Finite Group Databasebeta

← #391608

Subgroups — #391608

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23734C211yesmaximal, characteristic
2#11299C212yesmaximal, complemented
3#11213C212yesmaximal, complemented
4#11210C212yesmaximal, complemented
5#11185C212yesmaximal
6#11160C211yesmaximal, characteristic
7#11057C212yesmaximal
8#10959C211yesmaximal, characteristic
9#10904C212yesmaximal, complemented
10C2×#428C2212yes—
11C2×#421C2212yes—
12C2×#417C2212yes—
13C2×#410C2211yescharacteristic
14#2879C2211yescharacteristic
15#2873C2211yescharacteristic
16#2316C2212yes—
17#2297C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2096C412yescomplemented
20#2095C412yescomplemented
21#2094C412yescomplemented
22#2093C412yescomplemented
23C2×#344C2212yes—
24C2×#344C2212yes—
25C2×#343C2212yes—
26#1740C2211yescharacteristic
27#1736C2212yes—
28#1736C2212yes—
29#1733C2212yes—
30C2×#339C2212yes—
31C2×#339C2212yes—
32C2×#338C2212yes—
33C2×H(3,ℤ/4)C2212yes—
34#1728C2211yescharacteristic
35C22×#41C2311yescharacteristic
36C22×M16C2312yes—
37C22×M16C2312yes—
38C22×M16C2312yes—
39C2×#117C2312yes—
40C2×#117C2312yes—
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—41no—
43C22×C4⋊C4C2312yes—
44C22×C42C2311yescharacteristic
45#428—42no—
46#428—42no—
47C2×#107C2×C414yes—
48C2×#106C2×C414yes—
49#421—24no—
50#421—42no—
51C2×#104—42no—
52#417—42no—
53#417—42no—
54C2×#103C2×C414yes—
55C2×#102C2×C414yes—
56#410—42no—
57C2×#99—41no—
58#408—42no—
59#407—41no—
60#405—42no—
61C4×M16—41no—
62#391C2×C411yescharacteristic
63#391C2×C411yescharacteristic
64#390C2×C411yescharacteristic
65#390C2×C411yescharacteristic
66#389—42no—
67#389—42no—
68#386—41no—
69#384—41no—
70#384C2×C412yes—
71#384C2×C412yes—
72#380—42no—
73C4×C22⋊C4—41no—
74#344—24no—
75#344—42no—
76#343—42no—
77#342—41no—
78#340—42no—
79#340—42no—
80#339—42no—
81#339—42no—
82#339—42no—
83#338—24no—
84H(3,ℤ/4)—42no—
85#336C2×C412yes—
86#336C2×C412yes—
87C2×#41—22no—
88C2×#41—22no—
89C2×#41—22no—
90C2×#41—22no—
91C2×#41—22no—
92C2×#41—22no—
93C22×Q8C22×C411yescharacteristic
94C22×D4—21no—
95C22×D4C22×C411yescharacteristic
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4C2411yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16#4114yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C22×C414yes—
121#117—22no—
122#117—22no—
123#117—22no—
124#117—22no—
125#117—22no—
126#117—22no—
127#117—22no—
128#117—22no—
129#117—22no—
130#117—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—84no—
142C2×C4⋊C4C22×C414yes—
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158#107—44no—
159#106—44no—
160#104—48no—
161#103—44no—
162#102—44no—
163#99—44no—
164C22⋊C8—44no—
165#97—42no—
166C4×C8—42no—
167#95—41no—
168#95—41no—
169#95—44no—
170C24—21no—
171C24H(5,𝔽2)11yescharacteristic
172#41—42no—
173#41—42no—
174#41—42no—
175#41—42no—
176#41—42no—
177#41—42no—
178#41—42no—
179#41—42no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—42no—
191C2×D4—41no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—84no—
211C22×C4C23×C411yescharacteristic
212C22×C4C2×#4112yes—
213C22×C4H(5,𝔽2)12yes—
214C22×C4H(5,𝔽2)12yes—
215C22×C4H(5,𝔽2)12yes—
216C22×C4H(5,𝔽2)11yescharacteristic
217C22×C42−1+412yes—
218C22×C42−1+412yes—
219M16—24no—
220M16—24no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—164no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C42—22no—
261C42—22no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—81no—
275C23—81no—
276C23#51811yescharacteristic
277C23#54212yes—
278C23#56012yes—
279C23#56011yescharacteristic
280C23#56311yescharacteristic
281Q8—42no—
282Q8—42no—
283D4—42no—
284D4—42no—
285D4—82no—
286D4—82no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—84no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—162no—
314C2×C4C4×#4111yescharacteristic
315C2×C4#51812yes—
316C2×C4#51911yescharacteristic
317C2×C4#52012yes—
318C2×C4#53812yes—
319C2×C4#53912yes—
320C2×C4#53912yes—
321C2×C4#54012yes—
322C2×C4#54212yes—
323C2×C4#56011yescharacteristic
324C2×C4#56011yescharacteristic
325C2×C4#56212yes—
326C2×C4#56212yes—
327C2×C4#56212yes—
328C2×C4#56312yes—
329C2×C4#56312yes—
330C8—44no—
331C8—44no—
332C8—82no—
333C22—21no—
334C22—22no—
335C22—21no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—82no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—161no—
345C22#233211yescharacteristic
346C22#237611yescharacteristic
347C22#238011yescharacteristic
348C22#282912yes—
349C22#283812yes—
350C4—42no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—84no—
356C4—162no—
357C4—162no—
358C4#233512yes—
359C4#237812yes—
360C4#282712yes—
361C4#283712yes—
362C2—21no—
363C2—22no—
364C2—21no—
365C2—22no—
366C2—161no—
367C2#1464611yescharacteristic
368C2#2336011yescharacteristic
369C2#2336111yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11299C2semidirect
3. #11213C2semidirect
4. #11210C2semidirect
9. #10904C2semidirect
19. #2096C4semidirect
20. #2095C4semidirect
21. #2094C4semidirect
22. #2093C4semidirect

Maximal subgroups

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

subgroupindex
#237342
#112992
#112132
#112102
#111852
#111602
#110572
#109592
#109042