Finite Group Databasebeta

← #391522

Subgroups — #391522

Order 512.

795 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23734C211yesmaximal, characteristic
2#11300C212yesmaximal, complemented
3#11214C212yesmaximal, complemented
4#11213C212yesmaximal, complemented
5#11210C212yesmaximal, complemented
6#11057C212yesmaximal
7#11056C211yesmaximal, characteristic
8#10784C212yesmaximal
9#10751C211yesmaximal, characteristic
10C2×#568C2211yescharacteristic
11C2×#428C2212yes—
12#2905C2211yescharacteristic
13C2×#421C2212yes—
14C2×#417C2212yes—
15C2×#410C2211yescharacteristic
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
36#568—42no—
37#568—41no—
38#568Q814yes—
39C2×#131—42no—
40C22×M16—41no—
41C22×M16C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×#117C2312yes—
45C2×#117C2312yes—
46C2×#117C2311yescharacteristic
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2312yes—
49C22×C42C2311yescharacteristic
50#428—42no—
51#428—42no—
52C2×#107C2×C414yes—
53C2×#106C2×C414yes—
54#424—42no—
55#423—41no—
56#421—24no—
57#421—42no—
58C2×#104—42no—
59#417—42no—
60#417—42no—
61C2×#103C2×C414yes—
62C2×#102C2×C414yes—
63#410—22no—
64#410—41no—
65C2×#99—41no—
66#391C2×C411yescharacteristic
67#391C2×C411yescharacteristic
68#390C2×C411yescharacteristic
69#390C2×C411yescharacteristic
70#389—42no—
71#389—42no—
72#386—41no—
73#384—41no—
74#384C2×C412yes—
75#384C2×C412yes—
76#380—42no—
77C4×C22⋊C4—41no—
78#344—42no—
79#344—42no—
80#343—42no—
81#342—41no—
82#340—42no—
83#340—42no—
84#339—42no—
85#339—42no—
86#339—42no—
87#338—42no—
88H(3,ℤ/4)—42no—
89#336C2×C412yes—
90#336C2×C412yes—
91C2×#41—21no—
92C2×#41—21no—
93C2×#41—22no—
94C2×#41—22no—
95C2×#41—22no—
96C2×#41—22no—
97C2×#41C2×Q812yes—
98C2×#41#4112yes—
99C22×Q8C22×C411yescharacteristic
100C22×D4—21no—
101C22×D4C22×C411yescharacteristic
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4C2411yescharacteristic
106#131—48no—
107#131—48no—
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—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—44no—
128C2×M16C2×Q814yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—42no—
132C22×C8C22×C414yes—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170#107—44no—
171#106—44no—
172C4⋊C8—44no—
173#104—48no—
174#103—44no—
175#102—44no—
176#99—44no—
177#95—41no—
178#95—41no—
179#95—44no—
180C24—21no—
181C24H(5,𝔽2)11yescharacteristic
182#41—24no—
183#41—24no—
184#41—24no—
185#41—24no—
186#41—42no—
187#41—42no—
188#41—42no—
189#41—42no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8C4×Q812yes—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4C4×Q812yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4H(5,𝔽2)12yes—
226C22×C4H(5,𝔽2)12yes—
227C22×C4H(5,𝔽2)12yes—
228C22×C4H(5,𝔽2)11yescharacteristic
229C22×C4H(5,𝔽2)11yescharacteristic
230C22×C42−1+412yes—
231C22×C42−1+411yescharacteristic
232M16—24no—
233M16—24no—
234M16—48no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—48no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C4⋊C4—84no—
267C4⋊C4—164no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—22no—
276C42—22no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—81no—
291C23—81no—
292C23#51811yescharacteristic
293C23#53811yescharacteristic
294C23#54411yescharacteristic
295C23#56012yes—
296C23#56212yes—
297Q8—24no—
298Q8—42no—
299D4—24no—
300D4—44no—
301D4—44no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—48no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—84no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—162no—
331C2×C4C2×C4×Q811yescharacteristic
332C2×C4#51812yes—
333C2×C4#52012yes—
334C2×C4#52011yescharacteristic
335C2×C4#53812yes—
336C2×C4#53911yescharacteristic
337C2×C4#53912yes—
338C2×C4#54312yes—
339C2×C4#54311yescharacteristic
340C2×C4#54412yes—
341C2×C4#56011yescharacteristic
342C2×C4#56011yescharacteristic
343C2×C4#56011yescharacteristic
344C2×C4#56012yes—
345C2×C4#56212yes—
346C2×C4#56212yes—
347C2×C4#56311yescharacteristic
348C2×C4#56312yes—
349C8—44no—
350C8—44no—
351C8—82no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—161no—
364C22#232911yescharacteristic
365C22#237611yescharacteristic
366C22#238011yescharacteristic
367C22#282612yes—
368C22#284112yes—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—81no—
373C4—81no—
374C4—84no—
375C4—162no—
376C4—162no—
377C4#232912yes—
378C4#237612yes—
379C4#282611yescharacteristic
380C4#283411yescharacteristic
381C4#284111yescharacteristic
382C4#284211yescharacteristic
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—82no—
388C2#1463611yescharacteristic
389C2#2336011yescharacteristic
390C2#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. #11300C2semidirect
3. #11214C2semidirect
4. #11213C2semidirect
5. #11210C2semidirect
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
#113002
#112142
#112132
#112102
#110572
#110562
#107842
#107512