Finite Group Databasebeta

← #110491

Subgroups — #110491

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#20549C211yesmaximal, characteristic
3#17751C211yesmaximal, characteristic
4#17404C211yesmaximal, characteristic
5#8121C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#7856C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1995—22no—
16#1994—21no—
17#1994—21no—
18#1771—22no—
19#1767—21no—
20C4×C4⋊C8—21no—
21#1753—22no—
22#1733C414yes—
23#1561—22no—
24#1552—22no—
25#1549C412yescomplemented
26#1546C412yescomplemented
27#1472—22no—
28#1470C414yescomplemented
29#1467—22no—
30#1307—22no—
31#1292—22no—
32#1274—22no—
33C22×M16—21no—
34C22×M16C2×C412yes—
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×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117C2×C412yes—
51C2×#117C2×C412yes—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431C2×C412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80#407—21no—
81#407—21no—
82C2×C22⋊C8D412yes—
83C4×M16—21no—
84C4×M16—21no—
85C2×#97—21no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92#376D412yes—
93C43D411yescharacteristic
94C43D411yescharacteristic
95#339—24no—
96#339—24no—
97#336—22no—
98#336—22no—
99#332—24no—
100#332—24no—
101#324—28no—
102C22×Q8—21no—
103C22×Q8—21no—
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—41no—
113C2×M16—44no—
114C2×M16—82no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—24no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C4212yes—
158C2×C4⋊C4C4212yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42C22⋊C412yes—
167C2×C42C22⋊C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8C22⋊C414yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8C22⋊C414yes—
200#97—22no—
201#97—22no—
202#97—41no—
203#97—41no—
204#97—82no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—82no—
208#95—22no—
209#95—22no—
210C24C4×D411yescharacteristic
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—81no—
216C2×Q8—81no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—81no—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4#11711yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12611yescharacteristic
245M16—44no—
246M16—82no—
247M16—82no—
248M16—81no—
249M16—81no—
250M16—82no—
251M16—82no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—82no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—42no—
286C42—42no—
287C42—81no—
288C42—81no—
289C42#10114yes—
290C42C2×C22⋊C411yescharacteristic
291C42#11711yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12311yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42#12611yescharacteristic
305C42#12711yescharacteristic
306C23—22no—
307C23—41no—
308C23—41no—
309C23C4×C22⋊C411yescharacteristic
310C23#38611yescharacteristic
311C23#38811yescharacteristic
312Q8—82no—
313Q8—82no—
314Q8—82no—
315Q8—81no—
316Q8—81no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4C4×C22⋊C411yescharacteristic
350C2×C4C4×C22⋊C411yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#39011yescharacteristic
359C2×C4#39011yescharacteristic
360C2×C4#39011yescharacteristic
361C2×C4C2×#10112yes—
362C2×C4#41312yes—
363C8—44no—
364C8—44no—
365C8—82no—
366C8—82no—
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—82no—
377C22#143711yescharacteristic
378C22#190112yes—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—44no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4C4×#10112yes—
393C4#190411yescharacteristic
394C4#190511yescharacteristic
395C4#196611yescharacteristic
396C4#196711yescharacteristic
397C2—22no—
398C2—41no—
399C2—41no—
400C2#1095011yescharacteristic
401C2#1095411yescharacteristic
402C2#1096311yescharacteristic
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
25. #1549C4semidirect
26. #1546C4semidirect
28. #1470C4semidirect

Maximal subgroups

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

subgroupindex
#213292
#205492
#177512
#174042
#81212
#80042
#78562