Finite Group Databasebeta

← #484037

Subgroups — #484037

Order 512.

409 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33664C211yesmaximal, characteristic
2#18086C211yesmaximal, characteristic
3#18058C211yesmaximal, characteristic
4#12085C211yesmaximal, characteristic, complemented
5#12083C211yesmaximal, characteristic, complemented
6#12076C211yesmaximal, characteristic, complemented
7#12076C211yesmaximal, characteristic, complemented
8#12072C211yesmaximal, characteristic, complemented
9#12072C211yesmaximal, characteristic, complemented
10#12044C211yesmaximal, characteristic, complemented
11C8×Q32C211yesmaximal, characteristic, complemented
12#12034C211yesmaximal, characteristic
13#11676C211yesmaximal, characteristic
14#11594C211yesmaximal, characteristic
15#11409C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#2986—21no—
18#2984—21no—
19C2×C4×Q16C2211yescharacteristic
20#2236C411yescharacteristic
21#2236C411yescharacteristic
22#2227C411yescharacteristic
23#2227C411yescharacteristic
24#2209C411yescharacteristic
25#2209C411yescharacteristic
26#2188C411yescharacteristic
27#2188C411yescharacteristic
28C4×Q32C2211yescharacteristic
29C4×Q32C2211yescharacteristic
30C4×Q32C2211yescharacteristic
31C4×Q32C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1591C2211yescharacteristic
35#1591C2211yescharacteristic
36#1591C2211yescharacteristic
37#1591C2211yescharacteristic
38#1587—21no—
39#1587—21no—
40#1587—21no—
41#1587—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×Q16C2211yescharacteristic
46#1568C2211yescharacteristic
47#1564C2211yescharacteristic
48#1491—21no—
49#1485—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1332C2211yescharacteristic
58#1332C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1332C2211yescharacteristic
62#1332C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68#533—21no—
69C2×C4×Q8—21no—
70C2×Q32C2×C411yescharacteristic
71C2×Q32C2×C411yescharacteristic
72C2×Q32C2×C411yescharacteristic
73C2×Q32C2×C411yescharacteristic
74C2×M32—21no—
75#500C2×C411yescharacteristic
76#494C2×C411yescharacteristic
77#487—21no—
78#467—21no—
79#467C2×C411yescharacteristic
80#467C2×C411yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446—41no—
86#446—41no—
87#446—41no—
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92C8×Q8—41no—
93C4×Q16—21no—
94C4×Q16—21no—
95C4×Q16—21no—
96C4×Q16C2×C411yescharacteristic
97C4×Q16C2×C411yescharacteristic
98C4×Q16C2311yescharacteristic
99C4×Q16C2311yescharacteristic
100C4×Q16C2311yescharacteristic
101C4×Q16C2311yescharacteristic
102#433—41no—
103#433—41no—
104#432—21no—
105#432—41no—
106#431—21no—
107#431—41no—
108#431D411yescharacteristic
109C2×#107C2×C411yescharacteristic
110C2×#103—21no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#366C2×C411yescharacteristic
114#366C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#358C2×C411yescharacteristic
124#358C2×C411yescharacteristic
125#358C2×C411yescharacteristic
126#358C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335C2311yescharacteristic
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#334—21no—
139#334—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144Q8⋊C8—21no—
145Q8⋊C8—21no—
146Q8⋊C8—21no—
147Q8⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C22×Q8—21no—
153C2×Q16—22no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—41no—
157C2×Q16C22×C411yescharacteristic
158C2×Q16C22×C411yescharacteristic
159C2×Q16C22×C411yescharacteristic
160C2×Q16C22×C411yescharacteristic
161C2×M16—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8#4111yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166#125—41no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—41no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179#117—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C42—41no—
183C2×C42C2×D411yescharacteristic
184Q32—22no—
185Q32—22no—
186Q32—41no—
187Q32—41no—
188M32—42no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16C22×C411yescharacteristic
192C2×C16C22×C411yescharacteristic
193C2×C16C22×C411yescharacteristic
194C2×C16C22×C411yescharacteristic
195#107—21no—
196#107—21no—
197#107C22×C411yescharacteristic
198#107C22×C411yescharacteristic
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8D811yescharacteristic
214C4⋊C8D811yescharacteristic
215#103—21no—
216#103—21no—
217#103—21no—
218#103—21no—
219#103—21no—
220#103—21no—
221#103—21no—
222#103—21no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8D811yescharacteristic
226C22⋊C8D811yescharacteristic
227#97—21no—
228#97—41no—
229#97—41no—
230#97—41no—
231#97—41no—
232#97—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8—41no—
238C4×C8C22×C411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—81no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C4×D411yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4#13511yescharacteristic
259Q16—21no—
260Q16—21no—
261Q16—21no—
262Q16—21no—
263Q16—21no—
264Q16—21no—
265Q16—21no—
266Q16—42no—
267Q16—81no—
268Q16#13112yes—
269M16—42no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8C4×D411yescharacteristic
280C2×C8C4×D411yescharacteristic
281C2×C8C2×D811yescharacteristic
282C2×C8C2×D811yescharacteristic
283C2×C8C2×D811yescharacteristic
284C2×C8C2×D811yescharacteristic
285C2×C8C23×C411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42C4×D411yescharacteristic
312C42C2×D811yescharacteristic
313C42#13511yescharacteristic
314C42C22×D411yescharacteristic
315C16—22no—
316C16—21no—
317C16—21no—
318C16—21no—
319C16—21no—
320C16—41no—
321C23C4×D811yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—82no—
331Q8—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4C4×D811yescharacteristic
348C2×C4C4×D811yescharacteristic
349C2×C4C4×D811yescharacteristic
350C2×C4C2×C4×D411yescharacteristic
351C2×C4C22×D811yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C8—22no—
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C8—21no—
359C8—21no—
360C8—21no—
361C8—41no—
362C8—81no—
363C8—81no—
364C8—81no—
365C8—81no—
366C8C2×#13111yescharacteristic
367C8#56811yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22C2×C4×D811yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4#293311yescharacteristic
385C4#295611yescharacteristic
386C4#341711yescharacteristic
387C4#341711yescharacteristic
388C2—41no—
389C2#1799411yescharacteristic
390C2#3345811yescharacteristic
391C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radicalC23C4×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12085C2semidirect
5. #12083C2semidirect
6. #12076C2semidirect
7. #12076C2semidirect
8. #12072C2semidirect
9. #12072C2semidirect
10. #12044C2semidirect
11. C8×Q32C2semidirect

Maximal subgroups

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

subgroupindex
#336642
#180862
#180582
#120852
#120832
#120762
#120722
#120442
C8×Q322
#120342
#116762
#115942
#114092