Finite Group Databasebeta

← #120148

Subgroups — #120148

Order 512.

518 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21353C211yesmaximal, characteristic
2#20823C211yesmaximal, characteristic
3C2×#1302C211yesmaximal, characteristic
4#8034C211yesmaximal, characteristic
5C2×#1295C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#8017C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×#97C2211yescharacteristic
11C22×#97C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1940—22no—
16#1850—22no—
17#1847C412yescomplemented
18#1836C412yescomplemented
19#1808—21no—
20#1807—21no—
21#1776—22no—
22#1768—22no—
23#1302—22no—
24#1295—22no—
25#1295—22no—
26#1294—22no—
27#1294—21no—
28#1294—21no—
29C23×C8—21no—
30C23×C8—21no—
31C2×#117—21no—
32C22×C4⋊C4—21no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432Q812yes—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42C2×#107—22no—
43C2×#106—22no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—24no—
46C2×C4⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8D412yes—
49C2×#97—21no—
50C2×#97—21no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97—21no—
54C2×#97—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×#97—21no—
58C2×#97C2×C411yescharacteristic
59C2×#97C2×C411yescharacteristic
60C2×#97D412yes—
61C2×#97D412yes—
62C2×#97Q812yes—
63C2×C4×C8—21no—
64#384C2×C412yes—
65#383—21no—
66#382—21no—
67#382—21no—
68#382—21no—
69#382—21no—
70#382C2×C412yes—
71#382Q812yes—
72#381—21no—
73#380—21no—
74#380—21no—
75#380D411yescharacteristic
76#380D411yescharacteristic
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4D412yes—
80C4×C22⋊C4—21no—
81#376—21no—
82C2×#95—21no—
83C2×#95—21no—
84#340—22no—
85#340—22no—
86#336—24no—
87C23×C4—41no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4#4111yescharacteristic
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—44no—
98C22×C8—44no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102#117—81no—
103#117—81no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4—81no—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C22⋊C412yes—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—41no—
116C2×C42—22no—
117C2×C42—41no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×Q811yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130#107—44no—
131#106—44no—
132C4⋊C8—22no—
133C4⋊C8—48no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141#97—21no—
142#97—21no—
143#97—22no—
144#97—22no—
145#97—22no—
146#97—22no—
147#97—22no—
148#97—41no—
149#97—41no—
150#97—41no—
151#97C4⋊C412yes—
152#97C4⋊C412yes—
153C4×C8—41no—
154C4×C8—41no—
155#95—22no—
156#95—22no—
157#95—21no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166C24#12111yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4#12011yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12511yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4C4⋊Q811yescharacteristic
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—24no—
202C2×C8—22no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—44no—
208C2×C8—44no—
209C2×C8—42no—
210C2×C8—44no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—82no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C42—44no—
229C42—82no—
230C42C2×C4⋊C411yescharacteristic
231C42#11711yescharacteristic
232C42C4×Q811yescharacteristic
233C42C4×Q811yescharacteristic
234C42#12111yescharacteristic
235C42#12111yescharacteristic
236C42C22⋊Q811yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42C22⋊Q811yescharacteristic
239C42#12311yescharacteristic
240C42#12411yescharacteristic
241C42#12511yescharacteristic
242C42#12511yescharacteristic
243C42#12611yescharacteristic
244C42C4⋊Q811yescharacteristic
245C42C4⋊Q811yescharacteristic
246C23—22no—
247C23—22no—
248C23#38611yescharacteristic
249C23#39611yescharacteristic
250C23#39811yescharacteristic
251C23#47111yescharacteristic
252C23#47111yescharacteristic
253C23#47211yescharacteristic
254C23#47311yescharacteristic
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—162no—
292C2×C4—162no—
293C2×C4#38011yescharacteristic
294C2×C4#38711yescharacteristic
295C2×C4#38711yescharacteristic
296C2×C4#38911yescharacteristic
297C2×C4#38911yescharacteristic
298C2×C4#39111yescharacteristic
299C2×C4#39411yescharacteristic
300C2×C4#39511yescharacteristic
301C2×C4#39611yescharacteristic
302C2×C4#39811yescharacteristic
303C2×C4#39811yescharacteristic
304C2×C4#40011yescharacteristic
305C8—24no—
306C8—42no—
307C8—44no—
308C8—82no—
309C8—82no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22#144111yescharacteristic
319C22#188412yes—
320C22#190612yes—
321C22#190711yescharacteristic
322C22#190811yescharacteristic
323C22#206311yescharacteristic
324C22#206411yescharacteristic
325C22#208911yescharacteristic
326C22#209011yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4—162no—
342C4—162no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2#943112yes—
348C2#943212yes—
349C2#1121211yescharacteristic
350C2#1121511yescharacteristic
351C2#1121611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×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
17. #1847C4semidirect
18. #1836C4semidirect

Maximal subgroups

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

subgroupindex
#213532
#208232
C2×#13022
#80342
C2×#12952
#80262
#80172