Finite Group Databasebeta

← #511438

Subgroups — #511438

Order 512.

472 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28676C211yesmaximal, characteristic, complemented
2#28267C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13451C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic
9#13263C211yesmaximal, characteristic, complemented
10#13165C211yesmaximal, characteristic, complemented
11#13147C211yesmaximal, characteristic
12#13130C211yesmaximal, characteristic
13#13111C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic
15#12935C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2580—21no—
20#2567—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31#2087—21no—
32#2087—21no—
33#2065—21no—
34#2065—21no—
35#1987—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1846C2211yescharacteristic
45#1804—21no—
46C4×#107C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4⋊Q8—21no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#500—41no—
68#500—41no—
69C8×Q8—41no—
70C8×Q8—41no—
71C4×D8—41no—
72C4×D8—41no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#399—41no—
92#395—41no—
93#395—41no—
94#393—41no—
95#391—41no—
96#390—21no—
97#390—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#385—41no—
105#385—41no—
106#384—41no—
107#382—21no—
108C4×C4⋊C4—21no—
109#367—22no—
110#366—22no—
111#366—41no—
112#366—41no—
113C4⋊C16—41no—
114#357—22no—
115#357—41no—
116#357—41no—
117#357Q814yescomplemented
118C4×C16—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×Q8—41no—
127C22×Q8D811yescharacteristic
128C22×D4—21no—
129C23×C4—41no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8C2×Q812yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×D4—42no—
145C4×D4—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4D811yescharacteristic
161C2×C4⋊C4D811yescharacteristic
162C2×C4⋊C4D811yescharacteristic
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16C2×Q812yes—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—42no—
197#107C2×Q812yes—
198#107#4112yes—
199#106—81no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203#102—42no—
204#102—81no—
205#102—81no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221C24—41no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—82no—
226C2×Q8—161no—
227C2×Q8—161no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—82no—
242C22×C4—161no—
243C22×C4—161no—
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C22×D411yescharacteristic
251D8—22no—
252D8—22no—
253D8—41no—
254D8—41no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8C22×Q811yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C2×C8H(5,𝔽2)11yescharacteristic
267C2×C8H(5,𝔽2)11yescharacteristic
268C2×C82−1+411yescharacteristic
269C2×C82−1+411yescharacteristic
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—82no—
288C4⋊C4—161no—
289C4⋊C4—161no—
290C4⋊C4—161no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C4⋊C4—161no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—161no—
298C22⋊C4—161no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—82no—
306C16—22no—
307C16—41no—
308C16—41no—
309C16—41no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23C22×D811yescharacteristic
318Q8—44no—
319Q8—164no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
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×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—164no—
362C2×C4#50812yes—
363C2×C4#51012yes—
364C2×C4#53411yescharacteristic
365C2×C4#54811yescharacteristic
366C2×C4D4×Q811yescharacteristic
367C2×C4#57511yescharacteristic
368C2×C4#57711yescharacteristic
369C2×C4#57811yescharacteristic
370C8—41no—
371C8—41no—
372C8—41no—
373C8#54211yescharacteristic
374C8#54411yescharacteristic
375C8#55411yescharacteristic
376C8#56311yescharacteristic
377C22—82no—
378C22—82no—
379C22—82no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22#318611yescharacteristic
385C22#335811yescharacteristic
386C22Q8×D811yescharacteristic
387C22C2×#50811yescharacteristic
388C22C2×#51011yescharacteristic
389C22#341511yescharacteristic
390C22#341511yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—82no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4#272911yescharacteristic
405C4#320111yescharacteristic
406C4#337711yescharacteristic
407C4#338811yescharacteristic
408C2—162no—
409C2#3256411yescharacteristic
410C2#3375611yescharacteristic
411C2#3376511yescharacteristic
412C2#3392811yescharacteristic
413C2#3394211yescharacteristic
414C2#3394711yescharacteristic
415C2#3395211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28676C2semidirect
3. #13539C2semidirect
4. #13515C2semidirect
5. #13507C2semidirect
7. #13451C2semidirect
9. #13263C2semidirect
10. #13165C2semidirect
13. #13111C2semidirect
117. #357Q8semidirect

Maximal subgroups

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

subgroupindex
#286762
#282672
#135392
#135152
#135072
#134762
#134512
#133632
#132632
#131652
#131472
#131302
#131112
#129352