Finite Group Databasebeta

← #148736

Subgroups — #148736

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21163C211yesmaximal, characteristic
2C2×#1833C211yesmaximal, characteristic
3C2×#1389C211yesmaximal, characteristic
4#12569C211yesmaximal, characteristic
5C2×#1381C211yesmaximal, characteristic
6#12556C211yesmaximal, characteristic
7#12545C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C23×C16C2211yescharacteristic
10C22×#108C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#427C2211yescharacteristic
14#2905—21no—
15C2×#405C2211yescharacteristic
16#1946C414yescomplemented
17#1945—22no—
18#1833—22no—
19#1833—22no—
20C2×#344—21no—
21#1733—21no—
22C2×H(3,ℤ/4)—21no—
23#1389—22no—
24#1387—22no—
25#1381—22no—
26#1380—22no—
27#1374—22no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16D411yescharacteristic
31C23×C8C2311yescharacteristic
32C2×#117—21no—
33C2×#117—21no—
34C2×#117D411yescharacteristic
35C2×M32—21no—
36C2×M32—21no—
37C2×M32—21no—
38C2×M32—21no—
39C2×M32—21no—
40C2×M32Q812yes—
41C22×C16—21no—
42C22×C16—21no—
43C22×C16—21no—
44C22×C16—21no—
45C22×C16—21no—
46C22×C16—21no—
47C22×C16—21no—
48C22×C16Q812yes—
49C2×#108—21no—
50C2×#108—21no—
51C2×#108—21no—
52C2×#108—21no—
53C2×#108—21no—
54C2×#108—21no—
55C2×#108—21no—
56C2×#108—21no—
57C2×#108—21no—
58C2×#108—21no—
59C2×#108C2×C412yes—
60C2×#108D412yes—
61#427—21no—
62#427—21no—
63#427—21no—
64#427—21no—
65#427—21no—
66#427C2×C412yes—
67#427D412yes—
68#427Q812yes—
69C2×#107—21no—
70C2×#107D412yes—
71C2×#106—21no—
72C2×#106—21no—
73#424—24no—
74#424—42no—
75#423—42no—
76#405—21no—
77#405—21no—
78#405—21no—
79#405—21no—
80C2×#97C2×C412yes—
81C2×C4×C8D412yes—
82#344—24no—
83#339—24no—
84H(3,ℤ/4)—24no—
85C23×C4C2×D411yescharacteristic
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8C22⋊C412yes—
107C22×C8C22×C411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×Q811yescharacteristic
113C22×C8C2×Q811yescharacteristic
114C22×C8C2×Q811yescharacteristic
115C22×C8#4111yescharacteristic
116C22×C8#4111yescharacteristic
117C22×C8#4111yescharacteristic
118C22×C8#4111yescharacteristic
119C22×C8#4111yescharacteristic
120C22×C8#4111yescharacteristic
121#117—21no—
122#117—21no—
123#117—21no—
124#117—21no—
125#117—22no—
126#117—22no—
127#117—22no—
128#117—41no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C22⋊C412yes—
142M32—22no—
143M32—22no—
144M32—41no—
145M32—41no—
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16—41no—
151C2×C16—41no—
152C2×C16—41no—
153C2×C16—41no—
154C2×C16—41no—
155C2×C16—41no—
156#108—22no—
157#108—22no—
158#108—22no—
159#108—22no—
160#108—41no—
161#108—41no—
162#108—41no—
163#108—41no—
164#107—22no—
165#107—22no—
166#107—41no—
167#107C4⋊C414yes—
168#106—22no—
169#106—22no—
170#106—22no—
171#106—41no—
172C4⋊C8—48no—
173#97—22no—
174#97—22no—
175C4×C8—22no—
176C4×C8—22no—
177C24#13511yescharacteristic
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4#12311yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4C4⋊Q811yescharacteristic
188C22×C4#13511yescharacteristic
189C22×C4#13511yescharacteristic
190C22×C4#13511yescharacteristic
191C22×C4AGL(1,ℤ/8)11yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4#13711yescharacteristic
194C22×C4#13711yescharacteristic
195M16—41no—
196M16—41no—
197M16—41no—
198M16—41no—
199M16—42no—
200M16—42no—
201M16—42no—
202M16—42no—
203M16—81no—
204M16—81no—
205M16—81no—
206M16—81no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8C2×C4⋊C411yescharacteristic
217C2×C8#11711yescharacteristic
218C2×C8C4×D411yescharacteristic
219C2×C8C4×D411yescharacteristic
220C2×C8C4×Q811yescharacteristic
221C2×C8C4×Q811yescharacteristic
222C2×C8#12111yescharacteristic
223C2×C8#12111yescharacteristic
224C2×C8#12111yescharacteristic
225C2×C8#12111yescharacteristic
226C2×C8#12111yescharacteristic
227C2×C8C22⋊Q811yescharacteristic
228C2×C8C22⋊Q811yescharacteristic
229C2×C8C22⋊Q811yescharacteristic
230C2×C8C22⋊Q811yescharacteristic
231C2×C8C22⋊Q811yescharacteristic
232C2×C8C22⋊Q811yescharacteristic
233C2×C8C22⋊Q811yescharacteristic
234C2×C8#12311yescharacteristic
235C2×C8#12311yescharacteristic
236C2×C8#12511yescharacteristic
237C2×C8#12511yescharacteristic
238C2×C8#12511yescharacteristic
239C2×C8#12511yescharacteristic
240C2×C8#12611yescharacteristic
241C2×C8#12611yescharacteristic
242C2×C8C4⋊Q811yescharacteristic
243C2×C8C4⋊Q811yescharacteristic
244C4⋊C4—24no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C22⋊C4—42no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—81no—
256C42—22no—
257C42—22no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—81no—
262C16—42no—
263C16—42no—
264C23—42no—
265C23#41611yescharacteristic
266C23#45211yescharacteristic
267C23#46311yescharacteristic
268C23#47811yescharacteristic
269C23#48511yescharacteristic
270C23#48811yescharacteristic
271C23#49911yescharacteristic
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—162no—
277C2×C4—162no—
278C2×C4#38011yescharacteristic
279C2×C4#38611yescharacteristic
280C2×C4#39111yescharacteristic
281C2×C4#39411yescharacteristic
282C2×C4#39511yescharacteristic
283C2×C4#39811yescharacteristic
284C2×C4#40011yescharacteristic
285C2×C4#41611yescharacteristic
286C2×C4#41811yescharacteristic
287C2×C4#41911yescharacteristic
288C2×C4#44911yescharacteristic
289C2×C4#44911yescharacteristic
290C2×C4#45211yescharacteristic
291C2×C4#46311yescharacteristic
292C2×C4#46411yescharacteristic
293C2×C4#46411yescharacteristic
294C2×C4#47811yescharacteristic
295C2×C4#47911yescharacteristic
296C2×C4#47911yescharacteristic
297C2×C4#48211yescharacteristic
298C2×C4#48211yescharacteristic
299C2×C4#48511yescharacteristic
300C2×C4#48811yescharacteristic
301C2×C4#48911yescharacteristic
302C2×C4#48911yescharacteristic
303C2×C4#49911yescharacteristic
304C2×C4#50111yescharacteristic
305C2×C4#50111yescharacteristic
306C8—42no—
307C8—82no—
308C8—82no—
309C8#38711yescharacteristic
310C8#38711yescharacteristic
311C8#38911yescharacteristic
312C8#38911yescharacteristic
313C8#39611yescharacteristic
314C8#39611yescharacteristic
315C8#39811yescharacteristic
316C8#39811yescharacteristic
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22#180411yescharacteristic
325C22#187311yescharacteristic
326C22#199111yescharacteristic
327C22#204211yescharacteristic
328C22#205911yescharacteristic
329C22#208511yescharacteristic
330C22#210111yescharacteristic
331C4—82no—
332C4—84no—
333C4—162no—
334C4#144111yescharacteristic
335C4#180511yescharacteristic
336C4#187511yescharacteristic
337C4#199311yescharacteristic
338C4#204211yescharacteristic
339C4#205911yescharacteristic
340C4#208511yescharacteristic
341C4#210111yescharacteristic
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2#1120811yescharacteristic
349C2#1282912yes—
subgroupG/H
centerC2×C4#418
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#1804
prime radicalC24#135
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1946C4semidirect

Maximal subgroups

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

subgroupindex
#211632
C2×#18332
C2×#13892
#125692
C2×#13812
#125562
#125452