Finite Group Databasebeta

← #189869

Subgroups — #189869

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21818C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#8714C218yesmaximal
5#8026C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#8014C211yesmaximal, characteristic
8#8014C211yesmaximal, characteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2356—22no—
13#2356—22no—
14#2355C414yescomplemented
15#2355C414yescomplemented
16#2313—22no—
17#2310—21no—
18C2×#391—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283C2214yes—
22C2×#382C2211yescharacteristic
23#2274—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1984—22no—
26#1984—22no—
27#1817C2214yes—
28#1441—22no—
29C4⋊#95—21no—
30#1434C2214yes—
31#1294C2214yes—
32#1294C2214yes—
33#1294C2214yes—
34#1294C2214yes—
35C23×Q8D411yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8C2×C412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8D412yes—
44C2×#117—21no—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4D411yescharacteristic
50C22×C42—21no—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53#431—22no—
54#431—22no—
55#431—22no—
56#431—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C22⋊C8C2312yes—
60C2×C22⋊C8C2312yes—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63#391—24no—
64#391—24no—
65#391—44no—
66#387—28no—
67#387—28no—
68#384C2×C412yes—
69#384C2×C412yes—
70#383—24no—
71#382—21no—
72#382—21no—
73#382—22no—
74#382—22no—
75#382C2×C418yes—
76#381—24no—
77#380—21no—
78#380—21no—
79#380C2312yes—
80#380C2312yes—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C418yes—
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4C2×C412yes—
90C4×C22⋊C4—24no—
91#376—24no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—21no—
95C2×#95C2312yes—
96C2×#95C2312yes—
97#336—28no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8C22⋊C412yes—
105C22×Q8C22⋊C412yes—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2411yescharacteristic
113C22×C8—22no—
114C22×C8—22no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—24no—
121C4×Q8—24no—
122C4×Q8—24no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117—42no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C414yes—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—28no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42—42no—
158C2×C42C22×C414yes—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D414yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4112yes—
165C2×C42#4112yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C4⋊C8—42no—
169C4⋊C8—42no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C4×C8—42no—
175C4×C8—42no—
176#95—28no—
177#95—28no—
178#95—22no—
179#95—22no—
180#95—28no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—48no—
188#95—48no—
189#95—42no—
190C24C22×D411yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4#12312yes—
233C22×C4#12312yes—
234C22×C4#12312yes—
235C22×C4#12312yes—
236C22×C4#13512yes—
237C22×C4#13512yes—
238C22×C4AGL(1,ℤ/8)12yes—
239C22×C4#13712yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—84no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—44no—
271C42—44no—
272C42—88no—
273C42—84no—
274C42#11712yes—
275C42#11712yes—
276C42#11712yes—
277C42#11712yes—
278C42#12312yes—
279C42#12312yes—
280C42#12312yes—
281C42#12312yes—
282C23—22no—
283C23—22no—
284C23C22×C22⋊C411yescharacteristic
285C23C2×#12311yescharacteristic
286C23C2×#12311yescharacteristic
287C23C2×#13511yescharacteristic
288C23C2×#13511yescharacteristic
289C23C2×AGL(1,ℤ/8)11yescharacteristic
290C23C2×#13711yescharacteristic
291Q8—48no—
292Q8—48no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4#38112yes—
333C2×C4#38112yes—
334C2×C4#38112yes—
335C2×C4#38112yes—
336C2×C4#41612yes—
337C2×C4#41612yes—
338C2×C4#41712yes—
339C2×C4#41712yes—
340C2×C4#48214yes—
341C2×C4#48214yes—
342C2×C4#48514yes—
343C2×C4#48514yes—
344C2×C4C2×#11711yescharacteristic
345C2×C4C2×#11711yescharacteristic
346C2×C4C2×#12311yescharacteristic
347C2×C4C2×#12311yescharacteristic
348C8—82no—
349C8—82no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22C2×#38111yescharacteristic
359C22#288412yes—
360C22C2×#41611yescharacteristic
361C22C2×#41711yescharacteristic
362C22#290112yes—
363C22C2×#48211yescharacteristic
364C22C2×#48211yescharacteristic
365C22C2×#48511yescharacteristic
366C22C2×#48511yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—22no—
373C4—44no—
374C4—44no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4—162no—
379C4#182514yes—
380C4#182514yes—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2#1761612yes—
386C2#1762812yes—
387C2#2069111yescharacteristic
388C2C2×#182511yescharacteristic
389C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2355C4semidirect
15. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#218182
#139422
#87142
#80262
#80142