Finite Group Databasebeta

← #6850694

Subgroups — #6850694

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#32074C218yesmaximal, complemented
5#29808C212yesmaximal, complemented
6#29144C211yesmaximal, characteristic, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27972C212yesmaximal, complemented
9#27889C212yesmaximal, complemented
10#24698C214yesmaximal, complemented
11#24134C214yesmaximal
12#21981C212yesmaximal, complemented
13#21691C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic
15C2×#554C2212yes—
16C2×#494C2211yescharacteristic
17C2×#475C2214yes—
18C2×#441C2214yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2567C2212yes—
25#2552—22no—
26#2536C2212yes—
27#2508—22no—
28#2502C2214yes—
29#2495—22no—
30#2489C2214yes—
31#2485—22no—
32#2440—22no—
33#2440—22no—
34#2356C2212yes—
35D4×C4⋊C4—22no—
36#2346—22no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39#2285C2212yes—
40C2×#384C2212yes—
41C42×D4C2211yescharacteristic
42#2096C2214yes—
43#2072C2218yes—
44#2056C2218yes—
45#2035C2218yes—
46#2026C2214yes—
47#2001C2218yes—
48#1993C2214yes—
49#1992C2214yes—
50#1971C2214yes—
51#1956C2211yescharacteristic
52#1956C2212yes—
53#1956C2212yes—
54#1956C2211yescharacteristic
55#1955C2212yes—
56#1948C2218yes—
57#1940C2212yes—
58#1927C2218yes—
59#1924C2214yes—
60#1872C2218yes—
61#1853C2218yes—
62#1844C2212yes—
63#1844C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1834C2212yes—
67#1805C2214yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×Q16C2314yes—
72#554—48no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2312yes—
77C2×#125—22no—
78C2×#125C2312yes—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8C2314yes—
81C2×C22⋊Q8C2314yes—
82C2×C4×Q8C2312yes—
83C2×C4×Q8C2312yes—
84C2×C4×D4—22no—
85C2×C4×D4D412yes—
86C2×C4×D4C2311yescharacteristic
87C2×#117—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4C2314yes—
91C22×C42C2312yes—
92#494—24no—
93#475—48no—
94#441—48no—
95#431—28no—
96C2×#106C2318yes—
97C2×#106C2314yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2314yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2312yes—
110C2×C4×C8C2311yescharacteristic
111#400—24no—
112#398—22no—
113#398—24no—
114#398—24no—
115#397—24no—
116#395—22no—
117#395—22no—
118#394—22no—
119#391C2312yes—
120#391C2312yes—
121#391C2314yes—
122#391C2314yes—
123#389—24no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#388—24no—
133#388—24no—
134#387—24no—
135#385—24no—
136#385—24no—
137#384—22no—
138#384—22no—
139#384—22no—
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#384C2314yes—
143#384C2312yes—
144#384C2312yes—
145#384C2312yes—
146#383—22no—
147#382—22no—
148#382—24no—
149#382C2312yes—
150#380—22no—
151#380—22no—
152#380—22no—
153#380—22no—
154#380—22no—
155#380C2314yes—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4D412yes—
160C4×C4⋊C4C2314yes—
161C4×C4⋊C4C2311yescharacteristic
162C4×C22⋊C4D412yes—
163C4×C22⋊C4D412yes—
164C4×C22⋊C4C2312yes—
165#376—22no—
166C43C2311yescharacteristic
167#340C2318yes—
168#340C2318yes—
169#340C2318yes—
170#336C2314yes—
171#336C2312yes—
172C22×Q8—22no—
173C22×Q8C2412yes—
174C22×Q8C2412yes—
175C22×D4C2×D411yescharacteristic
176C23×C4—24no—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2412yes—
180C2×Q16—28no—
181C2×Q16—44no—
182C2×Q16—44no—
183C22×C8C2412yes—
184C22×C8C2414yes—
185C22×C8C2412yes—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—28no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—84no—
192#125—44no—
193#125—48no—
194C22⋊Q8—416no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C4×Q8—28no—
198C4×Q8—28no—
199C4×D4—28no—
200C4×D4—24no—
201C4×D4—48no—
202#117—48no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—84no—
223C2×C4⋊C4—84no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4C2×D412yes—
234C2×C22⋊C4C2×D412yes—
235C2×C22⋊C4C2×D412yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2412yes—
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249#106—44no—
250#106—48no—
251C4⋊C8—28no—
252C4⋊C8—24no—
253#103—48no—
254#103—44no—
255#103—44no—
256#103—48no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259#97—24no—
260#97—42no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8#4114yes—
264#95—24no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95C2×D411yescharacteristic
275#95C2×D411yescharacteristic
276#95C2×D412yes—
277#95C2×D412yes—
278C24C22×D412yes—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—44no—
288C2×Q8—84no—
289C2×D4—24no—
290C2×D4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—48no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4#13718yes—
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C2511yescharacteristic
318Q16—416no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8C2×#4114yes—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—84no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—44no—
366C42—44no—
367C42—48no—
368C42C2×#4112yes—
369C42H(5,𝔽2)14yes—
370C422−1+412yes—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23C2×#13714yes—
378C23C23×D411yescharacteristic
379Q8—44no—
380Q8—44no—
381Q8—44no—
382Q8—44no—
383Q8—164no—
384D4—28no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—216no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—48no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—168no—
419C2×C4#53512yes—
420C2×C4#53512yes—
421C2×C4#54714yes—
422C2×C4C2×#13714yes—
423C2×C4C2×#13714yes—
424C2×C4#57714yes—
425C2×C4C22×#4111yescharacteristic
426C2×C4C2×H(5,𝔽2)12yes—
427C2×C4C2×2−1+411yescharacteristic
428C2×C4#58512yes—
429C8—24no—
430C8—44no—
431C8—42no—
432C22—24no—
433C22—24no—
434C22—24no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22C2×#53511yescharacteristic
440C22C2×#54711yescharacteristic
441C22#350011yescharacteristic
442C22C22×#13712yes—
443C22C2×#57712yes—
444C4—24no—
445C4—22no—
446C4—22no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—84no—
453C4—82no—
454C4—82no—
455C4—84no—
456C4#334418yes—
457C4#350612yes—
458C4#354712yes—
459C2—44no—
460C2#3949111yescharacteristic
461C2#6080011yescharacteristic
462C2#6081811yescharacteristic
463C2C2×#334412yes—
464C2#6254212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33297C2semidirect
4. #32074C2semidirect
5. #29808C2semidirect
6. #29144C2semidirect
7. #28512C2semidirect
8. #27972C2semidirect
9. #27889C2semidirect
10. #24698C2semidirect
12. #21981C2semidirect
13. #21691C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361802
#332972
#320742
#298082
#291442
#285122
#279722
#278892
#246982
#241342
#219812
#216912
#212652