Finite Group Databasebeta

← #10830099

Subgroups — #10830099

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic
4C2×#5026C211yesmaximal, characteristic
5#71791C212yesmaximal
6#71786C211yesmaximal, characteristic, complemented
7#71781C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71633C212yesmaximal
10#71628C212yesmaximal, complemented
11#71627C211yesmaximal, characteristic, complemented
12#24199—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—22no—
18C2×C3×#125—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#215C2211yescharacteristic
24C22×#215C2212yes—
25C2×C4×D12C2211yescharacteristic
26#5231—21no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5083—24no—
33#5026—24no—
34#4993—21no—
35#4989—22no—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic, complemented
47C2×#461—31no—
48C2×#440—31no—
49#2580—31no—
50#2043—32no—
51#2013—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1863—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—24no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×#125—216no—
67C3×#125—28no—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8D414yes—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#215—22no—
89C2×#215—22no—
90C2×#215—24no—
91C2×#215—24no—
92C2×#215—22no—
93C2×#215—42no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×Dic3⋊C4—22no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×D12—22no—
104C4×D12—22no—
105#835—22no—
106#835—24no—
107#835—24no—
108#835—42no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#809—22no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118C22×SD16—32no—
119#557—24no—
120#557—24no—
121C2×C4⋊Q8C2×D312yes—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#125—22no—
124C2×#125—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2×D312yes—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130#461—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—31no—
140C2×#97—31no—
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C2×C3×Q8C2×D412yes—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12#4112yes—
181C2×C4×D3—42no—
182C2×C4×D3—82no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4C2×D412yes—
193C3×C42—28no—
194C3×C42—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#215—28no—
200#215—48no—
201(C2×D3)⋊C4—42no—
202(C2×D3)⋊C4—44no—
203(C2×D3)⋊C4—82no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208Dic3⋊C4—82no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8#4112yes—
218C22×Q8#6711yescharacteristic
219C22×Q8C22×D312yes—
220C22×Q8C22×D311yescharacteristic
221C22×D4—31no—
222C23×C4—61no—
223C2×SD16—62no—
224C2×SD16—62no—
225C2×SD16—62no—
226C2×SD16—64no—
227C2×SD16—64no—
228C2×SD16—122no—
229C22×C8—31no—
230C22×C8—32no—
231C22×C8—31no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8#6714yes—
236#125—216no—
237#125—28no—
238C4×Q8—28no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8#6714yes—
245C4×D4—62no—
246C4×D4—62no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4#6712yes—
257C2×C4⋊C4#6711yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42C22×D312yes—
265C2×C42C22×D311yescharacteristic
266#107—62no—
267C4⋊C8—62no—
268#103—64no—
269#103—62no—
270#103—64no—
271#103—122no—
272#103—122no—
273#102—62no—
274#97—62no—
275#95—61no—
276#95—61no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C22×D3—22no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—28no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—42no—
290C3×Q8—42no—
291C3×Q8—41no—
292C3×Q8—41no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4#13714yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322D12—22no—
323D12—22no—
324D12—41no—
325D12—41no—
326C4×D3—84no—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C3⋊C8—42no—
330C24—61no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8C2×#6714yes—
344C2×Q8C2×#6712yes—
345C2×D4—32no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—122no—
357C22×C4—242no—
358C22×C4C2×#6712yes—
359C22×C4C2×#6712yes—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—68no—
364SD16—128no—
365C2×C8—32no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—62no—
369C2×C8—61no—
370C2×C8—62no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—62no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—242no—
388C4⋊C4C2×#6712yes—
389C22⋊C4—124no—
390C22⋊C4—122no—
391C22⋊C4—242no—
392C42—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42C2×#6712yes—
397C42C2×#6712yes—
398C22×C3C2×#12011yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3#55011yescharacteristic
401C22×C3C2×#13712yes—
402C22×C3#57712yes—
403C2×D3—44no—
404C2×D3—42no—
405C2×D3—82no—
406C2×D3—81no—
407C2×D3—81no—
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—44no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4#52511yescharacteristic
416C3×C4#53711yescharacteristic
417C3×C4#54611yescharacteristic
418C3×C4#54711yescharacteristic
419Dic3—82no—
420Dic3—81no—
421C23—62no—
422C23—122no—
423C23—121no—
424C23—122no—
425C23—121no—
426C23C22×#6711yescharacteristic
427Q8—28no—
428Q8—24no—
429Q8—22no—
430Q8—24no—
431Q8—22no—
432Q8—41no—
433Q8—42no—
434Q8—41no—
435Q8—42no—
436D4—62no—
437D4—62no—
438D4—121no—
439D4—121no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—244no—
461C2×C4#94214yes—
462C2×C4#95312yes—
463C2×C4#95312yes—
464C2×C4D3×#4111yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4#100911yescharacteristic
467C2×C4#101011yescharacteristic
468C2×C4C22×#6711yescharacteristic
469C2×C4C22×#6711yescharacteristic
470C8—62no—
471C8—122no—
472C8—121no—
473C2×C3#240711yescharacteristic
474C2×C3#300411yescharacteristic
475C2×C3#301011yescharacteristic
476C2×C3#337212yes—
477C2×C3#340012yes—
478D3—82no—
479C22—122no—
480C22—124no—
481C22—241no—
482C22—241no—
483C22—242no—
484C22C2×#94212yes—
485C22#583111yescharacteristic
486C22#583411yescharacteristic
487C22#583712yes—
488C22C2×#95311yescharacteristic
489C4—24no—
490C4—22no—
491C4—22no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—241no—
497C4—242no—
498C4#582411yescharacteristic
499C4#582811yescharacteristic
500C4#582911yescharacteristic
501C4#583311yescharacteristic
502C3#2419911yescharacteristic, complemented, Sylow
503C2—242no—
504C2#8368712yes—
505C2#8369212yes—
506C2#8405911yescharacteristic
507C2#8409711yescharacteristic
508C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3#24199
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
6. #71786C2semidirect
7. #71781C2semidirect
8. #71635C2semidirect
10. #71628C2semidirect
11. #71627C2semidirect
46. C2×#557D3semidirect
502. C3#24199semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50832
#825012
C2×#50262
#717912
#717862
#717812
#716352
#716332
#716282
#716272
#241993