Finite Group Databasebeta

← #305846

Subgroups — #305846

Order 512.

635 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28789C211yesmaximal, characteristic, complemented
2#26041C211yesmaximal, characteristic
3#26040C211yesmaximal, characteristic
4#22705C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic, complemented
6#10649C211yesmaximal, characteristic, complemented
7#10581C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9594C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic
12#8890C211yesmaximal, characteristic
13#8816C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15C8×#102C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#474—21no—
18C2×C8×D4C2211yescharacteristic
19#2552C2211yescharacteristic
20#2551C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22D4×C4⋊C4C2211yescharacteristic
23#2087—21no—
24#2086—21no—
25#2083—21no—
26#2059—21no—
27#2057—21no—
28#2057—21no—
29#1990—21no—
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1942—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1862—21no—
40#1862—21no—
41#1852—21no—
42#1852—21no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1808—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×#125—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76#478—44no—
77#474—44no—
78C8×D4—22no—
79C8×D4Q814yescomplemented
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107D411yescharacteristic
90C2×#107D411yescharacteristic
91C2×#106—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102D411yescharacteristic
101C2×#102D411yescharacteristic
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#398—41no—
108#395—41no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#385—41no—
116#385—41no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#380—21no—
123#380—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—41no—
140#340—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149D4⋊C8Q814yescomplemented
150C82—22no—
151C22×Q8—41no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—41no—
155C23×C4—41no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—81no—
163C22×C8—81no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171C4⋊Q8—81no—
172#125—42no—
173#125—81no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—81no—
179C4×D4—81no—
180C4×D4C2×Q812yes—
181C4×D4C2×Q812yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2411yescharacteristic
219#107—22no—
220#107—41no—
221#107—42no—
222#107—42no—
223#107—42no—
224#107—42no—
225#107—81no—
226#107—81no—
227#107—81no—
228#107—81no—
229#107—81no—
230#106—42no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241#102—42no—
242#102—42no—
243#102D814yes—
244#102Q1614yes—
245C22⋊C8—24no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—81no—
250C4×C8C2×Q812yes—
251C4×C8C2×Q812yes—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—161no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C4⋊Q812yes—
268C2×D4C2×D812yes—
269C2×D4C2×Q1612yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—161no—
292C22×C4—161no—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—84no—
320C2×C8C22⋊Q812yes—
321C2×C8C22⋊Q812yes—
322C2×C8C2×D812yes—
323C2×C8C2×Q1612yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—82no—
360C4⋊C4—161no—
361C4⋊C4C4⋊Q812yes—
362C4⋊C4C2×D812yes—
363C4⋊C4C2×Q1612yes—
364C4⋊C4#13512yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42C22×Q811yescharacteristic
376C42C22×Q811yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C23—22no—
380C23—22no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23D4211yescharacteristic
387Q8—164no—
388D4—22no—
389D4—22no—
390D4—22no—
391D4#50014yes—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—164no—
432C2×C4—164no—
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×C4⋊Q811yescharacteristic
435C2×C4#54411yescharacteristic
436C2×C4#54711yescharacteristic
437C2×C4D4×Q811yescharacteristic
438C2×C4#55411yescharacteristic
439C2×C4C22×D811yescharacteristic
440C2×C4C22×Q1611yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4#57511yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57811yescharacteristic
447C8—22no—
448C8—22no—
449C8—22no—
450C8—22no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—81no—
455C8—81no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22#244011yescharacteristic
464C22#327511yescharacteristic
465C22D4×D811yescharacteristic
466C22#328011yescharacteristic
467C22D4×Q1611yescharacteristic
468C22#329011yescharacteristic
469C22#329011yescharacteristic
470C4—21no—
471C4—21no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#307611yescharacteristic
484C4#308111yescharacteristic
485C4C2×#50011yescharacteristic
486C4#316411yescharacteristic
487C4#333411yescharacteristic
488C4#333811yescharacteristic
489C4#338111yescharacteristic
490C4#338911yescharacteristic
491C2—44no—
492C2#1988611yescharacteristic
493C2#1992911yescharacteristic
494C2#1996811yescharacteristic
495C2#1999311yescharacteristic
496C2#2604911yescharacteristic
497C2#2606511yescharacteristic
498C2#2608011yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28789C2semidirect
5. #10650C2semidirect
6. #10649C2semidirect
7. #10581C2semidirect
8. #10575C2semidirect
9. #9616C2semidirect
10. #9594C2semidirect
13. #8816C2semidirect
79. C8×D4Q8semidirect
149. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#287892
#260412
#260402
#227052
#106502
#106492
#105812
#105752
#96162
#95942
#88902
#88162
#86482
C8×#1022