Finite Group Databasebeta

← #314999

Subgroups — #314999

Order 512.

423 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22101C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic
3#10689C211yesmaximal, characteristic, complemented
4#10688C211yesmaximal, characteristic, complemented
5#10664C211yesmaximal, characteristic, complemented
6#10663C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9631C211yesmaximal, characteristic, complemented
9#9034C211yesmaximal, characteristic
10#9033C211yesmaximal, characteristic
11#8863C211yesmaximal, characteristic
12#8862C211yesmaximal, characteristic
13#8661C211yesmaximal, characteristic, complemented
14#8659C211yesmaximal, characteristic, complemented
15#8603C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17#2921C2211yescharacteristic
18C2×#433C2211yescharacteristic
19#2905C2211yescharacteristic
20#2396C2211yescharacteristic
21#1985C2211yescharacteristic
22#1950—21no—
23#1950—21no—
24#1940—21no—
25#1940C2211yescharacteristic
26#1924—21no—
27#1923—21no—
28#1916—21no—
29#1855—21no—
30#1855—21no—
31#1849C2211yescharacteristic
32#1848C2211yescharacteristic
33#1838C2211yescharacteristic
34#1837C2211yescharacteristic
35#1798C2211yescharacteristic
36#1770C2211yescharacteristic
37#1571C2211yescharacteristic
38#1570C2211yescharacteristic
39#1570C2211yescharacteristic
40#1570C2211yescharacteristic
41#1564C2211yescharacteristic
42#1564C2211yescharacteristic
43#1487C2211yescharacteristic
44#1487C2211yescharacteristic
45#1487C2211yescharacteristic
46#1487C2211yescharacteristic
47#1458C2211yescharacteristic
48#1458C2211yescharacteristic
49#1303C2211yescharacteristic
50#1303C2211yescharacteristic
51#1303C2211yescharacteristic
52#1303C2211yescharacteristic
53#1301C2211yescharacteristic
54#1301C2211yescharacteristic
55#1301C2211yescharacteristic
56#1301C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59C22×M16—21no—
60C2×C4×Q8C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C8×Q8—21no—
63C8×Q8—21no—
64C8×Q8—21no—
65C8×Q8—21no—
66#433—21no—
67#433—21no—
68#433—21no—
69#433—21no—
70#433—41no—
71#433—41no—
72#431C2311yescharacteristic
73#431C2311yescharacteristic
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76C2×#108—21no—
77C2×#108—21no—
78C2×#108—21no—
79C2×#108—21no—
80#424—41no—
81#424—41no—
82#423—21no—
83#423C2311yescharacteristic
84#423C2311yescharacteristic
85#423C2311yescharacteristic
86#423C2311yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93#407—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C4×M16—21no—
98C2×#97—21no—
99C2×#97C2311yescharacteristic
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102#400Q811yescharacteristic
103#399Q811yescharacteristic
104#398—21no—
105#396—21no—
106#395Q811yescharacteristic
107#393Q811yescharacteristic
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2311yescharacteristic
110C4×C22⋊C4—21no—
111#341—21no—
112#341—21no—
113#341—21no—
114#341—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#335—21no—
121#335—21no—
122#335—41no—
123#334—21no—
124#334—21no—
125#334—41no—
126#334—41no—
127#334—41no—
128Q8⋊C8—41no—
129Q8⋊C8—41no—
130Q8⋊C8—41no—
131Q8⋊C8—41no—
132#324—21no—
133#324—21no—
134#323—21no—
135#323—21no—
136C22×Q8C2×Q811yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—41no—
143C2×M16—81no—
144C2×M16—81no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Q811yescharacteristic
165C2×C4⋊C4C2×Q811yescharacteristic
166C2×C4⋊C4#4111yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4C2×Q811yescharacteristic
170C2×C22⋊C4#4111yescharacteristic
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2411yescharacteristic
175#108—81no—
176#108—81no—
177#108—81no—
178#108—81no—
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—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—42no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—81no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—42no—
212C22⋊C8—81no—
213#97—21no—
214#97—21no—
215#97—41no—
216#97—81no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8#4111yescharacteristic
225C4×C8#4111yescharacteristic
226C4×C8#4111yescharacteristic
227C4×C8#4111yescharacteristic
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Q811yescharacteristic
233#95C2×Q811yescharacteristic
234C24C22⋊Q811yescharacteristic
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254M16—41no—
255M16—41no—
256M16—41no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—41no—
261M16—41no—
262M16—82no—
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—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—82no—
288C2×C8#12311yescharacteristic
289C2×C8#12311yescharacteristic
290C2×C8#12311yescharacteristic
291C2×C8#12311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—42no—
308C22⋊C4—81no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C42H(5,𝔽2)11yescharacteristic
318C42H(5,𝔽2)11yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23C2×C22⋊Q811yescharacteristic
325Q8—22no—
326Q8—22no—
327Q8—41no—
328Q8—41no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
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—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4C2×#12311yescharacteristic
355C2×C4#53511yescharacteristic
356C2×C4#55311yescharacteristic
357C2×C4#55311yescharacteristic
358C2×C4#55411yescharacteristic
359C2×C4#55411yescharacteristic
360C2×C4#57711yescharacteristic
361C2×C4#57711yescharacteristic
362C2×C4#57711yescharacteristic
363C2×C4#57811yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C8—41no—
369C8—41no—
370C8—81no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8—81no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22#248711yescharacteristic
388C22#308211yescharacteristic
389C22#308411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4#309711yescharacteristic
402C4#309911yescharacteristic
403C4#322211yescharacteristic
404C4#322511yescharacteristic
405C2—21no—
406C2—21no—
407C2—81no—
408C2#2768911yescharacteristic
409C2#2769111yescharacteristic
410C2#2769411yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2487
prime radicalC24C22⋊Q8
LCSG ▷ C2×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
3. #10689C2semidirect
4. #10688C2semidirect
5. #10664C2semidirect
6. #10663C2semidirect
7. #9632C2semidirect
8. #9631C2semidirect
13. #8661C2semidirect
14. #8659C2semidirect

Maximal subgroups

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

subgroupindex
#221012
#219952
#106892
#106882
#106642
#106632
#96322
#96312
#90342
#90332
#88632
#88622
#86612
#86592
#86032