Finite Group Databasebeta

← #10830084

Subgroups — #10830084

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic
3C2×#4841C211yesmaximal, characteristic, complemented
4#74190C211yesmaximal, characteristic, complemented
5C3×#1804C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70214C212yesmaximal
8#70178C212yesmaximal, complemented
9#69552C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68907C212yesmaximal, complemented
12#24197—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936C2212yes—
15C22×#213C2211yescharacteristic
16C2×#926—21no—
17C2×#924—21no—
18C22×#204C2212yes—
19#5542—24no—
20#5542—24no—
21C2×#887C2212yes—
22C2×#879—22no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C3×#382C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#816C2212yes—
31#5014—24no—
32#4988C2212yes—
33#4984C2212yes—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4841—24no—
40#4677—21no—
41#4673—21no—
42#4668—22no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×#442—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2312yes—
63C22×C3×C8C2312yes—
64C3×C4×D4—22no—
65C3×C4×D4D414yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72#936—24no—
73#936—24no—
74C2×#213—22no—
75C2×#213—22no—
76C2×#213—22no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80#926—28no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86#924—28no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D412yes—
89C2×C4×Dic3—22no—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C2×#204—24no—
94C2×#204—24no—
95C2×#204—42no—
96#887—24no—
97#887—24no—
98#879—216no—
99C4×D12—22no—
100C4×D12—22no—
101#869—22no—
102#869D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×C4⋊C8—22no—
106C3×#102—42no—
107C3×#95—21no—
108C3×#95—21no—
109#832—22no—
110C3⋊#95—22no—
111#816—24no—
112#810—24no—
113#810—42no—
114#809—22no—
115#808—22no—
116#803—22no—
117C22×SD16—32no—
118C22×D8—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#461—64no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102C2×D312yes—
140C2×#97—31no—
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382C2×D311yescharacteristic
147#340C2×D312yes—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C23×D3—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4C2×D412yes—
158C22×C3×C4—21no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—44no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12#4112yes—
175C2×C4×D3—42no—
176C2×C4×D3—82no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—42no—
184C2×Dic6C2×D414yes—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—24no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#217—28no—
196#217—28no—
197#213—28no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—42no—
200(C2×D3)⋊C4—82no—
201C4⋊Dic3—22no—
202C4⋊Dic3—24no—
203C4⋊Dic3—24no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—82no—
207Dic3⋊C4—28no—
208Dic3⋊C4—24no—
209C4×Dic3—28no—
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—
218#204—48no—
219C22×Q8—31no—
220C22×Q8—32no—
221C22×D4—31no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—61no—
225C2×SD16—64no—
226C2×SD16—64no—
227C2×SD16—122no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—62no—
246C4×D4—62no—
247C4×D4D1214yes—
248#117—68no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4C22×D312yes—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—22no—
269#103—64no—
270#103—64no—
271#103—122no—
272#102—42no—
273#102—62no—
274#102—62no—
275#97—62no—
276#95—21no—
277#95—21no—
278#95—62no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—42no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—48no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
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—24no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321D12—22no—
322D12—22no—
323D12—41no—
324D12—41no—
325C4×D3—84no—
326Dic6—24no—
327Dic6—24no—
328Dic6—44no—
329Dic6—42no—
330Dic6—42no—
331C3×C8—42no—
332C3⋊C8—22no—
333C3⋊C8—41no—
334C24—21no—
335C24—61no—
336C2×Q8—34no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×Q8—122no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—32no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4C2×D1212yes—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—44no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—122no—
365C22×C4—242no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—128no—
370D8—68no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—32no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—68no—
384C4⋊C4—62no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—242no—
391C4⋊C4C2×D1212yes—
392C22⋊C4—24no—
393C22⋊C4—68no—
394C22⋊C4—68no—
395C22⋊C4—124no—
396C22⋊C4—122no—
397C22⋊C4—242no—
398C42—68no—
399C42—62no—
400C42—62no—
401C42C2×D1212yes—
402C42D3×D412yes—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3#57512yes—
412C22×C3#57712yes—
413C2×D3—44no—
414C2×D3—42no—
415C2×D3—82no—
416C2×D3—81no—
417C2×D3—81no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—42no—
421C3×C4—82no—
422C3×C4#52511yescharacteristic
423C3×C4#53711yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#54711yescharacteristic
426Dic3—44no—
427Dic3—42no—
428Dic3—42no—
429Dic3—81no—
430C23—22no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—62no—
436C23—121no—
437C23—122no—
438C23—122no—
439C23—121no—
440C23C2×D3×D411yescharacteristic
441Q8—64no—
442Q8—64no—
443Q8—124no—
444Q8—122no—
445Q8—122no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—62no—
451D4—62no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—64no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—244no—
474C2×C4#88212yes—
475C2×C4#88212yes—
476C2×C4C22×D1211yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C2×C4#100411yescharacteristic
479C2×C4#100711yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4D3×#4111yescharacteristic
482C8—42no—
483C8—62no—
484C8—121no—
485C2×C3—82no—
486C2×C3#240711yescharacteristic
487C2×C3#300511yescharacteristic
488C2×C3#301011yescharacteristic
489C2×C3#337612yes—
490C2×C3#340112yes—
491D3—82no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—82no—
496C22—82no—
497C22—124no—
498C22—122no—
499C22—241no—
500C22—242no—
501C22—241no—
502C22C2×#88211yescharacteristic
503C22#559911yescharacteristic
504C22#563311yescharacteristic
505C22#577012yes—
506C22#577312yes—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—82no—
511C4—124no—
512C4—122no—
513C4—122no—
514C4—241no—
515C4#551011yescharacteristic
516C4#552411yescharacteristic
517C4#561211yescharacteristic
518C4#563111yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#7401511yescharacteristic
523C2#7883611yescharacteristic
524C2#7892711yescharacteristic
525C2#8052812yes—
526C2#8115412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1804D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. C2×#5542C2semidirect
3. C2×#4841C2semidirect
4. #74190C2semidirect
5. C3×#1804C2semidirect
6. #70473C2semidirect
8. #70178C2semidirect
9. #69552C2semidirect
10. #68909C2semidirect
11. #68907C2semidirect
56. #1804D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×#50142
C2×#48412
#741902
C3×#18042
#704732
#702142
#701782
#695522
#689092
#689072
#241973