Finite Group Databasebeta

← #109869

Subgroups — #109869

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21988C211yesmaximal, characteristic
2#20537C211yesmaximal, characteristic
3#8044C211yesmaximal, characteristic
4#8031C211yesmaximal, characteristic
5C2×#1290C211yesmaximal, characteristic
6C2×#1287C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1854—22no—
18#1847C412yes—
19#1836C412yes—
20#1771—22no—
21#1770—21no—
22C8×C4⋊C4—21no—
23#1750—22no—
24#1296—21no—
25#1296—21no—
26#1295—22no—
27#1290C414yescomplemented
28#1287C414yescomplemented
29#1279—21no—
30#1279—21no—
31C23×C8—21no—
32C2×#117—21no—
33C2×#117D412yes—
34C22×C42C2311yescharacteristic
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432C2×C412yes—
43#432C2×C412yes—
44#432C2×C412yes—
45#432C2×C412yes—
46#432Q812yes—
47#431—21no—
48#431—21no—
49#431C2×C412yes—
50#431Q812yes—
51C2×#107—22no—
52C2×#106—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97C2×C412yes—
70C2×#97D412yes—
71C2×C4×C8—22no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74#384C2×C411yescharacteristic
75#384C2×C411yescharacteristic
76#384D411yescharacteristic
77#384D411yescharacteristic
78#382—21no—
79#382C2×C412yes—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#340—22no—
89#340—22no—
90#336—22no—
91#336—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—81no—
104C22×C8—81no—
105#117—82no—
106#117—81no—
107#117—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—41no—
122C2×C42—22no—
123C2×C42—41no—
124C2×C42C22×C411yescharacteristic
125C2×C42C22×C411yescharacteristic
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×Q811yescharacteristic
131C2×C42C2×Q811yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136#107—82no—
137#106—82no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—41no—
158#97—22no—
159#97—22no—
160#97—22no—
161#97—41no—
162#97—82no—
163#97C4214yes—
164#97C4⋊C414yes—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—82no—
168C4×C8C4⋊C412yes—
169C4×C8C4⋊C412yes—
170#95—22no—
171#95—22no—
172#95—41no—
173#95—41no—
174C24#12311yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#11711yescharacteristic
189C22×C4#11711yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×Q811yescharacteristic
195C22×C4C4×Q811yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C2×C8—24no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—82no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—82no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—42no—
243C42—42no—
244C42—81no—
245C42—81no—
246C42C2×C4211yescharacteristic
247C42C2×C4⋊C411yescharacteristic
248C42C2×C4⋊C411yescharacteristic
249C42#11711yescharacteristic
250C42C4×D411yescharacteristic
251C42C4×D411yescharacteristic
252C42C4×D411yescharacteristic
253C42C4×D411yescharacteristic
254C42C4×Q811yescharacteristic
255C42C4×Q811yescharacteristic
256C42C22⋊Q811yescharacteristic
257C42C22⋊Q811yescharacteristic
258C42#12311yescharacteristic
259C42#12311yescharacteristic
260C42#12411yescharacteristic
261C42C4⋊Q811yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#38611yescharacteristic
269C23#38711yescharacteristic
270C23#38711yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
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—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C4×C22⋊C411yescharacteristic
310C2×C4C4×C4⋊C411yescharacteristic
311C2×C4C4×C4⋊C411yescharacteristic
312C2×C4#38511yescharacteristic
313C2×C4#38511yescharacteristic
314C2×C4#38711yescharacteristic
315C2×C4#38711yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38911yescharacteristic
319C2×C4#38911yescharacteristic
320C2×C4#39111yescharacteristic
321C2×C4#47111yescharacteristic
322C2×C4#47111yescharacteristic
323C2×C4#47211yescharacteristic
324C2×C4#47311yescharacteristic
325C8—44no—
326C8—42no—
327C8—42no—
328C8—44no—
329C8—82no—
330C8—82no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#143611yescharacteristic
342C22#188111yescharacteristic
343C22#188111yescharacteristic
344C22#190711yescharacteristic
345C22#190811yescharacteristic
346C22#191111yescharacteristic
347C22#191211yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—161no—
361C4—161no—
362C4#193011yescharacteristic
363C4#193111yescharacteristic
364C4#195111yescharacteristic
365C4#195211yescharacteristic
366C2—41no—
367C2—41no—
368C2#937111yescharacteristic
369C2#937211yescharacteristic
370C2#937311yescharacteristic
371C2#937411yescharacteristic
372C2#1088811yescharacteristic
373C2#1089611yescharacteristic
374C2#1090011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C42 ▷ 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
27. #1290C4semidirect
28. #1287C4semidirect

Maximal subgroups

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

subgroupindex
#219882
#205372
#80442
#80312
C2×#12902
C2×#12872
#78932