Finite Group Databasebeta

← #211896

Subgroups — #211896

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22165C212yesmaximal, complemented
2#19157C211yesmaximal, characteristic
3#17956C211yesmaximal, characteristic, complemented
4#17328C211yesmaximal, characteristic, complemented
5#9721C211yesmaximal, characteristic, complemented
6#9626C211yesmaximal, characteristic, complemented
7#9590C212yesmaximal, complemented
8#9208C212yesmaximal, complemented
9#8669C211yesmaximal, characteristic, complemented
10C8×#106C211yesmaximal, 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×#106C2211yescharacteristic
26C8×C4⋊C4C2212yes—
27#1702C412yescomplemented
28#1701C412yescomplemented
29#1694C411yescharacteristic, complemented
30#1693C411yescharacteristic, complemented
31#1688C411yescharacteristic, complemented
32#1687C411yescharacteristic, complemented
33#1589—21no—
34#1589—21no—
35C8×D8—21no—
36C8×D8—21no—
37#1567C2211yescharacteristic
38C2×#334C2211yescharacteristic
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#446C2×C411yescharacteristic
69#446C2×C411yescharacteristic
70C8×Q8—21no—
71C4×D8—21no—
72C4×D8—21no—
73#436—22no—
74#436—22no—
75#436C2×C411yescharacteristic
76#436C2×C411yescharacteristic
77#435—21no—
78#435—21no—
79C8×D4—21no—
80C8×D4—22no—
81C8×D4—22no—
82C8×D4—22no—
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×#106—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#334C2×C411yescharacteristic
119#334C2×C411yescharacteristic
120#334C2×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—21no—
151C2×M16—21no—
152C2×M16—22no—
153C2×M16—42no—
154C2×M16C2×D411yescharacteristic
155C2×M16C2×D411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
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—21no—
185#107—21no—
186#106—24no—
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—21no—
223#97—21no—
224#97—21no—
225C4×C8—22no—
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—22no—
243#41—22no—
244#41—42no—
245#41—42no—
246#41—42no—
247#41—42no—
248C2×Q8—22no—
249C2×Q8C4×D411yescharacteristic
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4C4×D411yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264Q16—82no—
265SD16—82no—
266SD16—82no—
267D8—42no—
268D8—42no—
269D8—82no—
270M16—22no—
271M16—22no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—84no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—84no—
296C2×C8C4×D411yescharacteristic
297C2×C8C4×D411yescharacteristic
298C2×C8C4×D411yescharacteristic
299C2×C8C4×D411yescharacteristic
300C2×C8#12111yescharacteristic
301C2×C8#12111yescharacteristic
302C2×C8#12111yescharacteristic
303C2×C8#12111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4C4×D411yescharacteristic
311C4⋊C4C4×D411yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—42no—
321C42—42no—
322C42C23×C411yescharacteristic
323C42C22×D411yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—21no—
327C23—42no—
328C23#53211yescharacteristic
329Q8—22no—
330Q8—42no—
331Q8—42no—
332D4—22no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—82no—
339D4—82no—
340D4—82no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C4×#4111yescharacteristic
361C2×C4C2×#12111yescharacteristic
362C2×C4#52511yescharacteristic
363C2×C4#52911yescharacteristic
364C2×C4#57612yes—
365C2×C4#57712yes—
366C8—22no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—42no—
375C8—41no—
376C8—42no—
377C8—42no—
378C8—41no—
379C8—41no—
380C8—41no—
381C8—41no—
382C8—82no—
383C8#44312yes—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22C4×#12111yescharacteristic
393C22#317912yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—84no—
401C4—82no—
402C4#294612yes—
403C4C2×#44311yescharacteristic
404C4#295511yescharacteristic
405C4#305811yescharacteristic
406C4#306011yescharacteristic
407C2—22no—
408C2—81no—
409C2—162no—
410C2#2151711yescharacteristic
411C2#2153611yescharacteristic
412C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19157C2
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. #9721C2semidirect
6. #9626C2semidirect
7. #9590C2semidirect
8. #9208C2semidirect
9. #8669C2semidirect
10. C8×#106C2semidirect
11. #8257C2semidirect
27. #1702C4semidirect
28. #1701C4semidirect
29. #1694C4semidirect
30. #1693C4semidirect
31. #1688C4semidirect
32. #1687C4semidirect

Maximal subgroups

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

subgroupindex
#221652
#191572
#179562
#173282
#97212
#96262
#95902
#92082
#86692
C8×#1062
#82572