Finite Group Databasebeta

← #6798415

Subgroups — #6798415

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31353C218yesmaximal, complemented
5#29877C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27962C212yesmaximal, complemented
9#26113C214yesmaximal, complemented
10#25672C212yesmaximal, complemented
11#24972C214yesmaximal
12#21864C212yesmaximal, complemented
13C4×#487C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#533C2212yes—
16C2×#498C2212yes—
17C2×#487C2211yescharacteristic
18C2×#431C2212yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2571C2212yes—
24#2563—21no—
25#2552—21no—
26#2542—21no—
27#2538—21no—
28#2538C2212yes—
29#2497—22no—
30#2492—22no—
31#2441—22no—
32#2439C2214yes—
33#2429—22no—
34#2423C2214yes—
35#2369—21no—
36#2365—21no—
37#2355—21no—
38#2354—21no—
39#2354C2212yes—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2101C2218yes—
46#2088C2218yes—
47#2084C2214yes—
48#2077C2218yes—
49#2035C2218yes—
50#2024C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2211yescharacteristic
56#1956C2212yes—
57#1956C2211yescharacteristic
58#1954C2212yes—
59#1948C2214yes—
60#1939C2212yes—
61#1933C2214yes—
62#1923C2214yes—
63#1873C2218yes—
64#1865C2218yes—
65#1834C2214yes—
66#1834C2211yescharacteristic
67#1834C2212yes—
68#1825C2218yes—
69#1805C2218yes—
70#1803C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75#533—28no—
76#533—44no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8C2312yes—
93C2×C4×Q8C2314yes—
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C2×#117—22no—
98C2×#117C2314yes—
99C22×C42C2312yes—
100#498—44no—
101#487—24no—
102#431—44no—
103C2×#106C2318yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2318yes—
108C2×#103C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116#400—24no—
117#399—24no—
118#398—24no—
119#396—24no—
120#396—24no—
121#393—24no—
122#391—21no—
123#391—22no—
124#391—22no—
125#391—24no—
126#391C2312yes—
127#390—22no—
128#390—22no—
129#389—22no—
130#389—22no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—22no—
135#389—22no—
136#389—24no—
137#389C2312yes—
138#389C2314yes—
139#389C2314yes—
140#388—24no—
141#388—24no—
142#387—24no—
143#387—24no—
144#384—22no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#384C2312yes—
155#382—22no—
156#382C2314yes—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4D412yes—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2314yes—
182#340C2314yes—
183#340C2318yes—
184#336C2312yes—
185#336C2314yes—
186C22×Q8—22no—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8C2412yes—
190C22×D4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2412yes—
194C22×C8C2412yes—
195C22×C8C2414yes—
196C22×C8C2412yes—
197C4⋊Q8—24no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—82no—
206C4⋊Q8—82no—
207C4⋊Q8#4114yes—
208#125—48no—
209C22⋊Q8—28no—
210C22⋊Q8—44no—
211C22⋊Q8—48no—
212C22⋊Q8—84no—
213C22⋊Q8—84no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—44no—
217C4×Q8—48no—
218C4×Q8—44no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222#117—48no—
223#117—48no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2412yes—
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#106—48no—
279C4⋊C8—24no—
280C4⋊C8—44no—
281#103—28no—
282#103—44no—
283#103—48no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—24no—
289C4×C8—24no—
290C4×C8—42no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304C24C22×D412yes—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—44no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8C2×#4114yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2511yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—24no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—48no—
404C42—44no—
405C42—44no—
406C42—84no—
407C42#13514yes—
408C42#13714yes—
409C42C2×#4112yes—
410C42H(5,𝔽2)12yes—
411C422−1+414yes—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23C23×D411yescharacteristic
420Q8—28no—
421Q8—44no—
422Q8—84no—
423Q8—88no—
424Q8—164no—
425D4—44no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4#53612yes—
464C2×C4#53612yes—
465C2×C4#54612yes—
466C2×C4#54612yes—
467C2×C4C2×#13512yes—
468C2×C4C2×#13512yes—
469C2×C4C2×#13512yes—
470C2×C4C2×#13712yes—
471C2×C4C2×#13712yes—
472C2×C4C2×#13712yes—
473C2×C4#57712yes—
474C2×C4#57712yes—
475C2×C4C22×#4111yescharacteristic
476C2×C4C2×H(5,𝔽2)11yescharacteristic
477C2×C4C2×2−1+412yes—
478C2×C4#58512yes—
479C8—42no—
480C8—42no—
481C8—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22C2×#53611yescharacteristic
490C22C2×#54611yescharacteristic
491C22#350011yescharacteristic
492C22C22×#13511yescharacteristic
493C22C22×#13711yescharacteristic
494C22C2×#57711yescharacteristic
495C22C2×#57711yescharacteristic
496C4—22no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—44no—
501C4—42no—
502C4—42no—
503C4—84no—
504C4—82no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4#323114yes—
509C4#330414yes—
510C4#332914yes—
511C4#351512yes—
512C4#354712yes—
513C2—82no—
514C2#3895511yescharacteristic
515C2C2×#323111yescharacteristic
516C2#6088111yescharacteristic
517C2C2×#330411yescharacteristic
518C2C2×#332911yescharacteristic
519C2#6253111yescharacteristic
520C2#6254211yescharacteristic
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. #33289C2semidirect
4. #31353C2semidirect
5. #29877C2semidirect
6. #29141C2semidirect
7. #28687C2semidirect
8. #27962C2semidirect
9. #26113C2semidirect
10. #25672C2semidirect
12. #21864C2semidirect
13. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361342
#332892
#313532
#298772
#291412
#286872
#279622
#261132
#256722
#249722
#218642
C4×#4872
#212642