Finite Group Databasebeta

← #10829997

Subgroups — #10829997

Order 768.

1050 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×#5721C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic
4C2×C4×#216C211yesmaximal, characteristic
5#72073C212yesmaximal
6#72065C212yesmaximal
7#71785C211yesmaximal, characteristic
8#70485C211yesmaximal, characteristic
9#70472C211yesmaximal, characteristic
10#70284C212yesmaximal
11#70235C212yesmaximal
12#24191—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17C2×#944C2212yes—
18C22×#216C2211yescharacteristic
19#5721—24no—
20#5721—24no—
21C2×#890—21no—
22C2×#889—22no—
23C2×#887—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#462—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29#5229C2212yes—
30C2×#835C2212yes—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#816C2212yes—
34C4×#216—24no—
35#4982—22no—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4×C3⋊C8C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#557—31no—
47C2×#462D311yescharacteristic
48C2×C4×Q16—31no—
49#2582—31no—
50#2043—32no—
51#2016—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1863—32no—
56#1803—31no—
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×Dic6D411yescharacteristic
60C22×Dic6C2311yescharacteristic
61C2×C3×Q16—24no—
62C2×C3×Q16—24no—
63C2×C3×Q16—42no—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77#944—24no—
78#944—24no—
79C2×#216—22no—
80C2×#216—22no—
81C2×#216—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3C2312yes—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98#890—28no—
99#889—216no—
100#887—28no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×#103—24no—
107#835—22no—
108#835—24no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#816—42no—
114#810—22no—
115#807—22no—
116C4×C3⋊C8—22no—
117C22×Q16—31no—
118C22×Q16C2×D312yes—
119#557—64no—
120#557—64no—
121C2×C4⋊Q8—32no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#125—62no—
124C2×#125—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—31no—
128C2×C4×Q8—61no—
129C2×C4×Q8C2×D311yescharacteristic
130#462—24no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—31no—
136C2×#103—31no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#103C2×D312yes—
140C2×C4×C8—31no—
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C2×C3×Q8—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8C2×D414yes—
161C2×C3×Q8#4112yes—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3C2×D412yes—
172C22×Dic3C2×D412yes—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6Q1614yescomplemented
181C2×Dic6C2×D412yes—
182C3×Q16—48no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—42no—
191C3×C42—42no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194#216—28no—
195C4⋊Dic3—28no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—42no—
201C4⋊Dic3—81no—
202C4⋊Dic3C2×D412yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—24no—
206Dic3⋊C4—24no—
207Dic3⋊C4—82no—
208C4×Dic3—28no—
209C4×Dic3—24no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8#4112yes—
218C22×Q8—31no—
219C22×Q8—31no—
220C22×Q8C22×D311yescharacteristic
221C22×Q8C22×D312yes—
222C2×Q16—24no—
223C2×Q16—24no—
224C2×Q16—42no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C22×C8—31no—
229C22×C8—31no—
230C22×C8C22×D312yes—
231C4⋊Q8—22no—
232C4⋊Q8—64no—
233C4⋊Q8—64no—
234C4⋊Q8#6714yes—
235#125—616no—
236#125—68no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—34no—
240C4×Q8—68no—
241C4×Q8—64no—
242C4×Q8—64no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C42—21no—
262C2×C42—32no—
263C2×C42—62no—
264C2×C42—61no—
265C2×C42C22×D311yescharacteristic
266#107—62no—
267C4⋊C8—22no—
268#103—24no—
269#103—62no—
270#103—62no—
271#103—64no—
272#103—122no—
273#103—122no—
274C4×C8—62no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—24no—
282C3×Q8—24no—
283C3×Q8—42no—
284C3×Q8—42no—
285C3×Q8—41no—
286C3×Q8—41no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4#12012yes—
299C2×C3×C4#12012yes—
300C2×C3×C4C2×Q1612yes—
301C2×C3×C4#13712yes—
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C2×C3×C42−1+411yescharacteristic
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3C2×Q1614yes—
323Dic6—28no—
324Dic6—22no—
325Dic6—22no—
326Dic6—41no—
327Dic6—41no—
328C3×C8—42no—
329C3⋊C8—22no—
330C3⋊C8—41no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—34no—
342C2×Q8—32no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8#23712yes—
351C2×Q8C2×#6714yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—62no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363Q16—48no—
364Q16—68no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—32no—
369C2×C8—61no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—61no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—32no—
379C4⋊C4—42no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4—68no—
383C4⋊C4—64no—
384C4⋊C4—62no—
385C4⋊C4—64no—
386C4⋊C4—61no—
387C4⋊C4—62no—
388C4⋊C4—62no—
389C4⋊C4—122no—
390C4⋊C4—241no—
391C4⋊C4—242no—
392C42—42no—
393C42—68no—
394C42—64no—
395C42—62no—
396C42—62no—
397C42D3×D412yes—
398C42C2×#6712yes—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3#55011yescharacteristic
402C22×C3C22×Q1611yescharacteristic
403C22×C3C2×#13711yescharacteristic
404C22×C3#57711yescharacteristic
405C22×C3#57811yescharacteristic
406C3×C4—22no—
407C3×C4—44no—
408C3×C4—42no—
409C3×C4—81no—
410C3×C4#45114yes—
411C3×C4#52511yescharacteristic
412C3×C4#53711yescharacteristic
413C3×C4#54611yescharacteristic
414C3×C4#54711yescharacteristic
415Dic3—24no—
416Dic3—42no—
417Dic3—42no—
418Dic3—42no—
419Dic3—82no—
420C23C2×D3×D411yescharacteristic
421Q8—22no—
422Q8—24no—
423Q8—24no—
424Q8—22no—
425Q8—41no—
426Q8—42no—
427Q8—42no—
428Q8—41no—
429Q8—68no—
430Q8—62no—
431Q8—62no—
432Q8—121no—
433Q8—121no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—34no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4D3×Q1612yes—
461C2×C4#91812yes—
462C2×C4#93712yes—
463C2×C4#93712yes—
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×#23711yescharacteristic
466C2×C4#100411yescharacteristic
467C2×C4D3×#4111yescharacteristic
468C2×C4#101011yescharacteristic
469C2×C4C22×#6711yescharacteristic
470C8—42no—
471C8—62no—
472C8—121no—
473C2×C3#240711yescharacteristic
474C2×C3C2×#45111yescharacteristic
475C2×C3#301311yescharacteristic
476C2×C3#336311yescharacteristic
477C2×C3#337411yescharacteristic
478C2×C3#339511yescharacteristic
479C2×C3#340011yescharacteristic
480C22#559811yescharacteristic
481C22#559911yescharacteristic
482C22C2×D3×Q1611yescharacteristic
483C22C2×#91811yescharacteristic
484C22#577811yescharacteristic
485C22#577911yescharacteristic
486C22C2×#93711yescharacteristic
487C4—22no—
488C4—44no—
489C4—42no—
490C4—64no—
491C4—81no—
492C4—122no—
493C4—122no—
494C4—122no—
495C4—242no—
496C4#518614yes—
497C4#560611yescharacteristic
498C4#560811yescharacteristic
499C4#580611yescharacteristic
500C4#582411yescharacteristic
501C3#2419111yescharacteristic, complemented, Sylow
502C2#8046211yescharacteristic
503C2#8046511yescharacteristic
504C2#8046811yescharacteristic
505C2#8047011yescharacteristic
506C2#8270111yescharacteristic
507C2C2×#518611yescharacteristic
508C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#462C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#462D3
hypocenterC3#24191
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
180. C2×Dic6Q16semidirect
501. C3#24191semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×C3×#4622
#824822
C2×C4×#2162
#720732
#720652
#717852
#704852
#704722
#702842
#702352
#241913