Finite Group Databasebeta

← #498739

Subgroups — #498739

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21601C211yesmaximal, characteristic
2#13264C211yesmaximal, characteristic, complemented
3#13263C211yesmaximal, characteristic, complemented
4#13159C211yesmaximal, characteristic, complemented
5#13159C211yesmaximal, characteristic, complemented
6#13121C211yesmaximal, characteristic
7#13120C211yesmaximal, characteristic
8#13120C211yesmaximal, characteristic
9#13120C211yesmaximal, characteristic
10#13105C211yesmaximal, characteristic, complemented
11#13104C211yesmaximal, characteristic, complemented
12#13104C211yesmaximal, characteristic, complemented
13#13104C211yesmaximal, characteristic, complemented
14#12845C211yesmaximal, characteristic
15#12784C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×#405C2211yescharacteristic
18#2396C411yescharacteristic
19C2×C4⋊C16C2211yescharacteristic
20C2×C4⋊C16C2211yescharacteristic
21C2×C4⋊C16C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×#346C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1985C411yescharacteristic
34#1940C411yescharacteristic
35#1917C411yescharacteristic
36#1917C411yescharacteristic
37#1851C411yescharacteristic
38#1851C411yescharacteristic
39#1798C411yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C22⋊C4C2211yescharacteristic
44C8×C22⋊C4C2211yescharacteristic
45#1374C2211yescharacteristic
46#1374C2211yescharacteristic
47#1374C2211yescharacteristic
48#1374C2211yescharacteristic
49#1374C2211yescharacteristic
50#1374C2211yescharacteristic
51#1374C2211yescharacteristic
52#1374C2211yescharacteristic
53#1374C2211yescharacteristic
54#1374C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×M16C2×C411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C8×Q8—41no—
72C8×Q8—41no—
73C2×C4⋊C8C2×C411yescharacteristic
74C2×C4⋊C8C2×C411yescharacteristic
75C2×C4⋊C8C2×C411yescharacteristic
76C2×C4⋊C8C2×C411yescharacteristic
77C2×C4⋊C8C2×C411yescharacteristic
78C2×C4⋊C8C2×C411yescharacteristic
79C2×C22⋊C8C2×C411yescharacteristic
80C2×C22⋊C8C2×C411yescharacteristic
81#405—41no—
82#405—41no—
83C2×#97C2×C411yescharacteristic
84C2×#97C2×C411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#400C811yescharacteristic
92#399C811yescharacteristic
93#398C811yescharacteristic
94#398C811yescharacteristic
95#396C811yescharacteristic
96#396C811yescharacteristic
97#395C811yescharacteristic
98#393C811yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C22⋊C4C2×C411yescharacteristic
103C4×C22⋊C4C2×C411yescharacteristic
104C4⋊C16—41no—
105C4⋊C16—41no—
106C4⋊C16—41no—
107C4⋊C16—41no—
108C4⋊C16—41no—
109C4⋊C16—41no—
110C22⋊C16—41no—
111C22⋊C16—41no—
112C22⋊C16—41no—
113C22⋊C16—41no—
114#346—41no—
115C4×C16—41no—
116C4×C16—41no—
117C4×C16—41no—
118#336C2×C411yescharacteristic
119#336C2×C411yescharacteristic
120#336C2×C411yescharacteristic
121#336C2×C411yescharacteristic
122#336C2×C411yescharacteristic
123#336C2×C411yescharacteristic
124#336C2×C411yescharacteristic
125#336C2×C411yescharacteristic
126#336C2×C411yescharacteristic
127#336C2×C411yescharacteristic
128C22×Q8C2×C811yescharacteristic
129C23×C4C22×C411yescharacteristic
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C22×C8C22×C411yescharacteristic
144C22×C8C22×C411yescharacteristic
145C22×C8C22×C411yescharacteristic
146C22×C8C2411yescharacteristic
147C4×Q8—41no—
148C4×Q8—41no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4C2×C811yescharacteristic
152C2×C4⋊C4C2×C811yescharacteristic
153C2×C4⋊C4C2×C811yescharacteristic
154C2×C4⋊C4C2×C811yescharacteristic
155C2×C4⋊C4C2×C811yescharacteristic
156C2×C4⋊C4C2×C811yescharacteristic
157C2×C4⋊C4C2×C811yescharacteristic
158C2×C4⋊C4C2×C811yescharacteristic
159C2×C4⋊C4C2×C811yescharacteristic
160C2×C4⋊C4C2×C811yescharacteristic
161C2×C4⋊C4C2×C811yescharacteristic
162C2×C22⋊C4C2×C811yescharacteristic
163C2×C22⋊C4C2×C811yescharacteristic
164C2×C22⋊C4C2×C811yescharacteristic
165C2×C22⋊C4C2×C811yescharacteristic
166C2×C22⋊C4C2×C811yescharacteristic
167C2×C22⋊C4C2×C811yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
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×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—41no—
207#97—41no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95C2×C811yescharacteristic
221#95C2×C811yescharacteristic
222#95C2×C811yescharacteristic
223#95C2×C811yescharacteristic
224#95C2×C811yescharacteristic
225#95C2×C811yescharacteristic
226#95C2×C811yescharacteristic
227#95C2×C811yescharacteristic
228#95C2×C811yescharacteristic
229#95C2×C811yescharacteristic
230C24C22×C811yescharacteristic
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4C22×C811yescharacteristic
241C22×C4C22×C811yescharacteristic
242C22×C4C22×C811yescharacteristic
243C22×C4C22×C811yescharacteristic
244C22×C4C22×C811yescharacteristic
245C22×C4C22×C811yescharacteristic
246C22×C4C22×C811yescharacteristic
247C22×C4C22×C811yescharacteristic
248C22×C4C22×C811yescharacteristic
249C22×C4C22×C811yescharacteristic
250C22×C4C22×C811yescharacteristic
251C22×C4C22×C811yescharacteristic
252C22×C4C22×C811yescharacteristic
253C22×C4C23×C411yescharacteristic
254M16—44no—
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×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C8H(5,𝔽2)11yescharacteristic
283C2×C82−1+411yescharacteristic
284C2×C82−1+411yescharacteristic
285C2×C82−1+411yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C16—41no—
316C16—41no—
317C16—41no—
318C16—41no—
319C16—41no—
320C16—41no—
321C16—41no—
322C16—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23C23×C811yescharacteristic
331Q8—44no—
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—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4#50412yes—
373C2×C4#50412yes—
374C2×C4#50412yes—
375C2×C4#51811yescharacteristic
376C2×C4#51911yescharacteristic
377C2×C4#52011yescharacteristic
378C2×C4#56811yescharacteristic
379C2×C4#56811yescharacteristic
380C2×C4#56811yescharacteristic
381C8—41no—
382C8—41no—
383C8—41no—
384C8—41no—
385C8—41no—
386C8—41no—
387C8—41no—
388C8#56211yescharacteristic
389C8#56211yescharacteristic
390C8#56311yescharacteristic
391C8#56311yescharacteristic
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22#297211yescharacteristic
400C22#298211yescharacteristic
401C22#298811yescharacteristic
402C22C2×#50411yescharacteristic
403C22C2×#50411yescharacteristic
404C22C2×#50411yescharacteristic
405C22#340711yescharacteristic
406C4—22no—
407C4—22no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4#237811yescharacteristic
420C4#297811yescharacteristic
421C4#298711yescharacteristic
422C4#299511yescharacteristic
423C2—81no—
424C2#2333611yescharacteristic
425C2#3349011yescharacteristic
426C2#3349311yescharacteristic
427C2#3349911yescharacteristic
428C2#3350011yescharacteristic
429C2#3350611yescharacteristic
430C2#3350811yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C8C24
commutator, socleC23C23×C8
prime radicalC24C22×C8
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13264C2semidirect
3. #13263C2semidirect
4. #13159C2semidirect
5. #13159C2semidirect
10. #13105C2semidirect
11. #13104C2semidirect
12. #13104C2semidirect
13. #13104C2semidirect

Maximal subgroups

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

subgroupindex
#216012
#132642
#132632
#131592
#131212
#131202
#131052
#131042
#128452
#127842