Finite Group Databasebeta

← #10950945

Subgroups — #10950945

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
1#80714C211yesmaximal, characteristic
2C2×#4887C211yesmaximal, characteristic, complemented
3C3×#2232C211yesmaximal, characteristic, complemented
4#71077C211yesmaximal, characteristic, complemented
5#70602C211yesmaximal, characteristic, complemented
6#69332C212yesmaximal, complemented
7#69328C212yesmaximal, complemented
8#69196C212yesmaximal, complemented
9#69194C212yesmaximal, complemented
10#69113C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33756—31nomaximal, Sylow
13C2×#912C2211yescharacteristic
14C22×Dic12C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16#5606—21no—
17C2×#897—21no—
18C3×#466C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20#5157C2211yescharacteristic
21C2×#827C2212yescomplemented
22#5116C2211yescharacteristic
23#5041—21no—
24C2×#810—21no—
25C2×Dic24C2212yescomplemented
26C2×D3×C16C2212yes—
27#4887—21no—
28#4887—21no—
29#4887C2212yes—
30#4887C2212yes—
31#4887C2211yescharacteristic
32#4880C2211yescharacteristic
33#4778—21no—
34#4772—21no—
35C3×#366C2212yescomplemented
36C3×#357C2212yescomplemented
37C3×C22⋊C16C2211yescharacteristic
38#4549C2211yescharacteristic
39(C2×D3)⋊C16C2212yes—
40#4503C2212yescomplemented
41#4492C2212yescomplemented
42#4492C2212yescomplemented
43#4490C2212yescomplemented
44#3050—31no—
45C2×#467—31no—
46#2237—32no—
47#2235—31no—
48#2232D311yescharacteristic, complemented
49#2217—32no—
50#2216—32no—
51#2189—32no—
52#2188—31no—
53#2185—31no—
54C2×C22⋊C16—31no—
55C2×#237—21no—
56C22×C4×D3D411yescharacteristic
57C22×Dic6—21no—
58C2×C3×D8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62#938—21no—
63#937—41no—
64D4×Dic3—41no—
65C2×#214—21no—
66C2×(C2×D3)⋊C4—41no—
67C2×C4⋊Dic3—41no—
68#924—21no—
69C22×C3⋊C8D411yescharacteristic
70#912—42no—
71C2×Dic12—21no—
72C2×Dic12—42no—
73C2×Dic12C2312yes—
74C2×Dic12C2311yescharacteristic
75C2×D3×C8—42no—
76C2×D3×C8D411yescharacteristic
77C2×D3×C8D411yescharacteristic
78C2×D3×C8C2311yescharacteristic
79C2×D3×C8C2311yescharacteristic
80#897—22no—
81#897—22no—
82#897—21no—
83#897—41no—
84#897—41no—
85#892—21no—
86#883—41no—
87#67⋊C4—41no—
88C2×C3×C16C2312yes—
89C3×#107—21no—
90C3×#107C2311yescharacteristic
91C3×#107C2312yes—
92C3×#102—21no—
93#832—21no—
94#827—42no—
95#818C2311yescharacteristic
96#816—21no—
97C2×C3⋊C16C2312yes—
98#810—22no—
99#810—22no—
100#810—21no—
101#808—21no—
102Dic24—42no—
103D3×C16—42no—
104C2×#135—31no—
105C22×Q16—31no—
106C23×C8—31no—
107#525—61no—
108C2×C22⋊Q8—61no—
109C2×Q32—32no—
110C2×SD32—32no—
111C22×C16—32no—
112#467—31no—
113#467—32no—
114#467—31no—
115#467—32no—
116#467—61no—
117#467—61no—
118#466—31no—
119#466C2×D311yescharacteristic
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107C2×D311yescharacteristic
124#416—61no—
125C2×#103—61no—
126#367—32no—
127#366C2×D312yescomplemented
128#358—32no—
129#358—32no—
130#358—32no—
131#357C2×D312yescomplemented
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C23×D3D811yescharacteristic
136C2×C3×D4—21no—
137C2×C3×D4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C2×#67—41no—
141C2×#67—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3D811yescharacteristic
145#237—42no—
146#237—42no—
147C2×C4×D3—41no—
148C2×C4×D3—41no—
149C2×C4×D3D811yescharacteristic
150C2×C4×D3D811yescharacteristic
151C2×C4×D3C2×D411yescharacteristic
152C2×C4×D3C2×D411yescharacteristic
153C2×Dic6—22no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6—41no—
158C3×D8—22no—
159C2×C3×C8—22no—
160C2×C3×C8C2×D411yescharacteristic
161C2×C3×C8C2411yescharacteristic
162C3×C4⋊C4—21no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—41no—
165C3×C22⋊C4—41no—
166#217—41no—
167#217—41no—
168#214—42no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—42no—
173C4⋊Dic3—21no—
174C4⋊Dic3—41no—
175C4⋊Dic3—42no—
176C4⋊Dic3—42no—
177Dic3⋊C4—41no—
178C4×Dic3—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8C2×D411yescharacteristic
181C2×C3⋊C8C2×D411yescharacteristic
182Dic12—22no—
183Dic12—22no—
184Dic12—22no—
185Dic12—42no—
186D3×C8—22no—
187D3×C8—22no—
188D3×C8—41no—
189D3×C8—41no—
190C3×C16—22no—
191C3⋊C16—22no—
192C2×#41—61no—
193C22×Q8—61no—
194C23×C4—31no—
195#135—122no—
196C2×Q16—32no—
197C2×Q16—31no—
198C2×Q16—61no—
199C2×Q16—122no—
200C2×SD16—61no—
201C2×D8C22×D311yescharacteristic
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—62no—
211C22⋊Q8—62no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C22⋊Q8—121no—
215C22⋊Q8—121no—
216#121—21no—
217#121—61no—
218#120—121no—
219C4×D4—121no—
220C4×D4—121no—
221#117—61no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—121no—
224C2×C22⋊C4—121no—
225Q32—122no—
226SD32—122no—
227C2×C16—32no—
228C2×C16—122no—
229C2×C16C22×D312yes—
230#107—21no—
231#107—31no—
232#107C22×D312yes—
233#107C22×D311yescharacteristic
234#106—61no—
235#103—62no—
236#103—62no—
237#103—61no—
238#103—61no—
239#102—21no—
240#102—61no—
241C23×C3—41no—
242C23×C3C2×D811yescharacteristic
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—41no—
246C22×D3C2×D811yescharacteristic
247C22×D3C2×D811yescharacteristic
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—82no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—82no—
256C2×C3×C4C2×D811yescharacteristic
257C2×C3×C4C22×D411yescharacteristic
258#67—82no—
259#67—82no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—82no—
265C2×Dic3—82no—
266C2×Dic3C2×D811yescharacteristic
267C2×Dic3C2×D811yescharacteristic
268C4×D3—21no—
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—21no—
272C4×D3—41no—
273C4×D3—41no—
274Dic6—42no—
275Dic6—41no—
276Dic6—42no—
277Dic6—41no—
278Dic6—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—61no—
287C2×Q8—62no—
288C2×Q8—122no—
289C2×Q8—122no—
290C2×Q8—121no—
291C2×D4—21no—
292C2×D4—41no—
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—62no—
307Q16—62no—
308Q16—62no—
309Q16—122no—
310SD16—122no—
311D8—22no—
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—22no—
324C4⋊C4—41no—
325C4⋊C4—61no—
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—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#50812yes—
346C22×C3C22×D811yescharacteristic
347C2×D3—21no—
348C2×D3—21no—
349C2×D3—21no—
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—44no—
353C2×D3#51012yes—
354C3×C4—21no—
355C3×C4—82no—
356C3×C4—81no—
357C3×C4#53411yescharacteristic
358C3×C4#57511yescharacteristic
359Dic3—21no—
360Dic3—21no—
361Dic3—81no—
362Dic3—82no—
363Dic3—81no—
364C23—31no—
365C23—31no—
366C23—41no—
367C23—62no—
368C23—62no—
369C23—121no—
370C23D3×D811yescharacteristic
371Q8—121no—
372Q8—122no—
373Q8—122no—
374Q8—121no—
375Q8—242no—
376D4—41no—
377D4—41no—
378D4—82no—
379D4—242no—
380D4—242no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—82no—
392C2×C4—121no—
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#100912yes—
406C2×C3—22no—
407C2×C3—161no—
408C2×C3#318611yescharacteristic
409C2×C3C2×#50811yescharacteristic
410C2×C3C2×#51011yescharacteristic
411D3—22no—
412D3—41no—
413C22—21no—
414C22—21no—
415C22—32no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—124no—
425C22#491212yes—
426C22C2×D3×D811yescharacteristic
427C4—21no—
428C4—61no—
429C4—61no—
430C4—82no—
431C4—81no—
432C4—241no—
433C4—241no—
434C4—242no—
435C4#565311yescharacteristic
436C4#577711yescharacteristic
437C3#3375611yescharacteristic, complemented, Sylow
438C2—22no—
439C2—62no—
440C2—121no—
441C2—161no—
442C2#8129911yescharacteristic
443C2C2×#491211yescharacteristic
444C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2232C2
socleC22×C3C22×D8
prime radical#80714C2
hypercenter#2232D3
hypocenterC3#33756
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
2. C2×#4887C2semidirect
3. C3×#2232C2semidirect
4. #71077C2semidirect
5. #70602C2semidirect
6. #69332C2semidirect
7. #69328C2semidirect
8. #69196C2semidirect
9. #69194C2semidirect
10. #69113C2semidirect
11. D3×C22⋊C16C2semidirect
21. C2×#827C22semidirect
25. C2×Dic24C22semidirect
35. C3×#366C22semidirect
36. C3×#357C22semidirect
40. #4503C22semidirect
41. #4492C22semidirect
42. #4492C22semidirect
43. #4490C22semidirect
48. #2232D3semidirect
127. #366C2×D3semidirect
131. #357C2×D3semidirect
437. C3#33756semidirect

Maximal subgroups

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

subgroupindex
#807142
C2×#48872
C3×#22322
#710772
#706022
#693322
#693282
#691962
#691942
#691132
D3×C22⋊C162
#337563