Finite Group Databasebeta

← #117871

Subgroups — #117871

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4C2×#1531C211yesmaximal, characteristic
5C2×#1302C211yesmaximal, characteristic
6#8040C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#97C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993—21no—
16#1990—21no—
17#1960—21no—
18#1956—21no—
19#1872—22no—
20#1872—21no—
21#1872—21no—
22#1836—22no—
23#1836—22no—
24#1805—22no—
25#1803—22no—
26C2×#340—21no—
27C2×#340—21no—
28#1531—22no—
29#1531C414yescomplemented
30C2×#330—22no—
31#1302—22no—
32#1296—22no—
33#1280—22no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D412yes—
72C2×#103—24no—
73C2×#103—22no—
74C2×#103—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—21no—
83C2×#97C2×C412yes—
84C2×#97D412yes—
85C2×#97D412yes—
86C2×C4×C8—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D412yes—
91#382—21no—
92#382—21no—
93#382—21no—
94#382—21no—
95C2×#95—21no—
96C2×#95—21no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#330—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—41no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4⋊C412yes—
126C4⋊Q8C4⋊C412yes—
127#125—22no—
128#125—22no—
129#125—41no—
130#125—41no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—22no—
148#117—22no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C22⋊C4—41no—
173C2×C42—41no—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192#103—84no—
193#103—82no—
194#103—82no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—41no—
206#97—41no—
207#97—41no—
208#97C22⋊C414yes—
209C4×C8—41no—
210C4×C8—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215C24#12111yescharacteristic
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4#11711yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12711yescharacteristic
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—81no—
275C2×C8—81no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—42no—
301C42—42no—
302C42C2×C22⋊C411yescharacteristic
303C42C2×C4⋊C411yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42#12111yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42#12311yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12311yescharacteristic
316C42#12411yescharacteristic
317C42#12411yescharacteristic
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23#38811yescharacteristic
323C23#39211yescharacteristic
324C23#39611yescharacteristic
325C23#47112yes—
326C23#47312yes—
327Q8—44no—
328Q8—44no—
329Q8—82no—
330Q8—82no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—164no—
369C2×C4—162no—
370C2×C4—162no—
371C2×C4#38011yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38511yescharacteristic
374C2×C4#38611yescharacteristic
375C2×C4#38811yescharacteristic
376C2×C4#39011yescharacteristic
377C2×C4#39311yescharacteristic
378C2×C4#39411yescharacteristic
379C2×C4#39411yescharacteristic
380C2×C4#39611yescharacteristic
381C2×C4#39711yescharacteristic
382C2×C4#39911yescharacteristic
383C2×C4#45412yes—
384C2×C4#45512yes—
385C8—44no—
386C8—82no—
387C8—82no—
388C8—82no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22#143911yescharacteristic
398C22#193012yes—
399C22#200311yescharacteristic
400C22#200411yescharacteristic
401C22#200811yescharacteristic
402C22#200911yescharacteristic
403C22#201911yescharacteristic
404C22#202011yescharacteristic
405C22#206412yes—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—82no—
417C4—82no—
418C4—162no—
419C4—161no—
420C4—161no—
421C4#186012yes—
422C4#204012yes—
423C2—22no—
424C2—22no—
425C2#1013111yescharacteristic
426C2#1013511yescharacteristic
427C2#1013711yescharacteristic
428C2#1014111yescharacteristic
429C2#1107911yescharacteristic
430C2#1111711yescharacteristic
431C2#1114511yescharacteristic
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
29. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#218602
C2×#18362
C2×#15312
C2×#13022
#80402
#79142