Finite Group Databasebeta

← #314998

Subgroups — #314998

Order 512.

463 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22136C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic, complemented
3#10689C211yesmaximal, characteristic, complemented
4#10687C211yesmaximal, characteristic, complemented
5#10661C211yesmaximal, characteristic, complemented
6#10659C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9631C211yesmaximal, characteristic, complemented
9#9036C211yesmaximal, characteristic
10#9035C211yesmaximal, characteristic
11#8863C211yesmaximal, characteristic, complemented
12#8862C211yesmaximal, characteristic, complemented
13#8660C211yesmaximal, characteristic, complemented
14#8658C211yesmaximal, characteristic, complemented
15#8603C211yesmaximal, characteristic, complemented
16C2×C8×D4C2211yescharacteristic
17#2921C2211yescharacteristic
18C2×#433C2211yescharacteristic
19#2905C2211yescharacteristic
20#2392C2211yescharacteristic
21#1955—21no—
22#1955C2211yescharacteristic
23#1950—21no—
24#1950—21no—
25#1940C2211yescharacteristic
26#1924—21no—
27#1869—21no—
28#1855—21no—
29#1855—21no—
30#1850—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#1486C2211yescharacteristic
44#1486C2211yescharacteristic
45#1486C2211yescharacteristic
46#1486C2211yescharacteristic
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—
60C23×C8—21no—
61C2×C4×D4—21no—
62C2×C4×D4C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C8×D4—21no—
65C8×D4—21no—
66C8×D4—21no—
67C8×D4—21no—
68#433—21no—
69#433—21no—
70#433—21no—
71#433—21no—
72#433—41no—
73#433—41no—
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78C2×#108—21no—
79C2×#108—21no—
80C2×#108—21no—
81C2×#108—21no—
82#424—41no—
83#424—41no—
84#423—21no—
85#423C2311yescharacteristic
86#423C2311yescharacteristic
87#423C2311yescharacteristic
88#423C2311yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94#407—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101C4×M16—21no—
102C2×#97—21no—
103C2×#97C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#398Q811yescharacteristic
106#397Q811yescharacteristic
107#396—21no—
108#394—21no—
109#394Q811yescharacteristic
110#392Q811yescharacteristic
111C4×C4⋊C4C2311yescharacteristic
112C4×C22⋊C4—21no—
113#341—21no—
114#341—21no—
115#341—21no—
116#341—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—
128D4⋊C8—41no—
129D4⋊C8—41no—
130D4⋊C8—41no—
131D4⋊C8—41no—
132#324—21no—
133#324—21no—
134#323—21no—
135#323—21no—
136C22×D4—21no—
137C22×D4C2×Q811yescharacteristic
138C23×C4—21no—
139C23×C4C2×D411yescharacteristic
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—41no—
145C2×M16—81no—
146C2×M16—81no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—81no—
153C22×C8—81no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—21no—
162C4×D4—81no—
163C4×D4—81no—
164#117—21no—
165#117—21no—
166#117—21no—
167#117—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×Q811yescharacteristic
171C2×C4⋊C4C2×Q811yescharacteristic
172C2×C4⋊C4#4111yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4C2×Q811yescharacteristic
179C2×C22⋊C4#4111yescharacteristic
180C2×C42—21no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2411yescharacteristic
183#108—81no—
184#108—81no—
185#108—81no—
186#108—81no—
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—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—42no—
208C4⋊C8—81no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—42no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—81no—
221C22⋊C8—81no—
222C22⋊C8—81no—
223#97—21no—
224#97—21no—
225#97—41no—
226#97—81no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—41no—
232C4×C8#4111yescharacteristic
233C4×C8#4111yescharacteristic
234C4×C8#4111yescharacteristic
235C4×C8#4111yescharacteristic
236#95—21no—
237#95—21no—
238#95C2×Q811yescharacteristic
239#95C2×Q811yescharacteristic
240C24—21no—
241C24C22⋊Q811yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—81no—
247C2×D4—81no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×Q811yescharacteristic
268C22×C4C2×#4111yescharacteristic
269M16—41no—
270M16—41no—
271M16—41no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16—82no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—82no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8#12311yescharacteristic
306C2×C8#12311yescharacteristic
307C2×C8#12311yescharacteristic
308C2×C8#12311yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—81no—
316C4⋊C4—81no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C42—21no—
328C42—21no—
329C42—41no—
330C42—41no—
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C42H(5,𝔽2)11yescharacteristic
334C42H(5,𝔽2)11yescharacteristic
335C23—22no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C2×C22⋊Q811yescharacteristic
348D4—22no—
349D4—22no—
350D4—41no—
351D4—41no—
352D4—84no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×#12311yescharacteristic
380C2×C4#53511yescharacteristic
381C2×C4#55311yescharacteristic
382C2×C4#55311yescharacteristic
383C2×C4#55411yescharacteristic
384C2×C4#55411yescharacteristic
385C2×C4#57611yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C2×C4#57711yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—21no—
393C8—41no—
394C8—41no—
395C8—81no—
396C8—81no—
397C8—81no—
398C8—81no—
399C8—81no—
400C8—81no—
401C8—81no—
402C8—81no—
403C8—81no—
404C8—81no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22#248711yescharacteristic
420C22#308111yescharacteristic
421C22#308411yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4#309611yescharacteristic
433C4#309911yescharacteristic
434C4#322111yescharacteristic
435C4#322411yescharacteristic
436C2—21no—
437C2—21no—
438C2—81no—
439C2—82no—
440C2#2768811yescharacteristic
441C2#2769111yescharacteristic
442C2#2769411yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2487
prime radical#392Q8
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
2. #21995C2semidirect
3. #10689C2semidirect
4. #10687C2semidirect
5. #10661C2semidirect
6. #10659C2semidirect
7. #9632C2semidirect
8. #9631C2semidirect
11. #8863C2semidirect
12. #8862C2semidirect
13. #8660C2semidirect
14. #8658C2semidirect
15. #8603C2semidirect

Maximal subgroups

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

subgroupindex
#221362
#219952
#106892
#106872
#106612
#106592
#96322
#96312
#90362
#90352
#88632
#88622
#86602
#86582
#86032