Finite Group Databasebeta

← #108665

Subgroups — #108665

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13806C211yesmaximal, characteristic
2C4×#343C211yesmaximal, characteristic
3C2×#1300C212yesmaximal
4C2×#1288C212yesmaximal
5#7912C211yesmaximal, characteristic
6C2×#432C2212yes—
7C2×C4×M16C2212yes—
8C2×#383C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×#376C2211yescharacteristic
11C2×#343—21no—
12C2×#343—21no—
13#1436—21no—
14#1436—21no—
15#1434—21no—
16#1434C411yescharacteristic
17#1434C411yescharacteristic
18C4×#95—21no—
19C4×#95C411yescharacteristic
20C4×#95C411yescharacteristic
21#1300—24no—
22#1288C418yescomplemented
23#1280—21no—
24#1280—21no—
25C22×M16—22no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2311yescharacteristic
34#432—22no—
35#432—22no—
36#432—22no—
37#432C2×C414yes—
38C2×C4⋊C8—22no—
39C2×C22⋊C8—22no—
40C4×M16—22no—
41C4×M16—22no—
42C4×M16—22no—
43C4×M16C2×C414yes—
44C2×#97—22no—
45C2×#97—22no—
46C2×C4×C8—22no—
47#383—22no—
48#383C2×C414yes—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51#376—21no—
52#376—21no—
53#376D412yes—
54#376Q812yes—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95C2×C411yescharacteristic
65C2×#95C2×C411yescharacteristic
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C411yescharacteristic
68C2×#95D411yescharacteristic
69C2×#95D411yescharacteristic
70#343—24no—
71#343—24no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C22×C8—42no—
90C22×C8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—81no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42—81no—
113C2×C42C22×C412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×Q811yescharacteristic
116C2×C42#4112yes—
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4112yes—
120C2×C42#4112yes—
121C4⋊C8—24no—
122C4⋊C8—42no—
123C22⋊C8—24no—
124C22⋊C8—42no—
125#97—24no—
126#97—24no—
127#97—42no—
128#97—42no—
129C4×C8—24no—
130C4×C8—42no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—24no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—81no—
146#95—81no—
147#95C4212yes—
148#95C4212yes—
149#95C4⋊C412yes—
150#95C4⋊C412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#10412yes—
207C22×C4#10412yes—
208C22×C4C2×C4211yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4#11711yescharacteristic
211C22×C4#11711yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4C4×D412yes—
215C22×C4C4×D412yes—
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4#12312yes—
221C22×C4#12312yes—
222M16—28no—
223M16—44no—
224M16—44no—
225M16—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—82no—
231C2×C8—82no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C42—44no—
235C42—44no—
236C42—81no—
237C42—81no—
238C42—81no—
239C42—81no—
240C42—84no—
241C42—84no—
242C42#11712yes—
243C42#11712yes—
244C42C4×D412yes—
245C42C4×Q812yes—
246C42#12412yes—
247C42#12512yes—
248C42#12612yes—
249C42#12612yes—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23C4×C22⋊C411yescharacteristic
255C23#38011yescharacteristic
256C23#38111yescharacteristic
257C23C2×#10411yescharacteristic
258C23C2×#10411yescharacteristic
259C23#42111yescharacteristic
260C23#42111yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4#37611yescharacteristic
331C2×C4C4×C4⋊C411yescharacteristic
332C2×C4#38211yescharacteristic
333C2×C4#38311yescharacteristic
334C2×C4#38712yes—
335C2×C4#38712yes—
336C2×C4#38812yes—
337C2×C4#38812yes—
338C8—48no—
339C8—84no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22#143411yescharacteristic
353C22C4×#10411yescharacteristic
354C22#175911yescharacteristic
355C22#180611yescharacteristic
356C22#180611yescharacteristic
357C22#182611yescharacteristic
358C22#182611yescharacteristic
359C22#182711yescharacteristic
360C22#182711yescharacteristic
361C22#182911yescharacteristic
362C22#183011yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—44no—
372C4—42no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—82no—
377C4—81no—
378C4—81no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#877211yescharacteristic
384C2#877311yescharacteristic
385C2#877411yescharacteristic
386C2#877511yescharacteristic
387C2#1080811yescharacteristic
388C2#1081611yescharacteristic
389C2#1083011yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
22. #1288C4semidirect

Maximal subgroups

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

subgroupindex
#138062
C4×#3432
C2×#13002
C2×#12882
#79122