Finite Group Databasebeta

← #10829995

Subgroups — #10829995

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×C3×#557C211yesmaximal, characteristic
2C2×#5090C211yesmaximal, characteristic
3#82482C211yesmaximal, characteristic
4C2×C4×#216C211yesmaximal, characteristic
5#71793C212yesmaximal
6#71785C211yesmaximal, characteristic
7#71781C212yesmaximal
8#71634C211yesmaximal, characteristic
9#71632C212yesmaximal
10#71628C212yesmaximal
11#71626C211yesmaximal, characteristic
12#24191—31nomaximal, Sylow
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×#125—22no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C22×#216C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5229—21no—
27C2×#835C2212yes—
28C2×#835C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5090—24no—
33C4×#216—24no—
34#4982—22no—
35#4972—21no—
36C2×#810C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#557D311yescharacteristic
47C2×#462—31no—
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×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×Dic6C2311yescharacteristic
61C3×C4⋊Q8—24no—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×#125—216no—
66C3×#125—28no—
67C3×C4×Q8—28no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—22no—
88C2×#216—22no—
89C2×#216—24no—
90C2×#216—24no—
91C2×#216—22no—
92C2×#216—42no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—21no—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×Dic6—22no—
104C4×Dic6—22no—
105#835—22no—
106#835—24no—
107#835—24no—
108#835—42no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#810—22no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116C4×C3⋊C8—22no—
117C22×Q16—31no—
118C22×Q16—32no—
119#557—24no—
120#557—24no—
121C2×C4⋊Q8C2×D312yes—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#125—21no—
124C2×#125—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—31no—
127C2×C4×Q8C2×D312yes—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×Q8C2×D311yescharacteristic
130#462—64no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—31no—
137C2×#103—31no—
138C2×#103—32no—
139C2×#103—32no—
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—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8Q1614yescomplemented
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×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6#4112yes—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D412yes—
192C3×C42—28no—
193C3×C42—24no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198#216—28no—
199#216—48no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—42no—
205C4⋊Dic3—81no—
206Dic3⋊C4—42no—
207Dic3⋊C4—82no—
208C4×Dic3—42no—
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—31no—
219C22×Q8#6711yescharacteristic
220C22×Q8C22×D312yes—
221C22×Q8C22×D311yescharacteristic
222C2×Q16—62no—
223C2×Q16—62no—
224C2×Q16—62no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C22×C8—31no—
229C22×C8—32no—
230C22×C8—31no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8#6714yes—
235#125—216no—
236#125—28no—
237C4×Q8—28no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8#6714yes—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4#6711yescharacteristic
259C2×C4⋊C4#6712yes—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—61no—
264C2×C42C22×D312yes—
265C2×C42C22×D311yescharacteristic
266#107—62no—
267C4⋊C8—62no—
268#103—64no—
269#103—62no—
270#103—64no—
271#103—62no—
272#103—122no—
273#103—122no—
274C4×C8—62no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C3×Q8—28no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—24no—
283C3×Q8—24no—
284C3×Q8—42no—
285C3×Q8—42no—
286C3×Q8—41no—
287C3×Q8—41no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C2×Q1614yes—
307C2×C3×C4C2×Q1612yes—
308C2×C3×C4#13712yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4H(5,𝔽2)11yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×Dic3—22no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324Dic6—22no—
325Dic6—22no—
326Dic6—41no—
327Dic6—41no—
328C3⋊C8—22no—
329C3⋊C8—41no—
330C3⋊C8—42no—
331C2×Q8—22no—
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×Q8—32no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8#21614yescomplemented
350C2×Q8C2×#6714yes—
351C2×Q8C2×#6712yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4C2×#6712yes—
359C22×C4C2×#6712yes—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C23×D311yescharacteristic
363Q16—68no—
364Q16—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—122no—
388C4⋊C4—122no—
389C4⋊C4—242no—
390C4⋊C4—241no—
391C4⋊C4C2×#6712yes—
392C42—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—122no—
397C42C2×#6712yes—
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—24no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4#45114yes—
413C3×C4#52511yescharacteristic
414C3×C4#53711yescharacteristic
415C3×C4#54611yescharacteristic
416C3×C4#54711yescharacteristic
417Dic3—42no—
418Dic3—82no—
419Dic3—81no—
420C23C22×#6711yescharacteristic
421Q8—28no—
422Q8—24no—
423Q8—22no—
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—42no—
428Q8—41no—
429Q8—42no—
430Q8—62no—
431Q8—62no—
432Q8—121no—
433Q8—121no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—62no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4#94212yes—
460C2×C4C2×#21614yes—
461C2×C4C2×#21612yes—
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×C3C2×#45111yescharacteristic
475C2×C3#301311yescharacteristic
476C2×C3#336311yescharacteristic
477C2×C3#337411yescharacteristic
478C2×C3#339511yescharacteristic
479C2×C3#340011yescharacteristic
480C22C2×#94211yescharacteristic
481C22C22×#21611yescharacteristic
482C22#583111yescharacteristic
483C22#583411yescharacteristic
484C22#583711yescharacteristic
485C22#583811yescharacteristic
486C22C2×#95311yescharacteristic
487C4—24no—
488C4—22no—
489C4—44no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—122no—
494C4—241no—
495C4—242no—
496C4#522814yes—
497C4#582411yescharacteristic
498C4#582811yescharacteristic
499C4#582911yescharacteristic
500C4#583311yescharacteristic
501C3#2419111yescharacteristic, complemented, Sylow
502C2#8368711yescharacteristic
503C2#8369611yescharacteristic
504C2#8369911yescharacteristic
505C2#8370111yescharacteristic
506C2C2×#522811yescharacteristic
507C2#8409711yescharacteristic
508C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#557C2
socle, prime radicalC23×C3C22×D4
hypercenterC2×#557D3
hypocenterC3#24191
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
162. C2×C3×Q8Q16semidirect
349. C2×Q8#216semidirect
501. C3#24191semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#50902
#824822
C2×C4×#2162
#717932
#717852
#717812
#716342
#716322
#716282
#716262
#241913