Finite Group Databasebeta

← #10950921

Subgroups — #10950921

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
1D3×#467C211yesmaximal, characteristic, complemented
2C2×#4883C211yesmaximal, characteristic
3C3×#2235C211yesmaximal, characteristic, complemented
4#71085C211yesmaximal, characteristic, complemented
5#70602C211yesmaximal, characteristic, complemented
6#69350C212yesmaximal, complemented
7#69347C212yesmaximal, complemented
8#69228C212yesmaximal, complemented
9#69223C212yesmaximal, complemented
10#69110C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33753—31nomaximal, Sylow
13C2×D3×Q16C2211yescharacteristic, complemented
14C22×D24C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16D3×C22⋊Q8—21no—
17C2×#895—21no—
18C3×#467C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20#5188C2211yescharacteristic, complemented
21C2×#828C2212yescomplemented
22#5116C2211yescharacteristic
23#5051—21no—
24C2×#809—21no—
25C2×#800C2212yescomplemented
26C2×D3×C16C2212yescomplemented
27#4887C2211yescharacteristic, complemented
28#4883—21no—
29#4883—21no—
30#4883C2212yes—
31#4883C2212yes—
32D3×#107C2211yescharacteristic, complemented
33#4813—21no—
34D3×#103—21no—
35C3×#367C2212yescomplemented
36C3×#358C2212yescomplemented
37C3×C22⋊C16C2211yescharacteristic
38#4549C2211yescharacteristic
39#4509C2212yescomplemented
40(C2×D3)⋊C16C2212yescomplemented
41#4492C2212yescomplemented
42#4491C2212yescomplemented
43#4490C2212yescomplemented
44C2×#467—31no—
45C2×#466—31no—
46#2235—31no—
47#2235D311yescharacteristic, complemented
48#2233—32no—
49#2217—32no—
50#2217—32no—
51#2187—31no—
52#2187—31no—
53#2185—32no—
54C2×C22⋊C16—31no—
55C2×D3×Q8—21no—
56C22×D12—21no—
57C22×D12—41no—
58C22×C4×D3D411yescharacteristic
59C2×C3×Q16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C22⋊Q8—21no—
62C2×C3×C4⋊C4—21no—
63#946—21no—
64C2×#216—21no—
65C2×(C2×D3)⋊C4—41no—
66#924—21no—
67C22×C3⋊C8D411yescharacteristic
68D3×Q16—42no—
69C2×Dic12C2311yescharacteristic
70C2×D24—21no—
71C2×D24—42no—
72C2×D24C2312yes—
73C2×D3×C8—42no—
74C2×D3×C8D411yescharacteristic
75C2×D3×C8D411yescharacteristic
76C2×D3×C8C2311yescharacteristic
77C2×D3×C8C2311yescharacteristic
78#897—21no—
79#896—41no—
80#895—22no—
81#895—22no—
82#895—41no—
83#895—41no—
84D3×C4⋊C4—21no—
85D3×C4⋊C4—41no—
86D3×C22⋊C4—41no—
87#878—41no—
88C2×C3×C16C2312yes—
89C3×#107—21no—
90C3×#107C2311yescharacteristic
91C3×#107C2312yes—
92C3×#103—21no—
93#835—21no—
94#828—42no—
95#818C2311yescharacteristic
96#816—21no—
97C2×C3⋊C16C2312yes—
98#810—21no—
99#809—22no—
100#809—22no—
101#807—21no—
102#800—42no—
103D3×C16—42no—
104C22×Q16—31no—
105C22×D8—31no—
106C23×C8—31no—
107C2×C22⋊Q8—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×SD32—32no—
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×#107C2×D311yescharacteristic
124C2×#103—61no—
125C2×#102—61no—
126#367—32no—
127#367C2×D312yescomplemented
128#358—32no—
129#358C2×D312yescomplemented
130#357—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C23×D3—41no—
136C23×D3D811yescharacteristic
137C2×C3×Q8—21no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×Dic3D811yescharacteristic
141D3×Q8—42no—
142D3×Q8—42no—
143C2×D12—22no—
144C2×D12—42no—
145C2×D12—42no—
146C2×D12—42no—
147C2×D12—42no—
148C2×D12—41no—
149C2×D12—84no—
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—
161C2×C3×C8—22no—
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2411yescharacteristic
164C3×C4⋊C4—21no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—41no—
167C3×C4⋊C4—41no—
168C3×C22⋊C4—41no—
169#217—41no—
170#216—42no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—42no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—21no—
176C4⋊Dic3—41no—
177Dic3⋊C4—41no—
178Dic3⋊C4—41no—
179Dic3⋊C4—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8C2×D411yescharacteristic
182C2×C3⋊C8C2×D411yescharacteristic
183Dic12—22no—
184D24—22no—
185D24—22no—
186D24—42no—
187D3×C8—22no—
188D3×C8—22no—
189D3×C8—41no—
190D3×C8—41no—
191C3×C16—22no—
192C3⋊C16—22no—
193C22×Q8—61no—
194C22×D4—61no—
195C22×D4—121no—
196C23×C4—31no—
197C2×Q16—31no—
198C2×Q16—61no—
199C2×Q16—122no—
200C2×Q16C22×D311yescharacteristic
201C2×D8—32no—
202C2×D8—61no—
203C2×D8—122no—
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—21no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225C2×C22⋊C4—121no—
226SD32—122no—
227SD32—122no—
228C2×C16—32no—
229C2×C16—122no—
230C2×C16C22×D312yes—
231#107—21no—
232#107—31no—
233#107—61no—
234#107C22×D311yescharacteristic
235#107C22×D312yes—
236#103—21no—
237#103—61no—
238#103—61no—
239#103—61no—
240#102—62no—
241#102—62no—
242C23×C3C2×D811yescharacteristic
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—42no—
246C22×D3—82no—
247C22×D3—82no—
248C22×D3—82no—
249C22×D3SD3214yescomplemented
250C22×D3C2×D811yescharacteristic
251C22×D3C2×D811yescharacteristic
252C3×Q8—41no—
253C3×Q8—41no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—82no—
260C2×C3×C4C2×D811yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3C2×D811yescharacteristic
266C2×Dic3C2×D811yescharacteristic
267D12—42no—
268D12—42no—
269D12—82no—
270D12—84no—
271D12—84no—
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—121no—
289C2×Q8—21no—
290C2×Q8—61no—
291C2×Q8—121no—
292C2×Q8—122no—
293C2×Q8—122no—
294C2×D4—62no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—121no—
300C2×D4—244no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—41no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4D3×D411yescharacteristic
312Q16—22no—
313Q16—62no—
314Q16—122no—
315D8—62no—
316D8—62no—
317D8—122no—
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—22no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—61no—
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×C3C2×SD3212yes—
348C22×C3C22×D811yescharacteristic
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—44no—
354C2×D3—82no—
355C2×D3—82no—
356C2×D3—82no—
357C2×D3—164no—
358C2×D3C2×SD3212yes—
359C2×D3C2×SD3212yes—
360C3×C4—21no—
361C3×C4—82no—
362C3×C4—81no—
363C3×C4—81no—
364C3×C4#53411yescharacteristic
365C3×C4#57511yescharacteristic
366Dic3—21no—
367Dic3—21no—
368Dic3—81no—
369Dic3—81no—
370C23—31no—
371C23—31no—
372C23—34no—
373C23—121no—
374C23—121no—
375C23—122no—
376C23—242no—
377C23—242no—
378C23—242no—
379C23D3×D811yescharacteristic
380Q8—41no—
381Q8—41no—
382Q8—121no—
383Q8—121no—
384Q8—242no—
385D4—122no—
386D4—122no—
387D4—242no—
388D4—244no—
389D4—244no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—31no—
393C2×C4—31no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—82no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—242no—
408C2×C4—242no—
409C2×C4D3×D811yescharacteristic
410C2×C4C2×D3×D411yescharacteristic
411C8—21no—
412C8—61no—
413C8—61no—
414C8#100912yes—
415C2×C3—22no—
416C2×C3#318611yescharacteristic
417C2×C3C22×SD3211yescharacteristic
418C2×C3#341411yescharacteristic
419D3—22no—
420D3—22no—
421D3—162no—
422C22—21no—
423C22—21no—
424C22—32no—
425C22—32no—
426C22—61no—
427C22—61no—
428C22—61no—
429C22—61no—
430C22—124no—
431C22—242no—
432C22—242no—
433C22—242no—
434C22—484no—
435C22D3×SD3212yes—
436C22C2×D3×D811yescharacteristic
437C4—21no—
438C4—61no—
439C4—61no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—241no—
444C4—241no—
445C4#565311yescharacteristic
446C4#577711yescharacteristic
447C3#3375311yescharacteristic, complemented, Sylow
448C2—22no—
449C2—62no—
450C2—62no—
451C2—482no—
452C2#8129911yescharacteristic
453C2C2×D3×SD3211yescharacteristic
454C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2235C2
socleC22×C3C22×D8
prime radicalC2×#4883C2
hypercenter#2235D3
hypocenterC3#33753
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
1. D3×#467C2semidirect
3. C3×#2235C2semidirect
4. #71085C2semidirect
5. #70602C2semidirect
6. #69350C2semidirect
7. #69347C2semidirect
8. #69228C2semidirect
9. #69223C2semidirect
10. #69110C2semidirect
11. D3×C22⋊C16C2semidirect
13. C2×D3×Q16C22semidirect
20. #5188C22semidirect
21. C2×#828C22semidirect
25. C2×#800C22semidirect
26. C2×D3×C16C22semidirect
27. #4887C22semidirect
32. D3×#107C22semidirect
35. C3×#367C22semidirect
36. C3×#358C22semidirect
39. #4509C22semidirect
40. (C2×D3)⋊C16C22semidirect
41. #4492C22semidirect
42. #4491C22semidirect
43. #4490C22semidirect
47. #2235D3semidirect
127. #367C2×D3semidirect
129. #358C2×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
D3×#4672
C2×#48832
C3×#22352
#710852
#706022
#693502
#693472
#692282
#692232
#691102
D3×C22⋊C162
#337533