Finite Group Databasebeta

← #211856

Subgroups — #211856

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22165C212yesmaximal, complemented
2#19125C211yesmaximal, characteristic
3#17956C211yesmaximal, characteristic, complemented
4#17328C211yesmaximal, characteristic, complemented
5#9719C211yesmaximal, characteristic, complemented
6#9624C211yesmaximal, characteristic, complemented
7#9592C212yesmaximal, complemented
8#9210C212yesmaximal, complemented
9#8667C211yesmaximal, characteristic, complemented
10C8×#107C211yesmaximal, characteristic, complemented
11#8257C212yesmaximal, complemented
12#3169C2211yescharacteristic
13#3035C2212yes—
14#2965C2211yescharacteristic
15C8×#41C2211yescharacteristic
16#2931C2212yes—
17#2922C2211yescharacteristic
18#1980C2212yes—
19#1974—22no—
20#1950C2212yes—
21#1937—22no—
22#1876—22no—
23#1866—22no—
24#1806C2212yes—
25C4×#107C2211yescharacteristic
26C8×C4⋊C4C2212yes—
27#1682C412yescomplemented
28#1681C412yescomplemented
29#1674C411yescharacteristic, complemented
30#1673C411yescharacteristic, complemented
31#1668C411yescharacteristic, complemented
32#1667C411yescharacteristic, complemented
33#1589—21no—
34#1589—21no—
35C8×D8—21no—
36C8×D8—21no—
37#1567C2211yescharacteristic
38C2×#335C2211yescharacteristic
39#1540C2212yes—
40#1502C2212yes—
41#1492C2212yes—
42#1454—21no—
43#1453—21no—
44#1452C2211yescharacteristic
45#1451C2211yescharacteristic
46#1450—21no—
47C8×M16—21no—
48C8×M16C2211yescharacteristic
49C2×C82C2211yescharacteristic
50#1303C2212yes—
51#1301C2212yes—
52#1279C2212yes—
53C2×#135—22no—
54C2×#131—22no—
55#532C2312yes—
56C4×#41C2311yescharacteristic
57#500C2×C411yescharacteristic
58#493C2×C411yescharacteristic
59#486C2×C412yes—
60#466—21no—
61#466—21no—
62#464—22no—
63#463—22no—
64#462C2×C412yes—
65#461C2×C412yes—
66#460C2×C412yes—
67#459C2×C412yes—
68#446—21no—
69C8×Q8C2×C411yescharacteristic
70C8×Q8C2×C411yescharacteristic
71C4×D8—21no—
72C4×D8—21no—
73#436—21no—
74#436—22no—
75#436—22no—
76#435—22no—
77C8×D4—21no—
78C8×D4—21no—
79C8×D4—22no—
80C8×D4—22no—
81C8×D4C2×C411yescharacteristic
82C8×D4C2×C411yescharacteristic
83#433—21no—
84#433—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91C2×#108—22no—
92C2×#107—21no—
93#424D411yescharacteristic
94#424D411yescharacteristic
95#423C2312yes—
96C2×C4⋊C8C2312yes—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100#416—22no—
101#408—22no—
102#408D411yescharacteristic
103#408D411yescharacteristic
104C4×M16—21no—
105C4×M16C2311yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—22no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C4×C4⋊C4C2312yes—
112#341—22no—
113#340—21no—
114#339—22no—
115#336—22no—
116#335—21no—
117#335—21no—
118#335C2×C411yescharacteristic
119#335C2×C411yescharacteristic
120#335C2×C412yes—
121#332C2×C412yes—
122#332C2×C412yes—
123#331C2×C412yes—
124#331C2×C412yes—
125#327—24no—
126D4⋊C8—22no—
127D4⋊C8—22no—
128#322—21no—
129#322—21no—
130#322—21no—
131#322—21no—
132#322—21no—
133C82—21no—
134C82D411yescharacteristic
135C82D411yescharacteristic
136C82D411yescharacteristic
137C82D411yescharacteristic
138C2×#41—22no—
139C2×#41C2×D411yescharacteristic
140#135—82no—
141#135—82no—
142C2×Q16—22no—
143C2×SD16—22no—
144C2×SD16—22no—
145C2×D8—22no—
146C2×D8—21no—
147C2×D8—21no—
148#131—82no—
149#131—82no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—42no—
153C2×M16C2×D411yescharacteristic
154C2×M16C2×D411yescharacteristic
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—42no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8#4111yescharacteristic
163C22×C8#4111yescharacteristic
164C4⋊Q8C22×C412yes—
165#127C22×C412yes—
166#124—22no—
167#121—22no—
168#121—22no—
169C4×Q8C22×C411yescharacteristic
170C4×Q8#4111yescharacteristic
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4C22×C411yescharacteristic
176C4×D4#4111yescharacteristic
177#117—21no—
178#117C2×D411yescharacteristic
179C2×C4⋊C4—22no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42C2411yescharacteristic
183#108—82no—
184#107—24no—
185#107—21no—
186#107—21no—
187C4⋊C8—22no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8C22×C411yescharacteristic
199C4⋊C8C22×C411yescharacteristic
200C4⋊C8C22×C412yes—
201C4⋊C8C22×C412yes—
202#104—82no—
203#104—82no—
204#104—82no—
205#103—22no—
206#103—22no—
207#102—22no—
208#102—22no—
209#102—22no—
210#102—22no—
211C22⋊C8—22no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8#4112yes—
222#97—22no—
223#97—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—82no—
231C4×C8C22×C411yescharacteristic
232C4×C8C22×C411yescharacteristic
233C4×C8C2×D411yescharacteristic
234C4×C8C2×D411yescharacteristic
235C4×C8C2×D411yescharacteristic
236C4×C8C2×D411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8#4111yescharacteristic
240C4×C8#4111yescharacteristic
241#95—22no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—21no—
246#41—42no—
247#41—42no—
248#41—42no—
249#41—42no—
250C2×Q8—22no—
251C2×Q8C4×D411yescharacteristic
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4C4×D411yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266Q16—82no—
267SD16—82no—
268SD16—82no—
269D8—42no—
270D8—42no—
271D8—82no—
272M16—21no—
273M16—21no—
274M16—21no—
275M16—21no—
276M16—44no—
277M16—42no—
278M16—42no—
279M16—84no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—84no—
302C2×C8C4×D411yescharacteristic
303C2×C8C4×D411yescharacteristic
304C2×C8C4×D411yescharacteristic
305C2×C8C4×D411yescharacteristic
306C2×C8#12111yescharacteristic
307C2×C8#12111yescharacteristic
308C2×C8#12111yescharacteristic
309C2×C8#12111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4C4×D411yescharacteristic
317C4⋊C4C4×D411yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—42no—
327C42—42no—
328C42C23×C411yescharacteristic
329C42C22×D411yescharacteristic
330C42C2×#4111yescharacteristic
331C42C2×#4111yescharacteristic
332C23—21no—
333C23—42no—
334C23#53211yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—42no—
338Q8—42no—
339D4—21no—
340D4—21no—
341D4—41no—
342D4—41no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—82no—
348D4—82no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C4×#4111yescharacteristic
370C2×C4C2×#12111yescharacteristic
371C2×C4#52511yescharacteristic
372C2×C4#52911yescharacteristic
373C2×C4#57611yescharacteristic
374C2×C4#57611yescharacteristic
375C2×C4#57611yescharacteristic
376C2×C4#57811yescharacteristic
377C8—22no—
378C8—21no—
379C8—21no—
380C8—21no—
381C8—21no—
382C8—21no—
383C8—21no—
384C8—21no—
385C8—41no—
386C8—42no—
387C8—42no—
388C8—42no—
389C8—41no—
390C8—41no—
391C8—41no—
392C8—41no—
393C8—82no—
394C8#44312yes—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22C4×#12111yescharacteristic
404C22#317511yescharacteristic
405C22#317611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—84no—
413C4—82no—
414C4#294811yescharacteristic
415C4#295111yescharacteristic
416C4C2×#44311yescharacteristic
417C4#295511yescharacteristic
418C4#305811yescharacteristic
419C4#306111yescharacteristic
420C2—21no—
421C2—21no—
422C2—81no—
423C2—162no—
424C2#2151711yescharacteristic
425C2#2152411yescharacteristic
426C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19125C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22165C2semidirect
3. #17956C2semidirect
4. #17328C2semidirect
5. #9719C2semidirect
6. #9624C2semidirect
7. #9592C2semidirect
8. #9210C2semidirect
9. #8667C2semidirect
10. C8×#107C2semidirect
11. #8257C2semidirect
27. #1682C4semidirect
28. #1681C4semidirect
29. #1674C4semidirect
30. #1673C4semidirect
31. #1668C4semidirect
32. #1667C4semidirect

Maximal subgroups

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

subgroupindex
#221652
#191252
#179562
#173282
#97192
#96242
#95922
#92102
#86672
C8×#1072
#82572