Finite Group Databasebeta

← #10950949

Subgroups — #10950949

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×#5188C211yesmaximal, characteristic, complemented
2#80758C211yesmaximal, characteristic
3C3×#2188C211yesmaximal, characteristic, complemented
4#72063C211yesmaximal, characteristic, complemented
5#71504C212yesmaximal, complemented
6#71503C212yesmaximal, complemented
7#71078C211yesmaximal, characteristic, complemented
8#69228C212yesmaximal, complemented
9#69214C212yesmaximal, complemented
10D3×C22⋊C16C211yesmaximal, characteristic, complemented
11#69095C211yesmaximal, characteristic, complemented
12#33756—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×#946—21no—
15C2×#919C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17#5627—21no—
18C2×C3×Q32C2212yescomplemented
19C3×#467C2211yescharacteristic
20C2×#835—21no—
21#5188—21no—
22#5188—21no—
23#5188C2212yes—
24#5188C2211yescharacteristic
25#5188C2212yes—
26C2×#828C2212yescomplemented
27#5115C2211yescharacteristic
28C2×#818C2211yescharacteristic
29#5048—21no—
30C2×D3×C16C2212yes—
31#4883C2211yescharacteristic
32#4880C2211yescharacteristic
33#4807—21no—
34#4804—21no—
35C3×#358C2212yescomplemented
36C3×C22⋊C16C2211yescharacteristic
37#4564C2212yescomplemented
38#4564C2212yescomplemented
39#4549C2211yescharacteristic
40#4509C2212yescomplemented
41(C2×D3)⋊C16C2212yes—
42#4504C2212yescomplemented
43#4490C2212yescomplemented
44#3050—31no—
45C2×#467—31no—
46#2237—32no—
47#2235—31no—
48#2232—31no—
49#2217—32no—
50#2216—32no—
51#2189—32no—
52#2188D311yescharacteristic, complemented
53#2185—31no—
54C2×C22⋊C16—31no—
55C22×C3×Q8—21no—
56C2×#239—21no—
57C22×C4×D3D411yescharacteristic
58C2×C3×Q16—21no—
59C2×C3×Q16—42no—
60C2×C3×Q16C2311yescharacteristic
61C2×C3×Q16C2312yes—
62C22×C3×C8C2311yescharacteristic
63C3×C22⋊Q8—21no—
64#946—22no—
65#946—22no—
66#946—21no—
67#946—41no—
68#946—41no—
69C2×#215—21no—
70#930—21no—
71C2×(C2×D3)⋊C4—41no—
72C2×C4⋊Dic3—21no—
73C2×Dic3⋊C4—41no—
74C22×C3⋊C8D411yescharacteristic
75#919—42no—
76C2×D24C2311yescharacteristic
77C2×D3×C8—42no—
78C2×D3×C8D411yescharacteristic
79C2×D3×C8D411yescharacteristic
80C2×D3×C8C2311yescharacteristic
81C2×D3×C8C2311yescharacteristic
82#895—21no—
83#894—41no—
84#893—41no—
85#892—21no—
86#882—41no—
87#67⋊C4—41no—
88C3×Q32—42no—
89C2×C3×C16C2312yes—
90C3×#107C2311yescharacteristic
91C3×#103—21no—
92#835—22no—
93#835—22no—
94#835—21no—
95#828—42no—
96#821—21no—
97#818—21no—
98#818C2311yescharacteristic
99#818C2312yes—
100C2×C3⋊C16C2312yes—
101#809—21no—
102#808—21no—
103D3×C16—42no—
104C2×#135—31no—
105C22×Q16C2×D311yescharacteristic
106C23×C8—31no—
107#525—61no—
108C2×C22⋊Q8—61no—
109C2×Q32C2×D312yescomplemented
110C2×SD32—32no—
111C22×C16—32no—
112#467—32no—
113#467—31no—
114#467—32no—
115#467—61no—
116#467—61no—
117#467C2×D311yescharacteristic
118#466—31no—
119#466—31no—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107—31no—
124#416—61no—
125C2×#103—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#358—32no—
130#358C2×D312yescomplemented
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C23×D3D811yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×Q8—41no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C2×#67—41no—
144C22×Dic3—41no—
145C22×Dic3D811yescharacteristic
146#239—42no—
147#239—42no—
148C2×D12—21no—
149C2×D12—41no—
150C2×D12—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
158C3×Q16—22no—
159C3×Q16—22no—
160C3×Q16—22no—
161C3×Q16—42no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2411yescharacteristic
165C3×C4⋊C4—21no—
166C3×C4⋊C4—41no—
167C3×C22⋊C4—41no—
168#215—42no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—42no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175C4⋊Dic3—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3—41no—
178Dic3⋊C4—41no—
179Dic3⋊C4—42no—
180Dic3⋊C4—42no—
181C4×Dic3—41no—
182C2×C3⋊C8—41no—
183C2×C3⋊C8C2×D411yescharacteristic
184C2×C3⋊C8C2×D411yescharacteristic
185D24—22no—
186D3×C8—22no—
187D3×C8—22no—
188D3×C8—41no—
189D3×C8—41no—
190C3×C16—22no—
191C3⋊C16—22no—
192C2×#41—61no—
193C22×Q8—21no—
194C23×C4—31no—
195#135—122no—
196C2×Q16—21no—
197C2×Q16—42no—
198C2×Q16C22×D312yes—
199C2×Q16C22×D311yescharacteristic
200C2×SD16—61no—
201C2×D8—31no—
202C22×C8—31no—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—122no—
208C22×C8C22×D311yescharacteristic
209#123—121no—
210C22⋊Q8—21no—
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C22⋊Q8—61no—
214C22⋊Q8—121no—
215C22⋊Q8—121no—
216#121—61no—
217#121—61no—
218#120—121no—
219C4×D4—121no—
220C4×D4—121no—
221#117—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225Q32—42no—
226SD32—122no—
227C2×C16—32no—
228C2×C16—122no—
229C2×C16C22×D312yes—
230#107—32no—
231#107—31no—
232#107—61no—
233#107C22×D311yescharacteristic
234#106—61no—
235#103—21no—
236#103—62no—
237#103—62no—
238#103—61no—
239#102—61no—
240#102—61no—
241C23×C3C2×D811yescharacteristic
242C22×D3—22no—
243C22×D3—22no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3C2×D811yescharacteristic
247C22×D3C2×D811yescharacteristic
248C3×Q8—41no—
249C3×Q8—42no—
250C3×Q8—41no—
251C3×Q8—42no—
252C3×Q8—82no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—82no—
259C2×C3×C4C2×D811yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261#67—82no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—82no—
266C2×Dic3C2×D811yescharacteristic
267C2×Dic3C2×D811yescharacteristic
268D12—41no—
269D12—41no—
270D12—82no—
271D12—82no—
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—21no—
275C4×D3—21no—
276C4×D3—41no—
277C4×D3—41no—
278C4×D3—82no—
279C3×C8—21no—
280C3×C8H(5,𝔽2)12yes—
281C3⋊C8—21no—
282C3⋊C8—21no—
283C24—31no—
284#41—122no—
285#41—122no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×D4—61no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—41no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4D3×D411yescharacteristic
306Q16—22no—
307Q16—22no—
308Q16—22no—
309Q16—42no—
310SD16—122no—
311D8—62no—
312C2×C8—22no—
313C2×C8—31no—
314C2×C8—31no—
315C2×C8—62no—
316C2×C8—62no—
317C2×C8—121no—
318C2×C8—121no—
319C2×C8—121no—
320C2×C8D3×D411yescharacteristic
321C2×C8C23×D311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—62no—
326C4⋊C4—122no—
327C4⋊C4—122no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—41no—
331C22⋊C4—121no—
332C22⋊C4—122no—
333C22⋊C4—122no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—121no—
338C16—22no—
339C16—62no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3#50812yes—
343C22×C3C22×D811yescharacteristic
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—44no—
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3#51012yes—
354C3×C4—21no—
355C3×C4—81no—
356C3×C4—81no—
357C3×C4—82no—
358C3×C4#53411yescharacteristic
359C3×C4#57511yescharacteristic
360Dic3—21no—
361Dic3—21no—
362Dic3—82no—
363Dic3—81no—
364C23—31no—
365C23—31no—
366C23—62no—
367C23—62no—
368C23—121no—
369C23—121no—
370C23D3×D811yescharacteristic
371Q8—42no—
372Q8—41no—
373Q8—41no—
374Q8—42no—
375Q8—82no—
376D4—121no—
377D4—121no—
378D4—242no—
379D4—242no—
380D4—242no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—82no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—242no—
399C2×C4—242no—
400C2×C4D3×D811yescharacteristic
401C2×C4C2×D3×D411yescharacteristic
402C8—21no—
403C8—61no—
404C8—61no—
405C8#100412yes—
406C2×C3—22no—
407C2×C3#318611yescharacteristic
408C2×C3C2×#50811yescharacteristic
409C2×C3C2×#51011yescharacteristic
410D3—22no—
411D3—41no—
412D3—161no—
413C22—21no—
414C22—21no—
415C22—32no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—124no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22#491912yes—
426C22C2×D3×D811yescharacteristic
427C4—21no—
428C4—61no—
429C4—61no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4—242no—
434C4—241no—
435C4#558911yescharacteristic
436C4#577311yescharacteristic
437C3#3375611yescharacteristic, complemented, Sylow
438C2—22no—
439C2—62no—
440C2—121no—
441C2—481no—
442C2#8000611yescharacteristic
443C2C2×#491911yescharacteristic
444C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2188C2
socleC22×C3C22×D8
prime radical#80758C2
hypercenter#2188D3
hypocenterC3#33756
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
1. C2×#5188C2semidirect
3. C3×#2188C2semidirect
4. #72063C2semidirect
5. #71504C2semidirect
6. #71503C2semidirect
7. #71078C2semidirect
8. #69228C2semidirect
9. #69214C2semidirect
10. D3×C22⋊C16C2semidirect
11. #69095C2semidirect
18. C2×C3×Q32C22semidirect
26. C2×#828C22semidirect
35. C3×#358C22semidirect
37. #4564C22semidirect
38. #4564C22semidirect
40. #4509C22semidirect
42. #4504C22semidirect
43. #4490C22semidirect
52. #2188D3semidirect
109. C2×Q32C2×D3semidirect
130. #358C2×D3semidirect
437. C3#33756semidirect

Maximal subgroups

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

subgroupindex
C2×#51882
#807582
C3×#21882
#720632
#715042
#715032
#710782
#692282
#692142
D3×C22⋊C162
#690952
#337563