Finite Group Databasebeta

← #110501

Subgroups — #110501

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21879C211yesmaximal, characteristic
2#20548C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#8121C211yesmaximal, characteristic
6#7976C211yesmaximal, characteristic
7#7864C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
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#1520C414yes—
24#1515—22no—
25#1506—22no—
26#1497—22no—
27#1493—21no—
28#1493—21no—
29#1479C414yescomplemented
30#1307—22no—
31#1288—22no—
32#1275—22no—
33C22×M16D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431C2×C412yes—
66#431D412yes—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×#101—24no—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8C2×C412yes—
82C4×M16—21no—
83C4×M16C2×C412yes—
84C2×#97—21no—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8D411yescharacteristic
88C2×C4×C8D411yescharacteristic
89#383D412yes—
90#382—21no—
91#381—21no—
92#380C2×C412yes—
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—41no—
110C2×M16—82no—
111C2×M16C22⋊C414yes—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124#125—41no—
125#125—41no—
126#125—41no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C4212yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183#101—84no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—22no—
193#97—41no—
194#97—41no—
195#97—41no—
196#97—82no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—82no—
202C4×C8C22⋊C412yes—
203C4×C8C22⋊C412yes—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C22⋊C414yes—
208C24C4×D411yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4C2×C4211yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241M16—44no—
242M16—42no—
243M16—42no—
244M16—82no—
245M16—82no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—82no—
275C42—42no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42C2×C22⋊C411yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12311yescharacteristic
291C42#12411yescharacteristic
292C42#12411yescharacteristic
293C42#12611yescharacteristic
294C42#12711yescharacteristic
295C23—22no—
296C23—41no—
297C23—41no—
298C23C4×C22⋊C411yescharacteristic
299C23#38611yescharacteristic
300C23#38811yescharacteristic
301Q8—44no—
302Q8—82no—
303Q8—82no—
304Q8—81no—
305Q8—81no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4C4×C22⋊C411yescharacteristic
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38611yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#39011yescharacteristic
344C2×C4#39011yescharacteristic
345C2×C4#39011yescharacteristic
346C8—42no—
347C8—42no—
348C8—44no—
349C8—82no—
350C8—82no—
351C8—82no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—82no—
361C22#143711yescharacteristic
362C22#190212yes—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—84no—
374C4—82no—
375C4—82no—
376C4#175712yes—
377C4#191111yescharacteristic
378C4#191211yescharacteristic
379C2—22no—
380C2—41no—
381C2—41no—
382C2#1094911yescharacteristic
383C2#1095211yescharacteristic
384C2#1096111yescharacteristic
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
29. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#218792
#205482
#175962
#174792
#81212
#79762
#78642