Finite Group Databasebeta

← #6850696

Subgroups — #6850696

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#32071C218yesmaximal, complemented
5#29808C212yesmaximal, complemented
6#29144C211yesmaximal, characteristic, complemented
7#28511C211yesmaximal, characteristic, complemented
8#27975C212yesmaximal, complemented
9#27892C212yesmaximal, complemented
10#24699C214yesmaximal, complemented
11#24135C214yesmaximal
12#21984C212yesmaximal, complemented
13#21691C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic
15C2×#556C2212yes—
16C2×#494C2211yescharacteristic
17C2×#477C2214yes—
18C2×#441C2214yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2568C2212yes—
25#2547—22no—
26#2536C2212yes—
27#2508—22no—
28#2499C2214yes—
29#2496—22no—
30#2496—22no—
31#2495—22no—
32#2489C2214yes—
33#2485—22no—
34#2365—22no—
35#2356C2212yes—
36#2346—22no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39#2285C2212yes—
40#2285C2212yes—
41C42×D4C2211yescharacteristic
42#2095C2214yes—
43#2074C2218yes—
44#2056C2218yes—
45#2036C2218yes—
46#2026C2214yes—
47#2001C2218yes—
48#1993C2214yes—
49#1992C2214yes—
50#1971C2214yes—
51#1956C2211yescharacteristic
52#1956C2212yes—
53#1956C2212yes—
54#1956C2211yescharacteristic
55#1955C2212yes—
56#1949C2218yes—
57#1940C2212yes—
58#1928C2218yes—
59#1924C2214yes—
60#1873C2218yes—
61#1854C2218yes—
62#1845C2212yes—
63#1844C2211yescharacteristic
64#1834C2212yes—
65#1834C2212yes—
66#1834C2212yes—
67#1805C2214yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×Q16C2314yes—
72#556—48no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2312yes—
77C2×#126—24no—
78C2×#125C2312yes—
79C2×#124—22no—
80C2×C22⋊Q8C2314yes—
81C2×C22⋊Q8C2314yes—
82C2×C4×Q8—22no—
83C2×C4×Q8C2312yes—
84C2×C4×Q8C2312yes—
85C2×C4×D4D412yes—
86C2×C4×D4C2311yescharacteristic
87C2×#117—22no—
88C2×#117C2314yes—
89C22×C42C2312yes—
90#494—24no—
91#477—48no—
92#441—48no—
93#431—28no—
94C2×#107C2318yes—
95C2×#106C2314yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2314yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2314yes—
101C2×#103C2314yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×#97C2312yes—
105C2×#97C2312yes—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2312yes—
108C2×C4×C8C2311yescharacteristic
109#400—22no—
110#400—22no—
111#400—24no—
112#398—22no—
113#397—24no—
114#397—24no—
115#397—24no—
116#394—22no—
117#391—24no—
118#391C2312yes—
119#391C2312yes—
120#391C2314yes—
121#391C2314yes—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#388—24no—
131#388—24no—
132#387—24no—
133#387—24no—
134#387—24no—
135#384—22no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2312yes—
139#384C2312yes—
140#384C2312yes—
141#383—22no—
142#382—22no—
143#382—24no—
144#382C2314yes—
145#382C2312yes—
146#380—22no—
147#380—22no—
148#380—22no—
149#380—22no—
150#380—22no—
151#380—22no—
152#380—22no—
153#380C2314yes—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
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×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2412yes—
179C2×Q16—28no—
180C2×Q16—44no—
181C2×Q16—44no—
182C22×C8C2412yes—
183C22×C8C2414yes—
184C22×C8C2412yes—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—28no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—48no—
191#126—416no—
192#125—44no—
193#124—48no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—48no—
199C4×D4—24no—
200C4×D4—44no—
201#117—48no—
202#117—48no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2412yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2414yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4C2×D412yes—
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2411yescharacteristic
242C2×C42C2412yes—
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245#107—48no—
246#106—44no—
247C4⋊C8—28no—
248C4⋊C8—24no—
249#103—48no—
250#103—44no—
251#103—44no—
252#103—48no—
253C22⋊C8—28no—
254C22⋊C8—44no—
255#97—24no—
256#97—42no—
257C4×C8—24no—
258C4×C8—22no—
259C4×C8#4114yes—
260#95—24no—
261#95—22no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95C2×D411yescharacteristic
273#95C2×D411yescharacteristic
274#95C2×D412yes—
275#95C2×D412yes—
276C24C22×D412yes—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—48no—
286C2×Q8—44no—
287C2×D4—22no—
288C2×D4—42no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C2511yescharacteristic
313Q16—416no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8C2×#4114yes—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—42no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—84no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—48no—
359C42—48no—
360C42—44no—
361C42—44no—
362C42C2×#4112yes—
363C42H(5,𝔽2)14yes—
364C422−1+412yes—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23C23×D411yescharacteristic
373Q8—44no—
374Q8—44no—
375Q8—44no—
376Q8—44no—
377Q8—88no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4#53512yes—
414C2×C4#53512yes—
415C2×C4#54714yes—
416C2×C4#57514yes—
417C2×C4#57814yes—
418C2×C4C22×#4111yescharacteristic
419C2×C4C2×H(5,𝔽2)12yes—
420C2×C4C2×2−1+411yescharacteristic
421C2×C4#58512yes—
422C8—24no—
423C8—44no—
424C8—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22C2×#53511yescharacteristic
433C22C2×#54711yescharacteristic
434C22#350011yescharacteristic
435C22C2×#57512yes—
436C22C2×#57812yes—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—82no—
444C4—82no—
445C4—84no—
446C4—82no—
447C4—84no—
448C4—82no—
449C4#334918yes—
450C4#350612yes—
451C4#354712yes—
452C2—82no—
453C2#3949111yescharacteristic
454C2#6080111yescharacteristic
455C2#6082111yescharacteristic
456C2C2×#334912yes—
457C2#6254512yes—
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. #32071C2semidirect
5. #29808C2semidirect
6. #29144C2semidirect
7. #28511C2semidirect
8. #27975C2semidirect
9. #27892C2semidirect
10. #24699C2semidirect
12. #21984C2semidirect
13. #21691C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361882
#332972
#320712
#298082
#291442
#285112
#279752
#278922
#246992
#241352
#219842
#216912
#212652