Finite Group Databasebeta

← #117111

Subgroups — #117111

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4#20799C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7C2×#1281C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#427—22no—
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#382C2211yescharacteristic
16#1993—22no—
17#1993C414yescomplemented
18#1990—22no—
19#1987—22no—
20#1981—22no—
21#1956—22no—
22#1847—22no—
23#1847—22no—
24#1824—22no—
25#1787—22no—
26C2×#340—22no—
27#1294—22no—
28#1294—22no—
29#1281—22no—
30C23×Q8—21no—
31C23×C8D412yes—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8D412yes—
43C2×C4⋊Q8Q812yes—
44C2×#117—21no—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C42C2311yescharacteristic
50#431—21no—
51#431—21no—
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#427—44no—
59#427—44no—
60C2×#107—24no—
61C2×#106—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×#103—28no—
65C2×C22⋊C8D412yes—
66C2×C22⋊C8D412yes—
67C2×#97—21no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97C2×C412yes—
72C2×#97D412yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#384D412yes—
76#384D412yes—
77#382—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382C2×C412yes—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C2×#95—21no—
92C2×#95—21no—
93#340—28no—
94#340—28no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—84no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C22⋊C414yes—
114C22×C8SD1614yes—
115C22×C8Q1614yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8C4⋊C414yes—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174#107—48no—
175#106—48no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180#103—88no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193#95—22no—
194#95—22no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201C24#12711yescharacteristic
202C2×Q8—24no—
203C2×Q8—48no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—84no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—84no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4C22⋊Q811yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4C2×SD1612yes—
244C22×C4C2×Q1612yes—
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4#13712yes—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8#10318yescomplemented
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C42—82no—
279C42—82no—
280C42C2×C4⋊C411yescharacteristic
281C42#11711yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C42#12411yescharacteristic
293C42#12411yescharacteristic
294C42#12611yescharacteristic
295C42#12611yescharacteristic
296C23—22no—
297C23—22no—
298C23#39011yescharacteristic
299C23#39211yescharacteristic
300C23#39911yescharacteristic
301C23#49211yescharacteristic
302C23#49411yescharacteristic
303C23#49611yescharacteristic
304C23#49711yescharacteristic
305Q8—48no—
306Q8—84no—
307Q8—84no—
308Q8—84no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—162no—
335C2×C4—162no—
336C2×C4#38011yescharacteristic
337C2×C4#38511yescharacteristic
338C2×C4#38511yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#39311yescharacteristic
343C2×C4#39411yescharacteristic
344C2×C4#39411yescharacteristic
345C2×C4#39611yescharacteristic
346C2×C4#39611yescharacteristic
347C2×C4#39711yescharacteristic
348C2×C4C2×#10312yes—
349C2×C4#41712yes—
350C2×C4#45012yes—
351C2×C4#45112yes—
352C2×C4#46012yes—
353C2×C4#46212yes—
354C2×C4#47512yes—
355C2×C4#47712yes—
356C2×C4#48312yes—
357C2×C4#48412yes—
358C2×C4#48712yes—
359C2×C4#48912yes—
360C8—28no—
361C8—82no—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22#143911yescharacteristic
372C22#195611yescharacteristic
373C22#195911yescharacteristic
374C22#200812yes—
375C22#200912yes—
376C22#201411yescharacteristic
377C22#201611yescharacteristic
378C22#209511yescharacteristic
379C22#209611yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—84no—
389C4—162no—
390C4—162no—
391C4—162no—
392C4#180312yes—
393C4#187212yes—
394C4#202412yes—
395C4#207212yes—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2#1022112yes—
401C2#1022412yes—
402C2#1105011yescharacteristic
403C2#1112911yescharacteristic
404C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
17. #1993C4semidirect
257. C2×C8#103semidirect

Maximal subgroups

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

subgroupindex
#218342
#218042
C2×#19932
#207992
#80262
C2×#12812