Finite Group Databasebeta

← #120979

Subgroups — #120979

Order 512.

435 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#10848C211yesmaximal, characteristic
3#8087C211yesmaximal, characteristic
4#8063C211yesmaximal, characteristic
5#8042C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#7961C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1950—21no—
16#1950—21no—
17#1855—22no—
18#1849—21no—
19#1838—21no—
20C2×#344—22no—
21#1302C414yescomplemented
22#1299—22no—
23#1296—21no—
24#1296—21no—
25#1292—22no—
26#1286—21no—
27#1286—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16D412yes—
31C2×#117—21no—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432C2×C412yes—
40#432Q812yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431C2×C412yes—
45C2×#108—22no—
46C2×#108—22no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423—21no—
55#423D411yescharacteristic
56#423D411yescharacteristic
57#423D412yes—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64#407—21no—
65#407—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C2×#97—21no—
74C2×#97C2×C412yes—
75C2×C4×C8—21no—
76#384—21no—
77#382—21no—
78#380—21no—
79#380Q812yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82#344—44no—
83#341—22no—
84#341—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—21no—
92C2×M16—21no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—82no—
104C22×C8—22no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—41no—
109#117—42no—
110#117—42no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C22⋊C4—22no—
115C2×C42—41no—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128#108—82no—
129#108—82no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8C4⋊C412yes—
147C4⋊C8C4⋊C412yes—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154#97—22no—
155#97—22no—
156#97—22no—
157#97—22no—
158#97—41no—
159#97—41no—
160C4×C8—22no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164#95—22no—
165#95—41no—
166C24C22⋊Q811yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4#11711yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×Q811yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4C22⋊Q811yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4#12311yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4#12411yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4#12511yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12511yescharacteristic
191C22×C4#12611yescharacteristic
192C22×C4C4⋊Q811yescharacteristic
193M16—42no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—44no—
198M16—41no—
199M16—42no—
200M16—41no—
201M16—41no—
202M16—42no—
203M16—41no—
204M16—81no—
205M16—81no—
206M16—82no—
207M16—82no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C22⋊C4—44no—
230C42—82no—
231C42C2×C4⋊C411yescharacteristic
232C42#11711yescharacteristic
233C42C4×D411yescharacteristic
234C42C4×D411yescharacteristic
235C42#12111yescharacteristic
236C42#12111yescharacteristic
237C42C22⋊Q811yescharacteristic
238C42#12311yescharacteristic
239C42#12311yescharacteristic
240C42#12311yescharacteristic
241C42#12411yescharacteristic
242C42#12511yescharacteristic
243C42#12511yescharacteristic
244C42#12611yescharacteristic
245C42#12611yescharacteristic
246C42#12611yescharacteristic
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23#38711yescharacteristic
252C23#39311yescharacteristic
253C23#39811yescharacteristic
254C2×C4—21no—
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—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
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—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#38211yescharacteristic
283C2×C4#38511yescharacteristic
284C2×C4#38711yescharacteristic
285C2×C4#38811yescharacteristic
286C2×C4#38811yescharacteristic
287C2×C4#38911yescharacteristic
288C2×C4#39411yescharacteristic
289C2×C4#39711yescharacteristic
290C2×C4#39811yescharacteristic
291C2×C4#40011yescharacteristic
292C2×C4#40011yescharacteristic
293C2×C4#40011yescharacteristic
294C2×C4#45311yescharacteristic
295C2×C4#45411yescharacteristic
296C2×C4#45511yescharacteristic
297C2×C4#45611yescharacteristic
298C8—44no—
299C8—42no—
300C8—42no—
301C8—82no—
302C8—82no—
303C8—82no—
304C8—81no—
305C8—81no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22#144211yescharacteristic
317C22#202711yescharacteristic
318C22#202811yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—162no—
329C4#185911yescharacteristic
330C4#186011yescharacteristic
331C4#203711yescharacteristic
332C4#203811yescharacteristic
333C4#208111yescharacteristic
334C4#208211yescharacteristic
335C2—21no—
336C2—21no—
337C2—41no—
338C2—41no—
339C2#1126611yescharacteristic
340C2#1127611yescharacteristic
341C2#1128311yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ 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
21. #1302C4semidirect

Maximal subgroups

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

subgroupindex
#219942
#108482
#80872
#80632
#80422
#80042
#79612