Finite Group Databasebeta

← #110629

Subgroups — #110629

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21389C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3#17464C211yesmaximal, characteristic
4#17314C211yesmaximal, characteristic
5#8068C211yesmaximal, characteristic
6#8006C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1988C412yes—
16#1981C412yes—
17#1873—22no—
18C4×#103—22no—
19#1531C414yescomplemented
20#1528—22no—
21#1526—22no—
22#1485—22no—
23#1481C414yescomplemented
24#1479—22no—
25#1450—22no—
26C8×M16—22no—
27C8×M16—22no—
28#1300—22no—
29#1293—22no—
30#1276—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432C2×C412yes—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8C2×C412yes—
69C2×C4⋊C8C2×C412yes—
70C2×#103—24no—
71#407—21no—
72#407—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16D412yes—
80C4×M16D412yes—
81C2×#97C2×C412yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×C412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90C43—21no—
91#340—24no—
92#330—28no—
93Q8⋊C8—28no—
94#322—24no—
95C82—24no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—28no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C22⋊C414yes—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C4212yes—
116C4⋊Q8C4212yes—
117#125—22no—
118#125—22no—
119#125—41no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—22no—
138#117—22no—
139#117—44no—
140#117—41no—
141#117—41no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8C22⋊C414yes—
175#103—48no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184#97—22no—
185#97—22no—
186#97—41no—
187#97C22⋊C414yes—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—41no—
196#95—22no—
197#95—41no—
198C24#12411yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—44no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4#11711yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12711yescharacteristic
229M16—28no—
230M16—28no—
231M16—82no—
232M16—82no—
233C2×C8—28no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—82no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—44no—
259C42—42no—
260C42—42no—
261C42—44no—
262C42—82no—
263C42C2×C4211yescharacteristic
264C42C2×C22⋊C411yescharacteristic
265C42C2×C22⋊C411yescharacteristic
266C42#11711yescharacteristic
267C42C4×D411yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42#12011yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12311yescharacteristic
277C42#12411yescharacteristic
278C42#12411yescharacteristic
279C23—22no—
280C23—41no—
281C23—41no—
282C23#38811yescharacteristic
283C23#38811yescharacteristic
284C23#39011yescharacteristic
285Q8—44no—
286Q8—44no—
287Q8—82no—
288Q8—82no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4C4×C22⋊C411yescharacteristic
314C2×C4C4×C22⋊C411yescharacteristic
315C2×C4C4×C22⋊C411yescharacteristic
316C2×C4#38611yescharacteristic
317C2×C4#38611yescharacteristic
318C2×C4#38611yescharacteristic
319C2×C4#38811yescharacteristic
320C2×C4#38811yescharacteristic
321C2×C4#38811yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#39011yescharacteristic
324C2×C4#39011yescharacteristic
325C2×C4#44314yes—
326C8—28no—
327C8—44no—
328C8—44no—
329C8—82no—
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—82no—
340C22#143711yescharacteristic
341C22#197412yes—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—84no—
351C4—82no—
352C4—82no—
353C4#176412yes—
354C4#188412yes—
355C4#190212yes—
356C4#193712yes—
357C2—22no—
358C2—41no—
359C2—41no—
360C2#1092911yescharacteristic
361C2#1095211yescharacteristic
362C2#1095911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1531C4semidirect
23. #1481C4semidirect

Maximal subgroups

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

subgroupindex
#213892
#176482
#174642
#173142
#80682
#80062
#78702