Finite Group Databasebeta

← #229587

Subgroups — #229587

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C212yesmaximal, complemented
2#19125C211yesmaximal, characteristic
3#18066C211yesmaximal, characteristic, complemented
4#17829C211yesmaximal, characteristic, complemented
5#9844C211yesmaximal, characteristic, complemented
6#9624C211yesmaximal, characteristic, complemented
7#9578C212yesmaximal, complemented
8#8996C212yesmaximal, complemented
9#8932C212yesmaximal, complemented
10C4×#335C211yesmaximal, characteristic, complemented
11C8×#104C211yesmaximal, characteristic, complemented
12#3169C2211yescharacteristic
13#3035C2212yes—
14C8×#41C2211yescharacteristic
15#2935C2212yes—
16#2925C2212yes—
17#1974—22no—
18#1962C2212yes—
19#1950C2212yes—
20#1929—22no—
21#1876—22no—
22C4×C4⋊C8C2212yes—
23#1764—22no—
24C4×#104C2211yescharacteristic
25C42×C8C2211yescharacteristic
26#1682C412yescomplemented
27#1681C412yescomplemented
28#1674C411yescharacteristic, complemented
29#1673C411yescharacteristic, complemented
30#1668C411yescharacteristic, complemented
31#1667C411yescharacteristic, complemented
32#1589—21no—
33#1581—22no—
34C8×D8—21no—
35#1572—22no—
36#1570C2212yes—
37#1568—21no—
38#1567C2211yescharacteristic
39#1564—21no—
40C2×#335C2211yescharacteristic
41C2×#335C2211yescharacteristic
42#1540C2212yes—
43#1492C2212yes—
44#1480C2212yes—
45#1476C2211yescharacteristic
46C8×M16C2211yescharacteristic
47#1301C2212yes—
48#1278C2211yescharacteristic
49#1277C2212yes—
50#1276C2211yescharacteristic
51C2×#135—22no—
52C2×#131—22no—
53#532C2312yes—
54C4×#41C2311yescharacteristic
55#500C2×C411yescharacteristic
56#493C2×C411yescharacteristic
57#486C2×C412yes—
58#466—21no—
59#466—21no—
60#464—22no—
61#463—22no—
62#462C2×C412yes—
63#461C2×C412yes—
64#460C2×C412yes—
65#459C2×C412yes—
66C8×Q8C2×C411yescharacteristic
67C8×Q8C2×C411yescharacteristic
68C4×D8—21no—
69C4×D8—21no—
70#436—22no—
71#435—42no—
72C8×D4—21no—
73C8×D4—21no—
74C8×D4—22no—
75C8×D4C2×C411yescharacteristic
76C8×D4C2×C411yescharacteristic
77#433—42no—
78#432—22no—
79#431—21no—
80#431—22no—
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83C2×#108—22no—
84#423C2312yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#104—22no—
88C2×#104—22no—
89C2×#104D411yescharacteristic
90C2×#104D411yescharacteristic
91#416—22no—
92#408—22no—
93#408—22no—
94#405—22no—
95C4×M16C2311yescharacteristic
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C43C2311yescharacteristic
103#341—22no—
104#339D411yescharacteristic
105#339D411yescharacteristic
106#336—22no—
107#335—21no—
108#335—21no—
109#335C2×C411yescharacteristic
110#335C2×C411yescharacteristic
111#335C2×C412yes—
112#335D411yescharacteristic
113#335D411yescharacteristic
114#335D412yes—
115#334—22no—
116#334—22no—
117#332C2×C412yes—
118#332C2×C412yes—
119#331C2×C412yes—
120#331C2×C412yes—
121Q8⋊C8—21no—
122D4⋊C8—21no—
123D4⋊C8—22no—
124D4⋊C8—22no—
125D4⋊C8—42no—
126#322—21no—
127C82—21no—
128C2×#41—22no—
129C2×#41C2×D411yescharacteristic
130#135—82no—
131#135—82no—
132C2×Q16—22no—
133C2×SD16—22no—
134C2×SD16—22no—
135C2×D8—22no—
136C2×D8—21no—
137C2×D8—21no—
138#131—44no—
139#131—82no—
140C2×M16—22no—
141C2×M16—42no—
142C2×M16C2×D411yescharacteristic
143C2×M16C2×D411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—42no—
150C22×C8#4111yescharacteristic
151C22×C8#4111yescharacteristic
152C4⋊Q8C22×C412yes—
153#127C22×C412yes—
154#124—22no—
155#121—22no—
156#121—22no—
157C4×Q8C22×C411yescharacteristic
158C4×Q8#4111yescharacteristic
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4C22×C411yescharacteristic
164C4×D4#4111yescharacteristic
165#117—21no—
166#117C2×D411yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2411yescharacteristic
173#108—82no—
174#107—21no—
175#107—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8C22×C411yescharacteristic
184C4⋊C8C22×C411yescharacteristic
185C4⋊C8C22×C412yes—
186C4⋊C8C22×C412yes—
187C4⋊C8C2×D412yes—
188C4⋊C8C2×D412yes—
189#104—22no—
190#104—22no—
191#104—44no—
192#104—82no—
193#103—22no—
194#103—22no—
195#102—22no—
196#102—22no—
197#102—22no—
198#102—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207#97—21no—
208#97—42no—
209C4×C8—24no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—42no—
218C4×C8—42no—
219C4×C8—82no—
220C4×C8C22×C411yescharacteristic
221C4×C8C22×C411yescharacteristic
222C4×C8C2×D411yescharacteristic
223C4×C8C2×D411yescharacteristic
224#41—21no—
225#41—21no—
226#41—21no—
227#41—21no—
228#41—42no—
229#41—42no—
230#41—42no—
231#41—42no—
232C2×Q8—22no—
233C2×Q8C4×D411yescharacteristic
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4C4×D411yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C42−1+411yescharacteristic
249Q16—82no—
250SD16—82no—
251SD16—82no—
252D8—42no—
253D8—81no—
254D8—82no—
255M16—21no—
256M16—21no—
257M16—21no—
258M16—42no—
259M16—42no—
260M16—84no—
261M16#13512yes—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—81no—
283C2×C8—84no—
284C2×C8C4×D411yescharacteristic
285C2×C8C4×D411yescharacteristic
286C2×C8C4×D411yescharacteristic
287C2×C8C4×D411yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4C4×D411yescharacteristic
293C4⋊C4C4×D411yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—42no—
308C42#13512yes—
309C42C23×C411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C42H(5,𝔽2)11yescharacteristic
313C23—21no—
314C23—42no—
315C23#54711yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—42no—
319Q8—42no—
320D4—21no—
321D4—21no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4#52011yescharacteristic
354C2×C4D4211yescharacteristic
355C2×C4#54811yescharacteristic
356C2×C4#55311yescharacteristic
357C2×C4C2×#13511yescharacteristic
358C2×C4C2×#13511yescharacteristic
359C2×C4#57611yescharacteristic
360C2×C4#57611yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8—22no—
365C8—42no—
366C8—42no—
367C8—42no—
368C8—41no—
369C8—41no—
370C8—42no—
371C8—42no—
372C8—41no—
373C8—81no—
374C8—82no—
375C8#44312yes—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22#236111yescharacteristic
385C22#333911yescharacteristic
386C22#334111yescharacteristic
387C4—24no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4C4×#13511yescharacteristic
397C4#294811yescharacteristic
398C4C2×#44311yescharacteristic
399C4#295611yescharacteristic
400C4#328011yescharacteristic
401C4#328711yescharacteristic
402C2—21no—
403C2—21no—
404C2—81no—
405C2—162no—
406C2#2301511yescharacteristic
407C2#2302711yescharacteristic
408C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19125C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23113C2semidirect
3. #18066C2semidirect
4. #17829C2semidirect
5. #9844C2semidirect
6. #9624C2semidirect
7. #9578C2semidirect
8. #8996C2semidirect
9. #8932C2semidirect
10. C4×#335C2semidirect
11. C8×#104C2semidirect
26. #1682C4semidirect
27. #1681C4semidirect
28. #1674C4semidirect
29. #1673C4semidirect
30. #1668C4semidirect
31. #1667C4semidirect

Maximal subgroups

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

subgroupindex
#231132
#191252
#180662
#178292
#98442
#96242
#95782
#89962
#89322
C4×#3352
C8×#1042