Finite Group Databasebeta

← #10830123

Subgroups — #10830123

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5732C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3#82481C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic
5#72072C212yesmaximal
6#72065C212yesmaximal, complemented
7#71786C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic
9#70473C211yesmaximal, characteristic, complemented
10#70279C212yesmaximal
11#70235C212yesmaximal, complemented
12#24201—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#947C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×#215C2211yescharacteristic
19#5732—24no—
20#5732—24no—
21C2×#898—22no—
22C2×#897—21no—
23C2×#892—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#462—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29#5229C2212yes—
30C2×#835C2211yescharacteristic
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#821C2212yes—
34#5026—24no—
35#4986—21no—
36#4979—21no—
37#4972C2212yes—
38C2×#809C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4936C2211yescharacteristic
41#4933—22no—
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#462D311yescharacteristic, complemented
48C2×#440—31no—
49#2581—31no—
50#2042—32no—
51#2015—32no—
52#1985—31no—
53#1960—31no—
54#1878—32no—
55#1863—32no—
56#1804—31no—
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6D411yescharacteristic
62C2×C3×Q16—24no—
63C2×C3×Q16—24no—
64C2×C3×Q16—42no—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yes—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C42—21no—
75C2×C3×C42C2311yescharacteristic
76#947—24no—
77#947—24no—
78Q8×Dic3—24no—
79Q8×Dic3—24no—
80C2×#215—22no—
81C2×#215—22no—
82C2×#215—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4D412yes—
92C2×C4×Dic3—22no—
93C2×C4×Dic3C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2311yescharacteristic
96#898—216no—
97#897—28no—
98#892—28no—
99C4×D12—22no—
100C4×D12D414yes—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8—22no—
104C3×#103—24no—
105#835—22no—
106#835—24no—
107#835—42no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#821—42no—
114#809—22no—
115#807—22no—
116#803—22no—
117C22×Q16C2×D312yes—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—24no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#103C2×D312yes—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8C2×D414yes—
162C2×C3×Q8#4112yes—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—22no—
171C22×Dic3C2×D412yes—
172C22×Dic3C2×D412yes—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12C2×D412yes—
179C2×C4×D3—24no—
180C2×C4×D3—44no—
181C2×Dic6—22no—
182C2×Dic6—42no—
183C3×Q16—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×C4⋊C4—42no—
192C3×C42—42no—
193C3×C42C2×D412yes—
194C3×C42C2×D412yes—
195#215—28no—
196(C2×D3)⋊C4—28no—
197(C2×D3)⋊C4—28no—
198(C2×D3)⋊C4—24no—
199C4⋊Dic3—28no—
200C4⋊Dic3—22no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3C2×D412yes—
205Dic3⋊C4—24no—
206Dic3⋊C4—44no—
207C4×Dic3—28no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8#4112yes—
216C22×Q8—31no—
217C22×Q8C22×D311yescharacteristic
218C22×Q8C22×D312yes—
219C22×D4—31no—
220C23×C4—61no—
221C2×Q16—24no—
222C2×Q16—24no—
223C2×Q16—42no—
224C2×SD16—62no—
225C2×SD16—62no—
226C2×SD16—62no—
227C22×C8—31no—
228C22×C8—31no—
229C22×C8C22×D312yes—
230C4⋊Q8—22no—
231C4⋊Q8#6714yes—
232#126—616no—
233#124—64no—
234#124—64no—
235C22⋊Q8—68no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×Q8—62no—
241C4×Q8—62no—
242C4×D4—34no—
243C4×D4—62no—
244#117—68no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—61no—
258C2×C42—21no—
259C2×C42—32no—
260C2×C42—62no—
261C2×C42C22×D311yescharacteristic
262#107—62no—
263C4⋊C8—22no—
264#103—24no—
265#103—62no—
266#103—64no—
267#103—122no—
268#102—62no—
269#102—122no—
270#97—62no—
271#95—62no—
272#95—62no—
273#95—62no—
274#95—61no—
275#95—61no—
276C23×C3C22×D411yescharacteristic
277C22×D3—22no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—42no—
281C22×D3—42no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—42no—
287C3×Q8—42no—
288C3×Q8—41no—
289C3×Q8—41no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
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—24no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318D12—22no—
319D12—22no—
320D12—41no—
321D12—41no—
322C4×D3—48no—
323Dic6—44no—
324C3×C8—42no—
325C3⋊C8—22no—
326C3⋊C8—41no—
327C24—61no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—62no—
339C2×Q8—122no—
340C2×Q8#23712yes—
341C2×Q8C2×#6714yes—
342C2×D4—32no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—61no—
354C22×C4—64no—
355C22×C4—62no—
356C22×C4—124no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—48no—
361SD16—68no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—32no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—61no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—32no—
376C4⋊C4—42no—
377C4⋊C4—68no—
378C4⋊C4—62no—
379C4⋊C4—64no—
380C4⋊C4—61no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—124no—
384C22⋊C4—68no—
385C22⋊C4—68no—
386C22⋊C4—64no—
387C42—42no—
388C42—68no—
389C42—62no—
390C42—62no—
391C42D3×D412yes—
392C42C2×#6712yes—
393C22×C3C2×#12011yescharacteristic
394C22×C3#54811yescharacteristic
395C22×C3#55011yescharacteristic
396C22×C3#57512yes—
397C22×C3#57812yes—
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—82no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—81no—
408C3×C4#52511yescharacteristic
409C3×C4#53711yescharacteristic
410C3×C4#54611yescharacteristic
411C3×C4#54711yescharacteristic
412Dic3—42no—
413Dic3—42no—
414Dic3—42no—
415Dic3—82no—
416C23—62no—
417C23—122no—
418C23—122no—
419C23—121no—
420C23—121no—
421C23C2×D3×D411yescharacteristic
422Q8—22no—
423Q8—24no—
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—42no—
428Q8—42no—
429Q8—41no—
430Q8—124no—
431D4—62no—
432D4—62no—
433D4—121no—
434D4—121no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—64no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—128no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4#93712yes—
457C2×C4#93712yes—
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×#23711yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4#101011yescharacteristic
463C2×C4C22×#6711yescharacteristic
464C8—42no—
465C8—62no—
466C8—121no—
467C2×C3#240711yescharacteristic
468C2×C3#300511yescharacteristic
469C2×C3#301111yescharacteristic
470C2×C3#337712yes—
471C2×C3#340212yes—
472D3—82no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—242no—
477C22—242no—
478C22#559811yescharacteristic
479C22#559911yescharacteristic
480C22#576612yes—
481C22#577712yes—
482C22C2×#93711yescharacteristic
483C4—22no—
484C4—22no—
485C4—44no—
486C4—42no—
487C4—81no—
488C4—122no—
489C4—122no—
490C4—122no—
491C4—242no—
492C4#560611yescharacteristic
493C4#560811yescharacteristic
494C4#580611yescharacteristic
495C4#582411yescharacteristic
496C3#2420111yescharacteristic, complemented, Sylow
497C2—242no—
498C2#8046712yes—
499C2#8047112yes—
500C2#8270111yescharacteristic
501C2#8388411yescharacteristic
502C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#462C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#462D3
hypocenterC3#24201
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
2. C2×C3×#462C2semidirect
3. #82481C2semidirect
6. #72065C2semidirect
7. #71786C2semidirect
9. #70473C2semidirect
11. #70235C2semidirect
47. C2×#462D3semidirect
496. C3#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×C3×#4622
#824812
C2×#50262
#720722
#720652
#717862
#704802
#704732
#702792
#702352
#242013