Finite Group Databasebeta

← #10830105

Subgroups — #10830105

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×#5567C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic
4C2×#5019C211yesmaximal, characteristic
5#72057C212yesmaximal
6#71627C211yesmaximal, characteristic, complemented
7#70642C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70213C212yesmaximal
10#69838C211yesmaximal, characteristic, complemented
11#69822C211yesmaximal, characteristic, complemented
12#24199—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#944C2212yes—
17C22×#214C2212yes—
18#5567—24no—
19#5567—24no—
20C2×#889—22no—
21C2×#887C2212yes—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6—21no—
25C2×C4×Dic6C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32#5221C2212yes—
33#5210—22no—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#5019—24no—
37#4972C2212yes—
38C2×#810C2212yes—
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#557—31no—
47C2×#461—31no—
48C2×#440D311yescharacteristic
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×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×Dic6D411yescharacteristic
61C22×Dic6C2312yes—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—22no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#217—22no—
79#944—24no—
80#944—24no—
81C2×#214—24no—
82C2×#214—24no—
83C2×#214—42no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4D412yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96#889—216no—
97#887—24no—
98#887—24no—
99#870—28no—
100#869—22no—
101#869D414yes—
102C4×Dic6—28no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#107—22no—
106C3×#103—22no—
107C3×#102—22no—
108C3×#97—22no—
109#835—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#816—24no—
114#816—42no—
115#810—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—32no—
118C22×SD16C2×D311yescharacteristic
119#557—64no—
120#557—64no—
121C2×C4⋊Q8—32no—
122C2×C4⋊Q8—31no—
123C2×#125—61no—
124C2×#125—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8—61no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4C2×D311yescharacteristic
130#461—64no—
131#440—24no—
132C2×#107C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#103C2×D311yescharacteristic
139C2×#102C2×D311yescharacteristic
140C2×#97C2×D311yescharacteristic
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—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8C2×D412yes—
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—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—42no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C2×Dic6—22no—
176C2×Dic6—24no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6C2×D414yes—
183C3×SD16—28no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8#4112yes—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4C2×D412yes—
195C3×C22⋊C4—42no—
196C3×C42—24no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—44no—
200#217—82no—
201#214—48no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—24no—
211Dic3⋊C4—82no—
212C4×Dic3—28no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C22×Q8—31no—
219C22×Q8—32no—
220C22×Q8C22×D311yescharacteristic
221C22×D4C22×D311yescharacteristic
222C23×C4—21no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16—64no—
227C2×SD16—64no—
228C2×SD16—122no—
229C22×C8—32no—
230C22×C8C22×D311yescharacteristic
231C22×C8C22×D311yescharacteristic
232C4⋊Q8—34no—
233C4⋊Q8—64no—
234C4⋊Q8—64no—
235C4⋊Q8—62no—
236#125—68no—
237#125—616no—
238C4×Q8—22no—
239C4×Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8#6714yes—
245C4×D4—22no—
246C4×D4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C4⋊C4C22×D311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C42—21no—
263C2×C42—32no—
264C2×C42—62no—
265C2×C42C22×D311yescharacteristic
266#107—22no—
267C4⋊C8—62no—
268#103—22no—
269#103—64no—
270#103—64no—
271#103—122no—
272#103—122no—
273#102—22no—
274#97—22no—
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—22no—
285C3×Q8—22no—
286C3×Q8—41no—
287C3×Q8—41no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—41no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—84no—
302C2×C3×C4#12012yes—
303C2×C3×C4#12012yes—
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C42−1+411yescharacteristic
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—24no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3#13714yes—
322Dic6—28no—
323Dic6—24no—
324Dic6—24no—
325Dic6—42no—
326Dic6—42no—
327C3×C8—22no—
328C3×C8—41no—
329C3⋊C8—42no—
330C24—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—34no—
337C2×Q8—64no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8C2×#6712yes—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4#23512yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—42no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—82no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4C23×D311yescharacteristic
363SD16—28no—
364SD16—128no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—62no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8#23512yes—
374C4⋊C4—24no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—68no—
379C4⋊C4—68no—
380C4⋊C4—68no—
381C4⋊C4—64no—
382C4⋊C4—62no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—62no—
386C4⋊C4—62no—
387C4⋊C4—242no—
388C4⋊C4C2×#6712yes—
389C22⋊C4—42no—
390C22⋊C4—124no—
391C22⋊C4—242no—
392C42—24no—
393C42—68no—
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—42no—
411C3×C4—42no—
412C3×C4—81no—
413C3×C4#52511yescharacteristic
414C3×C4#53711yescharacteristic
415C3×C4#54611yescharacteristic
416C3×C4#54711yescharacteristic
417Dic3—24no—
418Dic3—44no—
419Dic3—42no—
420Dic3—82no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C2×D3×D411yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—68no—
432Q8—64no—
433Q8—64no—
434Q8—122no—
435Q8—122no—
436D4—22no—
437D4—22no—
438D4—41no—
439D4—41no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—34no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—84no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4#93712yes—
463C2×C4#93712yes—
464C2×C4C2×#23511yescharacteristic
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C2×C4#100911yescharacteristic
468C2×C4#101011yescharacteristic
469C2×C4C22×#6711yescharacteristic
470C8—22no—
471C8—41no—
472C8—122no—
473C2×C3—82no—
474C2×C3#240711yescharacteristic
475C2×C3#300411yescharacteristic
476C2×C3#301011yescharacteristic
477C2×C3#337212yes—
478C2×C3#340012yes—
479C22—44no—
480C22—42no—
481C22—81no—
482C22—82no—
483C22—81no—
484C22#555211yescharacteristic
485C22#555311yescharacteristic
486C22#577512yes—
487C22D3×#13712yes—
488C22C2×#93711yescharacteristic
489C4—22no—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—64no—
494C4—81no—
495C4—124no—
496C4—122no—
497C4—242no—
498C4#555011yescharacteristic
499C4#555111yescharacteristic
500C4#579111yescharacteristic
501C4#583311yescharacteristic
502C3#2419911yescharacteristic, complemented, Sylow
503C2—82no—
504C2#7975512yes—
505C2#7976412yes—
506C2#8240211yescharacteristic
507C2#8394811yescharacteristic
508C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#440D3
hypocenterC3#24199
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#5567C2semidirect
6. #71627C2semidirect
7. #70642C2semidirect
8. #70284C2semidirect
10. #69838C2semidirect
11. #69822C2semidirect
502. C3#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C2×C3×#4402
#823882
C2×#50192
#720572
#716272
#706422
#702842
#702132
#698382
#698222
#241993