Finite Group Databasebeta

← #326503

Subgroups — #326503

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33264C211yesmaximal, characteristic, complemented
2#21660C211yesmaximal, characteristic
3#10369C211yesmaximal, characteristic, complemented
4#10368C211yesmaximal, characteristic
5#10367C211yesmaximal, characteristic, complemented
6#10366C211yesmaximal, characteristic
7#9757C211yesmaximal, characteristic
8#9757C211yesmaximal, characteristic
9#8792C211yesmaximal, characteristic, complemented
10#8791C211yesmaximal, characteristic, complemented
11#8677C211yesmaximal, characteristic, complemented
12#8676C211yesmaximal, characteristic
13#8675C211yesmaximal, characteristic, complemented
14#8674C211yesmaximal, characteristic
15#8268C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#2960C2211yescharacteristic
18#2931C2211yescharacteristic
19#2864C2211yescharacteristic
20#2864C2211yescharacteristic
21#1994C2211yescharacteristic
22#1962C2211yescharacteristic
23#1944—21no—
24#1944—21no—
25#1943—21no—
26#1943—21no—
27#1938—21no—
28#1938—21no—
29#1846C2211yescharacteristic
30#1846C2211yescharacteristic
31#1837C2211yescharacteristic
32#1835C2211yescharacteristic
33#1835C2211yescharacteristic
34#1765—21no—
35#1765—21no—
36C4×#97C2211yescharacteristic
37#1564C2211yescharacteristic
38#1564C2211yescharacteristic
39#1518C2211yescharacteristic
40#1518C2211yescharacteristic
41#1518C2211yescharacteristic
42#1518C2211yescharacteristic
43#1517C2211yescharacteristic
44#1517C2211yescharacteristic
45#1517C2211yescharacteristic
46#1517C2211yescharacteristic
47#1450C2211yescharacteristic
48#1450C2211yescharacteristic
49#1286C2211yescharacteristic
50#1286C2211yescharacteristic
51#1286C2211yescharacteristic
52#1286C2211yescharacteristic
53#1286C2211yescharacteristic
54#1286C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1276C2211yescharacteristic
58#1276C2211yescharacteristic
59C2×#131—21no—
60#533C2311yescharacteristic
61#532C2311yescharacteristic
62#488—21no—
63#488—21no—
64#487Q811yescharacteristic
65#487Q811yescharacteristic
66#486Q811yescharacteristic
67#486Q811yescharacteristic
68#446—21no—
69#446—21no—
70C8×Q8—21no—
71C8×Q8—21no—
72#436—21no—
73#436—21no—
74C8×D4—21no—
75C8×D4—21no—
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79#431C2311yescharacteristic
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84#424—21no—
85#423C2311yescharacteristic
86#423C2311yescharacteristic
87C2×#104—21no—
88C2×#104—21no—
89#416—21no—
90#416—21no—
91#408—21no—
92#405—21no—
93C2×#97D411yescharacteristic
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×#97D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#387—41no—
104#387—41no—
105#387—41no—
106#387—41no—
107#382C2311yescharacteristic
108#382C2311yescharacteristic
109#382C2311yescharacteristic
110#382C2311yescharacteristic
111C43C2311yescharacteristic
112#341—21no—
113#341—21no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—21no—
118#339—21no—
119#339—21no—
120#335—21no—
121#335—21no—
122#334—21no—
123#334—21no—
124#328—41no—
125#328—41no—
126#328—41no—
127#328—41no—
128#327—41no—
129#327—41no—
130#327—41no—
131#327—41no—
132#322—21no—
133#322—21no—
134C82—21no—
135C82—21no—
136C2×#41—21no—
137#131—42no—
138#131—42no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—41no—
142C22×C8—41no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8C2×Q811yescharacteristic
148C4⋊Q8C2×Q811yescharacteristic
149C4⋊Q8C2×Q811yescharacteristic
150#127C2×Q811yescharacteristic
151#125—21no—
152#124—21no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155#121—21no—
156#121—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×D4—21no—
160C4×D4—21no—
161#117—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2411yescharacteristic
175#107—42no—
176#107—42no—
177#106—42no—
178#106—42no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189#104—42no—
190#104—42no—
191#103—21no—
192#103—21no—
193#103—21no—
194#103—21no—
195#102—21no—
196#102—21no—
197#102—21no—
198#102—21no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205#97—22no—
206#97—22no—
207#97—41no—
208#97—41no—
209#97—41no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—41no—
215C4×C8C2×Q811yescharacteristic
216C4×C8C2×Q811yescharacteristic
217C4×C8#4111yescharacteristic
218C4×C8#4111yescharacteristic
219C4×C8#4111yescharacteristic
220C4×C8#4111yescharacteristic
221C4×C8#4111yescharacteristic
222C4×C8#4111yescharacteristic
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22×D411yescharacteristic
254C22×C4H(5,𝔽2)11yescharacteristic
255C22×C4H(5,𝔽2)11yescharacteristic
256M16—41no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—82no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—82no—
274C2×C8C22⋊Q811yescharacteristic
275C2×C8C22⋊Q811yescharacteristic
276C2×C8C22⋊Q811yescharacteristic
277C2×C8C22⋊Q811yescharacteristic
278C2×C8#12411yescharacteristic
279C2×C8#12411yescharacteristic
280C2×C8#12411yescharacteristic
281C2×C8#12411yescharacteristic
282C2×C8#13512yes—
283C2×C8#13512yes—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
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—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—81no—
313C42—81no—
314C42C22×Q811yescharacteristic
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C23—41no—
319C23#53511yescharacteristic
320Q8—41no—
321Q8—41no—
322Q8—81no—
323Q8—81no—
324D4—41no—
325D4—41no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×#12411yescharacteristic
357C2×C4#55411yescharacteristic
358C2×C4#55411yescharacteristic
359C2×C4#55511yescharacteristic
360C2×C4#55511yescharacteristic
361C2×C4C2×#13511yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4#57511yescharacteristic
364C2×C4#57511yescharacteristic
365C8—21no—
366C8—21no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8#48812yes—
378C8#48812yes—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#256311yescharacteristic
385C22#321211yescharacteristic
386C22#321211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—161no—
397C4—161no—
398C4—161no—
399C4—161no—
400C4#307611yescharacteristic
401C4#307611yescharacteristic
402C4C2×#48811yescharacteristic
403C4#314211yescharacteristic
404C2—21no—
405C2—21no—
406C2—161no—
407C2#2914011yescharacteristic
408C2#2915511yescharacteristic
409C2#2915511yescharacteristic
subgroupG/H
center, socleC22#2563
commutator, Frattini, cosocleC2×C42C24
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33264C2semidirect
3. #10369C2semidirect
5. #10367C2semidirect
9. #8792C2semidirect
10. #8791C2semidirect
11. #8677C2semidirect
13. #8675C2semidirect
15. #8268C2semidirect

Maximal subgroups

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

subgroupindex
#332642
#216602
#103692
#103682
#103672
#103662
#97572
#87922
#87912
#86772
#86762
#86752
#86742
#82682