Finite Group Databasebeta

← #149095

Subgroups — #149095

Order 512.

458 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21181C211yesmaximal, characteristic
2#12567C211yesmaximal, characteristic
3#12563C211yesmaximal, characteristic
4C2×#1382C211yesmaximal, characteristic
5#12556C211yesmaximal, characteristic
6C2×#1379C211yesmaximal, characteristic
7#10866C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C23×C16C2211yescharacteristic
10C22×#108C2211yescharacteristic
11#2911—21no—
12C2×#427C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#1952—22no—
17#1951—22no—
18#1742—21no—
19#1740—21no—
20C2×H(3,ℤ/4)—21no—
21#1386—22no—
22#1383—22no—
23#1382C414yescomplemented
24#1380—22no—
25#1379—22no—
26C22×M16—21no—
27C22×M16D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117D411yescharacteristic
33C2×M32—21no—
34C2×M32—21no—
35C2×M32—21no—
36C2×M32—21no—
37C2×M32—21no—
38C2×M32D412yes—
39C22×C16—21no—
40C22×C16—21no—
41C22×C16—21no—
42C22×C16—21no—
43C22×C16—21no—
44C22×C16—21no—
45C22×C16—21no—
46C22×C16C2×C412yes—
47C2×#108—21no—
48C2×#108—21no—
49C2×#108—21no—
50C2×#108C2×C412yes—
51C2×#108D412yes—
52C2×#108Q812yes—
53#428—24no—
54#428—42no—
55#427—21no—
56#427—21no—
57#427—21no—
58#427—21no—
59#427—21no—
60#427—21no—
61#427—21no—
62#427—21no—
63#427—21no—
64#427—21no—
65#427—42no—
66#427C2×C412yes—
67#427Q812yes—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107D412yes—
71C2×#106—21no—
72C2×#106—21no—
73C2×#106—21no—
74#405—21no—
75#405—21no—
76#405—21no—
77#405—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80#343—24no—
81#342—42no—
82H(3,ℤ/4)—42no—
83C23×C4C2×D411yescharacteristic
84C2×M16—21no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—21no—
88C2×M16—21no—
89C2×M16—22no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16Q1612yes—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8C22×C411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×Q811yescharacteristic
103C22×C8C2×Q811yescharacteristic
104C22×C8#4111yescharacteristic
105C22×C8#4111yescharacteristic
106C22×C8#4111yescharacteristic
107C22×C8#4111yescharacteristic
108C22×C8#4111yescharacteristic
109C22×C8#4111yescharacteristic
110C22×C8#4111yescharacteristic
111C22×C8#4111yescharacteristic
112#117—21no—
113#117—21no—
114#117—21no—
115#117—21no—
116#117—22no—
117#117—22no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4SD1612yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C42—21no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140M32—22no—
141M32—22no—
142M32—41no—
143M32—41no—
144C2×C16—22no—
145C2×C16—22no—
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—41no—
149C2×C16—41no—
150C2×C16—41no—
151C2×C16—41no—
152C2×C16—41no—
153C2×C16—41no—
154#108—22no—
155#108—22no—
156#108—22no—
157#108C4⋊C414yes—
158#107—22no—
159#107—22no—
160#107—22no—
161#107—22no—
162#107—44no—
163#107—41no—
164#107—41no—
165#106—22no—
166#106—22no—
167#106—44no—
168#106—41no—
169#106—41no—
170#106—41no—
171#106—41no—
172#97—41no—
173#97—41no—
174C4×C8—41no—
175C4×C8—41no—
176C24AGL(1,ℤ/8)11yescharacteristic
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C4×D411yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4C2×SD1611yescharacteristic
189C22×C4C2×Q1611yescharacteristic
190C22×C4#13511yescharacteristic
191C22×C4#13511yescharacteristic
192C22×C4AGL(1,ℤ/8)11yescharacteristic
193C22×C4AGL(1,ℤ/8)11yescharacteristic
194C22×C4#13711yescharacteristic
195M16—24no—
196M16—22no—
197M16—22no—
198M16—42no—
199M16—41no—
200M16—42no—
201M16—41no—
202M16—42no—
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8C2×C4⋊C411yescharacteristic
210C2×C8#11711yescharacteristic
211C2×C8#11711yescharacteristic
212C2×C8C4×D411yescharacteristic
213C2×C8C4×D411yescharacteristic
214C2×C8C4×Q811yescharacteristic
215C2×C8#12111yescharacteristic
216C2×C8#12111yescharacteristic
217C2×C8C22⋊Q811yescharacteristic
218C2×C8C22⋊Q811yescharacteristic
219C2×C8C22⋊Q811yescharacteristic
220C2×C8#12311yescharacteristic
221C2×C8#12311yescharacteristic
222C2×C8#12311yescharacteristic
223C2×C8#12311yescharacteristic
224C2×C8#12411yescharacteristic
225C2×C8#12411yescharacteristic
226C2×C8#12411yescharacteristic
227C2×C8#12511yescharacteristic
228C2×C8#12511yescharacteristic
229C2×C8#12511yescharacteristic
230C2×C8#12511yescharacteristic
231C2×C8#12511yescharacteristic
232C2×C8#12611yescharacteristic
233C2×C8#12611yescharacteristic
234C2×C8#12611yescharacteristic
235C2×C8#12611yescharacteristic
236C2×C8C4⋊Q811yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42—81no—
258C42—81no—
259C42—81no—
260C42—81no—
261C16—42no—
262C16—42no—
263C23—42no—
264C23#44011yescharacteristic
265C23#44911yescharacteristic
266C23#46911yescharacteristic
267C23#47711yescharacteristic
268C23#47811yescharacteristic
269C23#48111yescharacteristic
270C23#49011yescharacteristic
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—162no—
277C2×C4—162no—
278C2×C4—162no—
279C2×C4#38511yescharacteristic
280C2×C4#38711yescharacteristic
281C2×C4#38811yescharacteristic
282C2×C4#39311yescharacteristic
283C2×C4#39411yescharacteristic
284C2×C4#39811yescharacteristic
285C2×C4#40011yescharacteristic
286C2×C4C4×SD1611yescharacteristic
287C2×C4C4×Q1611yescharacteristic
288C2×C4#44211yescharacteristic
289C2×C4#44911yescharacteristic
290C2×C4#45011yescharacteristic
291C2×C4#45111yescharacteristic
292C2×C4#46511yescharacteristic
293C2×C4#46711yescharacteristic
294C2×C4#46811yescharacteristic
295C2×C4#47511yescharacteristic
296C2×C4#47611yescharacteristic
297C2×C4#47711yescharacteristic
298C2×C4#47811yescharacteristic
299C2×C4#47811yescharacteristic
300C2×C4#47911yescharacteristic
301C2×C4#48211yescharacteristic
302C2×C4#48411yescharacteristic
303C2×C4#48511yescharacteristic
304C2×C4#48711yescharacteristic
305C2×C4#48811yescharacteristic
306C2×C4#48911yescharacteristic
307C8—44no—
308C8—42no—
309C8#38211yescharacteristic
310C8#38711yescharacteristic
311C8#38811yescharacteristic
312C8#38911yescharacteristic
313C8#39711yescharacteristic
314C8#39811yescharacteristic
315C8#40011yescharacteristic
316C8#40011yescharacteristic
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22#186311yescharacteristic
325C22#191911yescharacteristic
326C22#193611yescharacteristic
327C22#202511yescharacteristic
328C22#203411yescharacteristic
329C22#208711yescharacteristic
330C22#210011yescharacteristic
331C4—82no—
332C4—162no—
333C4—162no—
334C4—162no—
335C4#144211yescharacteristic
336C4#186411yescharacteristic
337C4#192111yescharacteristic
338C4#193311yescharacteristic
339C4#202411yescharacteristic
340C4#203611yescharacteristic
341C4#208811yescharacteristic
342C4#210111yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#1129311yescharacteristic
350C2#1340112yes—
subgroupG/H
centerC2×C4#482
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#2034
prime radicalC24AGL(1,ℤ/8)
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
23. #1382C4semidirect

Maximal subgroups

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

subgroupindex
#211812
#125672
#125632
C2×#13822
#125562
C2×#13792
#108662