Finite Group Databasebeta

← #110876

Subgroups — #110876

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21879C211yesmaximal, characteristic
2#20548C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#8129C211yesmaximal, characteristic
6#7965C211yesmaximal, characteristic
7#7960C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2283C2211yescharacteristic
15#1997C412yescomplemented
16#1966C412yescomplemented
17#1901—22no—
18#1801—22no—
19#1770—21no—
20C8×C4⋊C4—21no—
21#1768—22no—
22#1753—22no—
23#1518—22no—
24#1511C414yes—
25#1506—22no—
26#1493—21no—
27#1493—21no—
28#1487C414yescomplemented
29#1481—22no—
30#1309—22no—
31#1287—22no—
32#1286—22no—
33C22×M16—21no—
34C22×M16D412yes—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423C2×C412yes—
73#423D412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×#101—24no—
80#407—21no—
81#407C2×C412yes—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16C2×C412yes—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89#383—21no—
90#382D412yes—
91#381C2×C412yes—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C22⋊C4—21no—
95#376D412yes—
96#343—24no—
97#336—22no—
98#336—22no—
99#328—28no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102C22×Q8—21no—
103C22×Q8C4212yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—82no—
113C2×M16—82no—
114C2×M16C22⋊C414yes—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128#125—41no—
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—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C4212yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188#101—84no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—41no—
197#97—82no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—41no—
202C4×C8—82no—
203C4×C8C22⋊C412yes—
204C4×C8C22⋊C412yes—
205#95—22no—
206#95—22no—
207#95—22no—
208#95C22⋊C414yes—
209C24C4×D411yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—81no—
225C22×C4C2×C4211yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4#11711yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12611yescharacteristic
243M16—44no—
244M16—42no—
245M16—42no—
246M16—82no—
247M16—82no—
248M16—82no—
249M16—82no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—82no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42C2×C22⋊C411yescharacteristic
285C42#11711yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12311yescharacteristic
296C42#12411yescharacteristic
297C42#12411yescharacteristic
298C42#12611yescharacteristic
299C42#12711yescharacteristic
300C23—22no—
301C23—41no—
302C23—41no—
303C23C4×C22⋊C411yescharacteristic
304C23#38611yescharacteristic
305C23#38811yescharacteristic
306Q8—44no—
307Q8—82no—
308Q8—82no—
309Q8—81no—
310Q8—81no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4C4×C22⋊C411yescharacteristic
341C2×C4C4×C22⋊C411yescharacteristic
342C2×C4#38611yescharacteristic
343C2×C4#38611yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#39011yescharacteristic
350C2×C4#39011yescharacteristic
351C2×C4#39011yescharacteristic
352C2×C4#47112yes—
353C2×C4#47312yes—
354C8—42no—
355C8—42no—
356C8—44no—
357C8—82no—
358C8—82no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—82no—
369C22#143711yescharacteristic
370C22#190812yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—84no—
381C4—82no—
382C4—82no—
383C4#191111yescharacteristic
384C4#191211yescharacteristic
385C4#193012yes—
386C4#197711yescharacteristic
387C4#197911yescharacteristic
388C2—22no—
389C2—41no—
390C2—41no—
391C2#1094111yescharacteristic
392C2#1094211yescharacteristic
393C2#1094911yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1997C4semidirect
16. #1966C4semidirect
28. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#218792
#205482
#175902
#174862
#81292
#79652
#79602