Finite Group Databasebeta

← #10950919

Subgroups — #10950919

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5157C211yesmaximal, characteristic
2D3×#467C211yesmaximal, characteristic, complemented
3C3×#2187C211yesmaximal, characteristic, complemented
4#72036C211yesmaximal, characteristic, complemented
5#71487C212yesmaximal, complemented
6#71483C212yesmaximal, complemented
7#71078C211yesmaximal, characteristic, complemented
8#69200C212yesmaximal, complemented
9#69196C212yesmaximal, complemented
10D3×C22⋊C16C211yesmaximal, characteristic, complemented
11#69098C211yesmaximal, characteristic, complemented
12#33753—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×#938—21no—
15C2×D3×Q16C2211yescharacteristic, complemented
16C22×D3×C8C2211yescharacteristic
17D3×C22⋊Q8—21no—
18C2×C3×SD32C2212yescomplemented
19C3×#467C2211yescharacteristic
20C2×#832—21no—
21#5188C2211yescharacteristic, complemented
22#5157—21no—
23#5157—21no—
24#5157C2212yes—
25#5157C2212yes—
26C2×#827C2212yescomplemented
27#5116C2211yescharacteristic
28C2×#818C2211yescharacteristic
29#5051—21no—
30C2×D3×C16C2212yescomplemented
31#4887C2211yescharacteristic, complemented
32D3×#107C2211yescharacteristic, complemented
33#4813—21no—
34D3×#103—21no—
35C3×#357C2212yescomplemented
36C3×C22⋊C16C2211yescharacteristic
37#4564C2212yescomplemented
38#4562C2212yescomplemented
39#4549C2211yescharacteristic
40(C2×D3)⋊C16C2212yescomplemented
41#4505C2212yescomplemented
42#4503C2212yescomplemented
43#4490C2212yescomplemented
44C2×#467—31no—
45C2×#466—31no—
46#2235—31no—
47#2235—31no—
48#2233—32no—
49#2217—32no—
50#2217—32no—
51#2187—31no—
52#2187D311yescharacteristic, complemented
53#2185—32no—
54C2×C22⋊C16—31no—
55C22×C3×D4—21no—
56C22×#67—41no—
57C2×D3×Q8—21no—
58C22×C4×D3D411yescharacteristic
59C2×C3×Q16C2311yescharacteristic
60C2×C3×D8—21no—
61C2×C3×D8—42no—
62C2×C3×D8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C3×C22⋊Q8—21no—
65C2×#217—41no—
66#946—21no—
67C2×#216—21no—
68#938—22no—
69#938—22no—
70#938—41no—
71#938—41no—
72C2×C4⋊Dic3—21no—
73#924—21no—
74C22×C3⋊C8D411yescharacteristic
75D3×Q16—42no—
76C2×Dic12C2311yescharacteristic
77C2×D3×C8—42no—
78C2×D3×C8D411yescharacteristic
79C2×D3×C8D411yescharacteristic
80C2×D3×C8C2311yescharacteristic
81C2×D3×C8C2311yescharacteristic
82#897—21no—
83#896—41no—
84D3×C4⋊C4—21no—
85D3×C4⋊C4—41no—
86D3×C22⋊C4—41no—
87#878—41no—
88C3×SD32—42no—
89C2×C3×C16C2312yes—
90C3×#107C2311yescharacteristic
91C3×#103—21no—
92#835—21no—
93#832—22no—
94#832—22no—
95#827—42no—
96#818—21no—
97#818C2311yescharacteristic
98#818C2312yes—
99#816—21no—
100C2×C3⋊C16C2312yes—
101#810—21no—
102#807—21no—
103D3×C16—42no—
104C22×Q16—31no—
105C22×D8C2×D311yescharacteristic
106C23×C8—31no—
107C2×C22⋊Q8—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×SD32C2×D312yescomplemented
111C22×C16—32no—
112#467—31no—
113#467—31no—
114#467—31no—
115#467—61no—
116#467—61no—
117#467C2×D311yescharacteristic
118#466—32no—
119#466—32no—
120#466—61no—
121#466—61no—
122C2×#107—31no—
123C2×#107—31no—
124C2×#103—61no—
125C2×#102—61no—
126#367—32no—
127#367—32no—
128#358—32no—
129#358—32no—
130#357—32no—
131#357C2×D312yescomplemented
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C24×C3—41no—
136C23×D3D811yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×D4—22no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C2×#67—42no—
144C2×#67—42no—
145C2×#67—84no—
146C22×Dic3—41no—
147C22×Dic3D811yescharacteristic
148D3×Q8—42no—
149D3×Q8—42no—
150C2×C4×D3—41no—
151C2×C4×D3—41no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3D811yescharacteristic
155C2×C4×D3D811yescharacteristic
156C2×C4×D3C2×D411yescharacteristic
157C2×C4×D3C2×D411yescharacteristic
158C2×Dic6—21no—
159C2×Dic6—41no—
160C3×Q16—22no—
161C3×D8—22no—
162C3×D8—22no—
163C3×D8—42no—
164C2×C3×C8—22no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2411yescharacteristic
167C3×C4⋊C4—21no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—41no—
170#217—41no—
171#217—42no—
172#217—42no—
173#216—42no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—41no—
176C4⋊Dic3—21no—
177C4⋊Dic3—22no—
178C4⋊Dic3—41no—
179C4⋊Dic3—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182Dic3⋊C4—41no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8C2×D411yescharacteristic
185C2×C3⋊C8C2×D411yescharacteristic
186Dic12—22no—
187D3×C8—22no—
188D3×C8—22no—
189D3×C8—41no—
190D3×C8—41no—
191C3×C16—22no—
192C3⋊C16—22no—
193C22×Q8—61no—
194C22×D4—21no—
195C22×D4—121no—
196C23×C4—31no—
197C2×Q16—31no—
198C2×Q16—61no—
199C2×Q16—122no—
200C2×Q16C22×D311yescharacteristic
201C2×D8—21no—
202C2×D8—42no—
203C2×D8C22×D312yes—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8—122no—
210C22×C8C22×D311yescharacteristic
211C22⋊Q8—21no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—121no—
216C22⋊Q8—121no—
217#121—62no—
218#121—62no—
219#121—121no—
220#121—121no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225C2×C22⋊C4—121no—
226SD32—42no—
227SD32—122no—
228C2×C16—32no—
229C2×C16—122no—
230C2×C16C22×D312yes—
231#107—31no—
232#107—32no—
233#107—61no—
234#107—61no—
235#107C22×D311yescharacteristic
236#103—21no—
237#103—61no—
238#103—61no—
239#103—61no—
240#102—62no—
241#102—62no—
242C23×C3—42no—
243C23×C3—82no—
244C23×C3—82no—
245C23×C3—82no—
246C23×C3C2×D811yescharacteristic
247C22×D3—41no—
248C22×D3—41no—
249C22×D3SD3214yescomplemented
250C22×D3C2×D811yescharacteristic
251C22×D3C2×D811yescharacteristic
252C3×Q8—41no—
253C3×Q8—41no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—82no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4C2×D811yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263#67—84no—
264#67—84no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—82no—
270C2×Dic3C2×D811yescharacteristic
271C2×Dic3C2×D811yescharacteristic
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—21no—
275C4×D3—21no—
276C4×D3—41no—
277C4×D3—41no—
278C4×D3—82no—
279C4×D3—82no—
280Dic6—41no—
281Dic6—41no—
282Dic6—82no—
283C3×C8—21no—
284C3×C8H(5,𝔽2)12yes—
285C3⋊C8—21no—
286C3⋊C8—21no—
287C24—31no—
288C24—41no—
289C2×Q8—21no—
290C2×Q8—61no—
291C2×Q8—121no—
292C2×Q8—122no—
293C2×Q8—122no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—244no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4D3×D411yescharacteristic
312Q16—22no—
313Q16—62no—
314Q16—122no—
315D8—22no—
316D8—22no—
317D8—42no—
318C2×C8—22no—
319C2×C8—31no—
320C2×C8—31no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—121no—
324C2×C8—121no—
325C2×C8—121no—
326C2×C8D3×D411yescharacteristic
327C2×C8C23×D311yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—62no—
331C4⋊C4—61no—
332C4⋊C4—121no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C4⋊C4—121no—
337C22⋊C4—41no—
338C22⋊C4—122no—
339C22⋊C4—122no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C16—22no—
344C16—62no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—164no—
351C22×C3C2×SD3212yes—
352C22×C3C22×D811yescharacteristic
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—21no—
357C2×D3—44no—
358C2×D3C2×SD3212yes—
359C2×D3C2×SD3212yes—
360C3×C4—21no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4#53411yescharacteristic
364C3×C4#57511yescharacteristic
365Dic3—21no—
366Dic3—21no—
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—31no—
371C23—31no—
372C23—34no—
373C23—42no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—121no—
378C23—121no—
379C23D3×D811yescharacteristic
380Q8—41no—
381Q8—41no—
382Q8—121no—
383Q8—121no—
384Q8—242no—
385D4—42no—
386D4—42no—
387D4—82no—
388D4—244no—
389D4—244no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—31no—
393C2×C4—31no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—242no—
407C2×C4—242no—
408C2×C4—242no—
409C2×C4D3×D811yescharacteristic
410C2×C4C2×D3×D411yescharacteristic
411C8—21no—
412C8—61no—
413C8—61no—
414C8#100412yes—
415C2×C3—22no—
416C2×C3—162no—
417C2×C3#318611yescharacteristic
418C2×C3C22×SD3211yescharacteristic
419C2×C3#341411yescharacteristic
420D3—22no—
421D3—22no—
422C22—21no—
423C22—21no—
424C22—32no—
425C22—32no—
426C22—61no—
427C22—61no—
428C22—61no—
429C22—61no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—124no—
434C22—164no—
435C22D3×SD3212yes—
436C22C2×D3×D811yescharacteristic
437C4—21no—
438C4—61no—
439C4—61no—
440C4—81no—
441C4—81no—
442C4—242no—
443C4—241no—
444C4—241no—
445C4#558911yescharacteristic
446C4#577311yescharacteristic
447C3#3375311yescharacteristic, complemented, Sylow
448C2—22no—
449C2—62no—
450C2—62no—
451C2—162no—
452C2#8000611yescharacteristic
453C2C2×D3×SD3211yescharacteristic
454C2#8203511yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2187C2
socleC22×C3C22×D8
prime radicalC2×#5157C2
hypercenter#2187D3
hypocenterC3#33753
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
2. D3×#467C2semidirect
3. C3×#2187C2semidirect
4. #72036C2semidirect
5. #71487C2semidirect
6. #71483C2semidirect
7. #71078C2semidirect
8. #69200C2semidirect
9. #69196C2semidirect
10. D3×C22⋊C16C2semidirect
11. #69098C2semidirect
15. C2×D3×Q16C22semidirect
18. C2×C3×SD32C22semidirect
21. #5188C22semidirect
26. C2×#827C22semidirect
30. C2×D3×C16C22semidirect
31. #4887C22semidirect
32. D3×#107C22semidirect
35. C3×#357C22semidirect
37. #4564C22semidirect
38. #4562C22semidirect
40. (C2×D3)⋊C16C22semidirect
41. #4505C22semidirect
42. #4503C22semidirect
43. #4490C22semidirect
52. #2187D3semidirect
110. C2×SD32C2×D3semidirect
131. #357C2×D3semidirect
249. C22×D3SD32semidirect
447. C3#33753semidirect

Maximal subgroups

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

subgroupindex
C2×#51572
D3×#4672
C3×#21872
#720362
#714872
#714832
#710782
#692002
#691962
D3×C22⋊C162
#690982
#337533