Finite Group Databasebeta

← #10830101

Subgroups — #10830101

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×#5721C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic
4C2×#5018C211yesmaximal, characteristic
5#72057C212yesmaximal
6#72048C212yesmaximal
7#71786C211yesmaximal, characteristic, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70473C211yesmaximal, characteristic, complemented
10#70284C212yesmaximal, complemented
11#70235C212yesmaximal, complemented
12#24199—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17C2×#944C2212yes—
18C22×#214C2211yescharacteristic
19#5721—24no—
20#5721—24no—
21C2×#890—21no—
22C2×#889—22no—
23C2×#887—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#461—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5221C2212yes—
32C2×#832C2211yescharacteristic
33#5210—22no—
34C2×#816C2212yes—
35#5018—24no—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#557—31no—
47C2×#461D311yescharacteristic
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×Q8C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2311yescharacteristic
62C2×C3×SD16—24no—
63C2×C3×SD16—24no—
64C2×C3×SD16—42no—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77Q8×Dic3—24no—
78Q8×Dic3—24no—
79#944—24no—
80#944—24no—
81C2×#214—22no—
82C2×#214—22no—
83C2×#214—22no—
84C2×C4⋊Dic3—22no—
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—24no—
108#835—42no—
109#832—22no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#816—42no—
114#810—22no—
115#807—22no—
116#803—22no—
117C22×SD16—31no—
118C22×SD16C2×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×D4C2×D311yescharacteristic
130#461—24no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103C2×D312yes—
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—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D414yes—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4#4112yes—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—42no—
166C22×C3×C4—82no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C2×Dic6—22no—
176C2×Dic6—24no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C2×D412yes—
183C3×SD16—48no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C22⋊C4—42no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194#217—44no—
195#217—82no—
196#214—28no—
197C4⋊Dic3—28no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
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×D312yes—
221C22×D4C22×D311yescharacteristic
222C23×C4—21no—
223C2×SD16—24no—
224C2×SD16—24no—
225C2×SD16—42no—
226C2×SD16—62no—
227C2×SD16—62no—
228C2×SD16—62no—
229C22×C8—31no—
230C22×C8—31no—
231C22×C8C22×D312yes—
232C4⋊Q8—22no—
233C4⋊Q8—64no—
234C4⋊Q8—64no—
235C4⋊Q8#6714yes—
236#125—616no—
237#125—68no—
238C4×Q8—34no—
239C4×Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×D4—22no—
246C4×D4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
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—64no—
271#103—122no—
272#103—122no—
273#102—62no—
274#97—62no—
275#95—62no—
276#95—61no—
277#95—61no—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3C22×D411yescharacteristic
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—42no—
287C3×Q8—42no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—41no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—84no—
300C2×C3×C4#12012yes—
301C2×C3×C4#12012yes—
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—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3#13714yes—
322Dic6—28no—
323Dic6—22no—
324Dic6—22no—
325Dic6—41no—
326Dic6—41no—
327C3×C8—42no—
328C3⋊C8—22no—
329C3⋊C8—41no—
330C24—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—32no—
337C2×Q8—64no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8C2×#6714yes—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4#23712yes—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—42no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—62no—
359C22×C4—82no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—48no—
364SD16—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—68no—
380C4⋊C4—68no—
381C4⋊C4—68no—
382C4⋊C4—64no—
383C4⋊C4—62no—
384C4⋊C4—64no—
385C4⋊C4—61no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—242no—
389C22⋊C4—42no—
390C22⋊C4—124no—
391C22⋊C4—242no—
392C42—68no—
393C42—64no—
394C42—62no—
395C42—62no—
396C42D3×D412yes—
397C42C2×#6712yes—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3#55011yescharacteristic
406C22×C3C2×#13712yes—
407C22×C3#57712yes—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—44no—
411C3×C4—81no—
412C3×C4#52511yescharacteristic
413C3×C4#53711yescharacteristic
414C3×C4#54611yescharacteristic
415C3×C4#54711yescharacteristic
416Dic3—24no—
417Dic3—42no—
418Dic3—42no—
419Dic3—42no—
420Dic3—82no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C2×D3×D411yescharacteristic
427Q8—24no—
428Q8—24no—
429Q8—42no—
430Q8—42no—
431Q8—68no—
432Q8—62no—
433Q8—62no—
434Q8—121no—
435Q8—121no—
436D4—22no—
437D4—22no—
438D4—41no—
439D4—41no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—34no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—84no—
456C2×C4—122no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
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—82no—
474C2×C3#240711yescharacteristic
475C2×C3#300411yescharacteristic
476C2×C3#301011yescharacteristic
477C2×C3#337212yes—
478C2×C3#340012yes—
479C22—42no—
480C22—44no—
481C22—81no—
482C22—82no—
483C22—81no—
484C22#559811yescharacteristic
485C22#559911yescharacteristic
486C22#576212yes—
487C22D3×#13712yes—
488C22C2×#93711yescharacteristic
489C4—22no—
490C4—22no—
491C4—44no—
492C4—64no—
493C4—81no—
494C4—122no—
495C4—122no—
496C4—122no—
497C4—242no—
498C4#560611yescharacteristic
499C4#560811yescharacteristic
500C4#580611yescharacteristic
501C4#582411yescharacteristic
502C3#2419911yescharacteristic, complemented, Sylow
503C2—82no—
504C2#8041812yes—
505C2#8042312yes—
506C2#8270111yescharacteristic
507C2#8386011yescharacteristic
508C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#461D3
hypocenterC3#24199
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
7. #71786C2semidirect
8. #70486C2semidirect
9. #70473C2semidirect
10. #70284C2semidirect
11. #70235C2semidirect
502. C3#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×C3×#4612
#823882
C2×#50182
#720572
#720482
#717862
#704862
#704732
#702842
#702352
#241993