38011yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38511yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39311yescharacteristic
340C2×C4#39411yescharacteristic
341C2×C4#39411yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#39711yescharacteristic
344C2×C4#39911yescharacteristic
345C2×C4#47212yes—
346C2×C4#47312yes—
347C8—44no—
348C8—82no—
349C8—82no—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—42no—
359C22#143911yescharacteristic
360C22#200912yes—
361C22#201012yes—
362C22#201811yescharacteristic
363C22#202011yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—84no—
375C4—82no—
376C4—82no—
377C4—162no—
378C4#181612yes—
379C4#207812yes—
380C2—41no—
381C2—41no—
382C2#1004212yes—
383C2#1004512yes—
384C2#1108311yescharacteristic
385C2#1110711yescharacteristic
386C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#218392
#213942
#213582
C2×#14972
#80312
C2×#12872
#79122