Finite Group Databasebeta

← #117114

Subgroups — #117114

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 401 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#20762C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7C2×#1281C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#97C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993—22no—
16#1993C414yescomplemented
17#1990—22no—
18#1987—22no—
19#1981—22no—
20#1956—22no—
21#1836—22no—
22#1836—22no—
23#1824—22no—
24#1787—22no—
25#1735—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—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×#103—28no—
61C2×C22⋊C8D412yes—
62C2×C22⋊C8D412yes—
63C2×#97—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×#97—21no—
67C2×#97C2×C412yes—
68C2×#97D412yes—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#384D412yes—
72#384D412yes—
73#382—21no—
74#382—21no—
75#382—21no—
76#382—21no—
77#382—21no—
78#382C2×C412yes—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C2×#95—21no—
88C2×#95—21no—
89#340—28no—
90#340—28no—
91#340—28no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—84no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C22⋊C414yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8C4⋊C414yes—
129#117—42no—
130#117—42no—
131#117—42no—
132#117—42no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
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—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×Q811yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175#103—88no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184C4×C8—41no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188#95—22no—
189#95—22no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194#95—41no—
195#95—41no—
196C24#12711yescharacteristic
197C2×Q8—24no—
198C2×Q8—48no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—84no—
205C2×Q8—84no—
206C2×Q8—84no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—84no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#13512yes—
239C22×C4#13512yes—
240C22×C4#13712yes—
241C22×C4#13712yes—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—82no—
274C42—82no—
275C42C2×C4⋊C411yescharacteristic
276C42#11711yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42#12311yescharacteristic
286C42#12311yescharacteristic
287C42#12411yescharacteristic
288C42#12411yescharacteristic
289C42#12611yescharacteristic
290C42#12611yescharacteristic
291C23—22no—
292C23—22no—
293C23#39011yescharacteristic
294C23#39211yescharacteristic
295C23#39911yescharacteristic
296C23#49511yescharacteristic
297C23#49511yescharacteristic
298C23#49711yescharacteristic
299C23#49711yescharacteristic
300Q8—48no—
301Q8—84no—
302Q8—84no—
303Q8—84no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4#38011yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#39311yescharacteristic
338C2×C4#39411yescharacteristic
339C2×C4#39411yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39711yescharacteristic
343C2×C4#41612yes—
344C2×C4#41912yes—
345C2×C4#45212yes—
346C2×C4#45212yes—
347C2×C4#46412yes—
348C2×C4#46412yes—
349C2×C4#47912yes—
350C2×C4#47912yes—
351C2×C4#48512yes—
352C2×C4#48512yes—
353C2×C4#48812yes—
354C2×C4#49112yes—
355C8—44no—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22#143911yescharacteristic
367C22#195811yescharacteristic
368C22#196011yescharacteristic
369C22#200812yes—
370C22#200912yes—
371C22#201511yescharacteristic
372C22#201511yescharacteristic
373C22#209711yescharacteristic
374C22#209711yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—84no—
384C4—162no—
385C4—162no—
386C4—162no—
387C4#180512yes—
388C4#187312yes—
389C4#202612yes—
390C4#207412yes—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2#1022212yes—
396C2#1022312yes—
397C2#1105211yescharacteristic
398C2#1112911yescharacteristic
399C2#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
16. #1993C4semidirect

Maximal subgroups

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

subgroupindex
#218342
#218042
C2×#19932
#207622
#80262
C2×#12812