Finite Group Databasebeta

← #10950946

Subgroups — #10950946

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#467C211yesmaximal, characteristic, complemented
2#83742C211yesmaximal, characteristic
3#72064C211yesmaximal, characteristic, complemented
4#72063C211yesmaximal, characteristic, complemented
5C2×#4549C211yesmaximal, characteristic, complemented
6#71086C212yesmaximal, complemented
7#71082C212yesmaximal, complemented
8#71081C212yesmaximal, complemented
9#71080C212yesmaximal, complemented
10#70602C211yesmaximal, characteristic, complemented
11#70599C211yesmaximal, characteristic, complemented
12#33756—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16#5791—21no—
17C2×#904C2211yescharacteristic
18C3×#467—21no—
19C3×#467—21no—
20C3×#467C2212yes—
21C3×#467C2212yes—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103—21no—
24C2×#829C2212yescomplemented
25C2×#828C2212yescomplemented
26#5116C2211yescharacteristic
27#5116C2211yescharacteristic
28#5115C2211yescharacteristic
29#5115C2211yescharacteristic
30#5114—21no—
31#5114—21no—
32#5113—21no—
33#5106C2211yescharacteristic
34C22×C3⋊C16C2212yes—
35#4564C2212yescomplemented
36#4564C2212yescomplemented
37#4549C2211yescharacteristic
38#4549C2211yescharacteristic
39#4549C2212yes—
40#4492C2212yescomplemented
41#4491C2212yescomplemented
42#4490C2212yescomplemented
43#4489C2212yescomplemented
44#3050—31no—
45C2×#467D311yescharacteristic, complemented
46#2237—32no—
47#2235—31no—
48#2232—31no—
49#2217—32no—
50#2216—32no—
51#2189—32no—
52#2188—31no—
53#2185—31no—
54C2×C22⋊C16—31no—
55C22×C3×Q8—21no—
56C23×C3×C4D411yescharacteristic
57C2×#235—21no—
58C2×C3×Q16—21no—
59C2×C3×Q16—42no—
60C2×C3×Q16C2312yes—
61C22×C3×C8—42no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8D411yescharacteristic
64C22×C3×C8D411yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—41no—
71C3×C22⋊Q8—41no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—41no—
74C2×C3×C22⋊C4—41no—
75#953—41no—
76#930—21no—
77#930—21no—
78#929—41no—
79C4×#67—41no—
80C4×#67—41no—
81#926—21no—
82#924—21no—
83#924—21no—
84C2×Dic12C2311yescharacteristic
85#904—42no—
86C2×D24C2311yescharacteristic
87C2×#204—21no—
88C3×#107—21no—
89C3×#107C2312yes—
90C3×#103—22no—
91C3×#103—22no—
92#829—42no—
93#828—42no—
94#821—21no—
95#821—21no—
96#818C2311yescharacteristic
97#818C2311yescharacteristic
98#817—21no—
99#816—21no—
100#816—21no—
101C2×C3⋊C16—42no—
102C2×C3⋊C16C2312yes—
103C2×C3⋊C16C2312yes—
104C2×#135—31no—
105C22×Q16C2×D311yescharacteristic
106C23×C8C2×D311yescharacteristic
107#525—61no—
108C2×C22⋊Q8—21no—
109C2×Q32—32no—
110C2×SD32—32no—
111C22×C16—32no—
112#467—21no—
113#467—21no—
114#467—31no—
115#467—31no—
116#467C2×D312yes—
117#467C2×D312yes—
118#466—31no—
119#466—31no—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107C2×D311yescharacteristic
124#416—61no—
125C2×#103—21no—
126#367—32no—
127#366—32no—
128#358—32no—
129#358—32no—
130#358—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—31no—
134C22⋊C16—32no—
135C24×C3D811yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C2×C3×Q8—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4D811yescharacteristic
145C22×C3×C4D811yescharacteristic
146C22×C3×C4D811yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153#235—42no—
154#235—42no—
155C2×D12—21no—
156C2×C4×D3—41no—
157C2×Dic6—21no—
158C3×Q16—22no—
159C3×Q16—22no—
160C3×Q16—42no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—41no—
165C2×C3×C8—41no—
166C2×C3×C8—41no—
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2411yescharacteristic
171C3×C4⋊C4—22no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—42no—
177#217—41no—
178#217—41no—
179#217—41no—
180(C2×D3)⋊C4—41no—
181(C2×D3)⋊C4—41no—
182C4⋊Dic3—21no—
183C4⋊Dic3—21no—
184Dic3⋊C4—41no—
185Dic3⋊C4—41no—
186C4×Dic3—41no—
187Dic12—22no—
188D24—22no—
189#204—42no—
190C3⋊C16—22no—
191C3⋊C16—22no—
192C2×#41—61no—
193C22×Q8—21no—
194C23×C4#6711yescharacteristic
195#135—122no—
196C2×Q16—21no—
197C2×Q16—31no—
198C2×Q16—42no—
199C2×Q16C22×D312yes—
200C2×SD16—61no—
201C2×D8—31no—
202C22×C8—42no—
203C22×C8#6711yescharacteristic
204C22×C8#6711yescharacteristic
205C22×C8#6711yescharacteristic
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C22×C8C22×D311yescharacteristic
209#123—121no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216#121—61no—
217#121—61no—
218#120—121no—
219C4×D4—121no—
220C4×D4—121no—
221#117—61no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—41no—
224C2×C22⋊C4—41no—
225Q32—122no—
226SD32—122no—
227C2×C16—32no—
228C2×C16—32no—
229C2×C16—122no—
230#107—21no—
231#107—31no—
232#107—31no—
233#107C22×D312yes—
234#106—61no—
235#103—22no—
236#103—22no—
237#103—61no—
238#103—61no—
239#102—61no—
240#102—61no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3C2×D811yescharacteristic
245C23×C3C2×D811yescharacteristic
246C23×C3C2×D811yescharacteristic
247C22×D3—41no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8—82no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—82no—
263C2×C3×C4—82no—
264C2×C3×C4C2×D811yescharacteristic
265C2×C3×C4C2×D811yescharacteristic
266C2×C3×C4C2×D811yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268#67—82no—
269#67—82no—
270#67—82no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274D12—41no—
275D12—41no—
276C4×D3—82no—
277Dic6—41no—
278Dic6—41no—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—21no—
282C3×C8H(5,𝔽2)12yes—
283C24#21311yescharacteristic
284#41—122no—
285#41—122no—
286C2×Q8—22no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—61no—
291C2×D4—61no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—121no—
300C22×C4#21311yescharacteristic
301C22×C4#21311yescharacteristic
302C22×C4#21311yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306Q16—22no—
307Q16—22no—
308Q16—42no—
309Q16—62no—
310SD16—122no—
311D8—62no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8C2×#6711yescharacteristic
319C2×C8C2×#6711yescharacteristic
320C2×C8C2×#6711yescharacteristic
321C2×C8C23×D311yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—121no—
338C16—62no—
339C16—62no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3#50812yes—
349C22×C3#51012yes—
350C22×C3C22×D811yescharacteristic
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—82no—
358C3×C4—82no—
359C3×C4#53411yescharacteristic
360C3×C4#57511yescharacteristic
361Dic3—81no—
362Dic3—81no—
363Dic3—81no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—121no—
368C23C2×#21311yescharacteristic
369C23C2×#21311yescharacteristic
370C23C2×#21311yescharacteristic
371Q8—42no—
372Q8—42no—
373Q8—82no—
374Q8—121no—
375Q8—121no—
376D4—121no—
377D4—121no—
378D4—242no—
379D4—242no—
380D4—242no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—242no—
398C2×C4C2×#21311yescharacteristic
399C2×C4C2×#21311yescharacteristic
400C2×C4C2×#21311yescharacteristic
401C2×C4C22×#6711yescharacteristic
402C8—21no—
403C8—21no—
404C8—21no—
405C8#100412yes—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3#318611yescharacteristic
410C2×C3C2×#50811yescharacteristic
411C2×C3C2×#51011yescharacteristic
412D3—161no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22#517912yes—
425C22#519412yes—
426C22C22×#21311yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—82no—
432C4—241no—
433C4—241no—
434C4—241no—
435C4#580411yescharacteristic
436C4#582211yescharacteristic
437C3#3375611yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—481no—
442C2#8262511yescharacteristic
443C2C2×#517911yescharacteristic
444C2C2×#519411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#467C2
socleC22×C3C22×D8
prime radical#83742C2
hypercenterC2×#467D3
hypocenterC3#33756
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×C3×#467C2semidirect
3. #72064C2semidirect
4. #72063C2semidirect
5. C2×#4549C2semidirect
6. #71086C2semidirect
7. #71082C2semidirect
8. #71081C2semidirect
9. #71080C2semidirect
10. #70602C2semidirect
11. #70599C2semidirect
24. C2×#829C22semidirect
25. C2×#828C22semidirect
35. #4564C22semidirect
36. #4564C22semidirect
40. #4492C22semidirect
41. #4491C22semidirect
42. #4490C22semidirect
43. #4489C22semidirect
45. C2×#467D3semidirect
437. C3#33756semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4672
#837422
#720642
#720632
C2×#45492
#710862
#710822
#710812
#710802
#706022
#705992
#337563