Finite Group Databasebeta

← #10950915

Subgroups — #10950915

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#466C211yesmaximal, characteristic, complemented
2C2×#4883C211yesmaximal, characteristic, complemented
3C3×#2232C211yesmaximal, characteristic, complemented
4#71075C211yesmaximal, characteristic, complemented
5#70599C211yesmaximal, characteristic, complemented
6#69330C212yesmaximal, complemented
7#69327C212yesmaximal, complemented
8#69195C212yesmaximal, complemented
9#69191C212yesmaximal, complemented
10#69108C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33752—31nomaximal, Sylow
13C2×D3×D8C2211yescharacteristic, complemented
14C22×D24C2211yescharacteristic, complemented
15C22×D3×C8C2211yescharacteristic
16D3×#121—21no—
17C2×#895—21no—
18C3×#466C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic, complemented
20#5157C2211yescharacteristic, complemented
21C2×#826C2212yescomplemented
22#5115C2211yescharacteristic
23#5038—21no—
24C2×#809—21no—
25C2×D48C2212yescomplemented
26C2×D3×C16C2212yescomplemented
27#4883—21no—
28#4883—21no—
29#4883C2212yes—
30#4883C2212yescomplemented
31#4883C2211yescharacteristic, complemented
32D3×#107C2211yescharacteristic, complemented
33#4775—21no—
34D3×#102—21no—
35C3×#366C2212yescomplemented
36C3×#357C2212yescomplemented
37C3×C22⋊C16C2211yescharacteristic, complemented
38#4549C2211yescharacteristic, complemented
39#4509C2212yescomplemented
40(C2×D3)⋊C16C2212yescomplemented
41#4491C2212yescomplemented
42#4491C2212yescomplemented
43#4489C2212yescomplemented
44C2×#466—31no—
45C2×#466—31no—
46#2232—31no—
47#2232—32no—
48#2232D311yescharacteristic, complemented
49#2215—32no—
50#2215—32no—
51#2184—31no—
52#2184—32no—
53#2184—31no—
54C2×C22⋊C16—31no—
55C2×D3×D4—21no—
56C2×D3×D4—41no—
57C22×D12—21no—
58C22×D12—41no—
59C22×C4×D3D411yescharacteristic
60C2×C3×D8C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#121—21no—
63C2×C3×C4⋊C4—21no—
64#939—41no—
65#938—21no—
66C2×#213—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—41no—
69C22×C3⋊C8D411yescharacteristic
70D3×D8—42no—
71C2×D24—21no—
72C2×D24—42no—
73C2×D24C2312yes—
74C2×D24C2311yescharacteristic
75C2×D3×C8—42no—
76C2×D3×C8D411yescharacteristic
77C2×D3×C8D411yescharacteristic
78C2×D3×C8C2311yescharacteristic
79C2×D3×C8C2311yescharacteristic
80#895—22no—
81#895—22no—
82#895—21no—
83#895—41no—
84#895—41no—
85D3×C4⋊C4—21no—
86#884—41no—
87D3×C22⋊C4—41no—
88C2×C3×C16C2312yes—
89C3×#107—21no—
90C3×#107C2311yescharacteristic
91C3×#107C2312yes—
92C3×#102—21no—
93#832—21no—
94#826—42no—
95#821—21no—
96#818C2311yescharacteristic
97C2×C3⋊C16C2312yes—
98#809—22no—
99#809—22no—
100#809—21no—
101#807—21no—
102D48—42no—
103D3×C16—42no—
104C22×D8—31no—
105C22×D8—31no—
106C23×C8—31no—
107C2×#121—61no—
108C2×#121—61no—
109C2×D16—32no—
110C2×D16—32no—
111C22×C16—32no—
112#466—31no—
113#466—31no—
114#466—32no—
115#466—31no—
116#466—32no—
117#466—61no—
118#466—61no—
119#466—61no—
120#466—61no—
121#466C2×D311yescharacteristic
122C2×#107—31no—
123C2×#107C2×D311yescharacteristic, complemented
124C2×#102—61no—
125C2×#102—61no—
126#366—32no—
127#366C2×D312yescomplemented
128#357—32no—
129#357—32no—
130#357—32no—
131#357C2×D312yescomplemented
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic, complemented
135C23×D3—41no—
136C23×D3—41no—
137C23×D3D811yescharacteristic
138C2×C3×D4—21no—
139C2×C3×D4—41no—
140C22×C3×C4—41no—
141C22×C3×C4C2×D411yescharacteristic
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C22×Dic3D811yescharacteristic
146D3×D4—42no—
147D3×D4—42no—
148D3×D4—84no—
149C2×D12—22no—
150C2×D12—21no—
151C2×D12—42no—
152C2×D12—42no—
153C2×D12—41no—
154C2×D12—42no—
155C2×D12—42no—
156C2×D12—41no—
157C2×D12—84no—
158C2×C4×D3—41no—
159C2×C4×D3—41no—
160C2×C4×D3—41no—
161C2×C4×D3D811yescharacteristic
162C2×C4×D3D811yescharacteristic
163C2×C4×D3C2×D411yescharacteristic
164C2×C4×D3C2×D411yescharacteristic
165C3×D8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2411yescharacteristic
169C3×C4⋊C4—21no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—41no—
172C3×C22⋊C4—41no—
173#217—41no—
174#213—42no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—42no—
178(C2×D3)⋊C4—42no—
179C4⋊Dic3—21no—
180Dic3⋊C4—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8C2×D411yescharacteristic
183C2×C3⋊C8C2×D411yescharacteristic
184D24—22no—
185D24—22no—
186D24—22no—
187D24—42no—
188D3×C8—22no—
189D3×C8—22no—
190D3×C8—41no—
191D3×C8—41no—
192C3×C16—22no—
193C3⋊C16—22no—
194C22×D4—61no—
195C22×D4—61no—
196C22×D4—121no—
197C22×D4—121no—
198C23×C4—31no—
199C2×D8—31no—
200C2×D8—32no—
201C2×D8—61no—
202C2×D8—61no—
203C2×D8—122no—
204C2×D8—122no—
205C2×D8C22×D311yescharacteristic
206C22×C8—31no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8—31no—
210C22×C8—31no—
211C22×C8—122no—
212C22×C8C22×D311yescharacteristic
213#121—21no—
214#121—61no—
215#121—62no—
216#121—62no—
217#121—61no—
218#121—61no—
219#121—121no—
220#121—121no—
221#121—121no—
222#121—121no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—121no—
226C2×C22⋊C4—121no—
227D16—122no—
228D16—122no—
229C2×C16—32no—
230C2×C16—122no—
231C2×C16C22×D312yes—
232#107—21no—
233#107—31no—
234#107—61no—
235#107C22×D311yescharacteristic
236#107C22×D312yes—
237#102—21no—
238#102—61no—
239#102—62no—
240#102—62no—
241#102—61no—
242#102—61no—
243C23×C3—41no—
244C23×C3C2×D811yescharacteristic
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—42no—
248C22×D3—41no—
249C22×D3—82no—
250C22×D3—82no—
251C22×D3—82no—
252C22×D3—82no—
253C22×D3—82no—
254C22×D3—82no—
255C22×D3D1614yescomplemented
256C22×D3C2×D811yescharacteristic
257C22×D3C2×D811yescharacteristic
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—82no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—82no—
266C2×C3×C4C2×D811yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268#67—82no—
269#67—82no—
270#67—82no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3C2×D811yescharacteristic
274C2×Dic3C2×D811yescharacteristic
275D12—42no—
276D12—41no—
277D12—42no—
278D12—41no—
279D12—82no—
280D12—84no—
281D12—84no—
282D12—82no—
283C4×D3—21no—
284C4×D3—21no—
285C4×D3—21no—
286C4×D3—21no—
287C4×D3—41no—
288C4×D3—41no—
289C4×D3—82no—
290C3×C8—21no—
291C3×C8H(5,𝔽2)12yes—
292C3⋊C8—21no—
293C3⋊C8—21no—
294C24—31no—
295C24—121no—
296C24—121no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—62no—
300C2×D4—61no—
301C2×D4—122no—
302C2×D4—122no—
303C2×D4—121no—
304C2×D4—121no—
305C2×D4—122no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—122no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—121no—
312C2×D4—244no—
313C2×D4—244no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—41no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4D3×D411yescharacteristic
324D8—22no—
325D8—62no—
326D8—62no—
327D8—62no—
328D8—122no—
329D8—122no—
330C2×C8—22no—
331C2×C8—31no—
332C2×C8—31no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—121no—
336C2×C8—121no—
337C2×C8—121no—
338C2×C8D3×D411yescharacteristic
339C2×C8C23×D311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4—61no—
344C4⋊C4—121no—
345C22⋊C4—41no—
346C22⋊C4—121no—
347C22⋊C4—122no—
348C22⋊C4—122no—
349C22⋊C4—121no—
350C22⋊C4—121no—
351C16—22no—
352C16—62no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3C2×D1612yes—
359C22×C3C22×D811yescharacteristic
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C2×D3—44no—
365C2×D3—82no—
366C2×D3—81no—
367C2×D3—82no—
368C2×D3—81no—
369C2×D3—82no—
370C2×D3—81no—
371C2×D3—164no—
372C2×D3—164no—
373C2×D3C2×D1612yes—
374C2×D3C2×D1612yes—
375C3×C4—21no—
376C3×C4—81no—
377C3×C4—82no—
378C3×C4#53411yescharacteristic
379C3×C4#57511yescharacteristic
380Dic3—21no—
381Dic3—21no—
382Dic3—81no—
383C23—31no—
384C23—31no—
385C23—34no—
386C23—41no—
387C23—121no—
388C23—121no—
389C23—122no—
390C23—121no—
391C23—242no—
392C23—242no—
393C23—242no—
394C23—242no—
395C23—242no—
396C23—242no—
397C23D3×D811yescharacteristic
398D4—41no—
399D4—41no—
400D4—82no—
401D4—122no—
402D4—121no—
403D4—122no—
404D4—121no—
405D4—242no—
406D4—242no—
407D4—244no—
408D4—244no—
409D4—242no—
410D4—242no—
411D4—242no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—82no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—242no—
428C2×C4D3×D811yescharacteristic
429C2×C4C2×D3×D411yescharacteristic
430C8—21no—
431C8—61no—
432C8—61no—
433C8#100912yes—
434C2×C3—22no—
435C2×C3—161no—
436C2×C3#318611yescharacteristic
437C2×C3C22×D1611yescharacteristic
438C2×C3#341411yescharacteristic
439D3—22no—
440D3—22no—
441D3—162no—
442D3—161no—
443C22—21no—
444C22—21no—
445C22—32no—
446C22—32no—
447C22—61no—
448C22—61no—
449C22—61no—
450C22—61no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—124no—
455C22—242no—
456C22—241no—
457C22—242no—
458C22—241no—
459C22—242no—
460C22—241no—
461C22—484no—
462C22—484no—
463C22D3×D1612yes—
464C22C2×D3×D811yescharacteristic
465C4—21no—
466C4—61no—
467C4—61no—
468C4—82no—
469C4—81no—
470C4—241no—
471C4#565311yescharacteristic
472C4#577711yescharacteristic
473C3#3375211yescharacteristic, complemented, Sylow
474C2—22no—
475C2—62no—
476C2—62no—
477C2—161no—
478C2—482no—
479C2—481no—
480C2#8129911yescharacteristic
481C2C2×D3×D1611yescharacteristic
482C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2232C2
socleC22×C3C22×D8
hypercenter#2232D3
hypocenterC3#33752
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2232

Complements

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

NH ≅ G/Ntype
1. D3×#466C2semidirect
2. C2×#4883C2semidirect
3. C3×#2232C2semidirect
4. #71075C2semidirect
5. #70599C2semidirect
6. #69330C2semidirect
7. #69327C2semidirect
8. #69195C2semidirect
9. #69191C2semidirect
10. #69108C2semidirect
11. D3×C22⋊C16C2semidirect
13. C2×D3×D8C22semidirect
14. C22×D24C22semidirect
19. C2×C3×#107C22semidirect
20. #5157C22semidirect
21. C2×#826C22semidirect
25. C2×D48C22semidirect
26. C2×D3×C16C22semidirect
30. #4883C22semidirect
31. #4883C22semidirect
32. D3×#107C22semidirect
35. C3×#366C22semidirect
36. C3×#357C22semidirect
37. C3×C22⋊C16C22semidirect
38. #4549C22semidirect
39. #4509C22semidirect
40. (C2×D3)⋊C16C22semidirect
41. #4491C22semidirect
42. #4491C22semidirect
43. #4489C22semidirect
48. #2232D3semidirect
123. C2×#107C2×D3semidirect
127. #366C2×D3semidirect
131. #357C2×D3semidirect
134. C22⋊C16C2×D3semidirect
255. C22×D3D16semidirect
473. C3#33752semidirect

Maximal subgroups

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

subgroupindex
D3×#4662
C2×#48832
C3×#22322
#710752
#705992
#693302
#693272
#691952
#691912
#691082
D3×C22⋊C162
#337523