Finite Group Databasebeta

← #495935

Subgroups — #495935

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic
2#21525C211yesmaximal, characteristic
3#20455C211yesmaximal, characteristic
4#13132C211yesmaximal, characteristic, complemented
5#13132C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#12993C211yesmaximal, characteristic
9#12920C211yesmaximal, characteristic
10#12868C211yesmaximal, characteristic
11#12674C211yesmaximal, characteristic, complemented
12C4×#367C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2323—21no—
22#2274—21no—
23#2237C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1935—21no—
39#1894—21no—
40#1808C2211yescharacteristic
41#1776—21no—
42#1776—21no—
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1763—21no—
49#1736—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×M16D411yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#467—22no—
73#467C2×C414yes—
74C4×Q16—22no—
75C4×Q16—22no—
76#427—22no—
77#427C2×C414yes—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92#405—22no—
93#405—22no—
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—21no—
99#389—21no—
100#389—41no—
101#388—41no—
102#388—41no—
103#387—41no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—41no—
111C4×C4⋊C4—41no—
112C4×C22⋊C4—41no—
113C4×C22⋊C4—41no—
114#367—22no—
115#367—22no—
116#367C2×C414yescomplemented
117#366C2×C414yescomplemented
118#358—22no—
119#358—22no—
120#358C2×C414yescomplemented
121#358C2×C414yescomplemented
122C22⋊C16—22no—
123C22⋊C16C2×C414yes—
124#346—22no—
125C4×C16—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×Q8—21no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C22×C412yes—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C22×C412yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8#4112yes—
154C22×C8C2411yescharacteristic
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—22no—
160#117—22no—
161#117—24no—
162#117—42no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4D811yescharacteristic
177C2×C4⋊C4D811yescharacteristic
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D811yescharacteristic
182C2×C22⋊C4D811yescharacteristic
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16C22×C412yes—
203C2×C16C22×C412yes—
204#107—24no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107C22×C412yes—
215#107C22×C412yes—
216#107C22×C412yes—
217#107#4112yes—
218#107#4112yes—
219#106—24no—
220#103—24no—
221#103—42no—
222#103—42no—
223#97—22no—
224#97—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8#4112yes—
228C4×C8#4112yes—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24C2×D811yescharacteristic
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—44no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—161no—
259C22×C4—161no—
260C22×C4C4×D412yes—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4#13512yes—
267C22×C4C22×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272M16—44no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C4×D412yes—
285C2×C8#12312yes—
286C2×C8C23×C411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—161no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—161no—
317C42—21no—
318C42—21no—
319C42—44no—
320C42—82no—
321C42—161no—
322C42—161no—
323C42—161no—
324C42—161no—
325C42—161no—
326C42#12312yes—
327C42#13712yes—
328C16—22no—
329C16—22no—
330C16—41no—
331C16—41no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23C4×D812yes—
339C23C22×D811yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—164no—
385C2×C4C4×D812yes—
386C2×C4#48012yes—
387C2×C4#48012yes—
388C2×C4C2×C4×D411yescharacteristic
389C2×C4C2×#12311yescharacteristic
390C2×C4#52511yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C8—22no—
395C8—22no—
396C8—41no—
397C8C4×#4111yescharacteristic
398C8C4×#4111yescharacteristic
399C8#52911yescharacteristic
400C8#52911yescharacteristic
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22C2×C4×D811yescharacteristic
409C22C2×#48011yescharacteristic
410C22#310011yescharacteristic
411C22#341611yescharacteristic
412C22#341611yescharacteristic
413C22#341611yescharacteristic
414C22#341711yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4C4×#12311yescharacteristic
429C4C4×#13711yescharacteristic
430C4#309111yescharacteristic
431C4#310811yescharacteristic
432C2—42no—
433C2C4×#48011yescharacteristic
434C2#3345511yescharacteristic
435C2#3346011yescharacteristic
436C2#3365111yescharacteristic
437C2#3365311yescharacteristic
438C2#3368111yescharacteristic
439C2#3368311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13132C2semidirect
5. #13132C2semidirect
6. #13112C2semidirect
7. #13110C2semidirect
11. #12674C2semidirect
12. C4×#367C2semidirect
13. #12638C2semidirect
14. C4×#358C2semidirect
23. #2237C4semidirect
116. #367C2×C4semidirect
117. #366C2×C4semidirect
120. #358C2×C4semidirect
121. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
#215252
#204552
#131322
#131122
#131102
#129932
#129202
#128682
#126742
C4×#3672
#126382
C4×#3582
#126002