Finite Group Databasebeta

← #10691455

Subgroups — #10691455

Order 768.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83298C211yesmaximal, characteristic
2C42×D12C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9233—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×D12—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C42×D3—21no—
13C3×#423C2211yescharacteristic
14C2×C3×#104—21no—
15C2×C3×#104—21no—
16C3×C43C2211yescharacteristic
17#5087C412yescomplemented
18#5072C412yescomplemented
19#5036—21no—
20#5033—21no—
21C4×#213—22no—
22C4×(C2×D3)⋊C4—21no—
23C4×(C2×D3)⋊C4—22no—
24C4×C4⋊Dic3—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4537—22no—
29#4483—22no—
30#3100—31no—
31C42×D4—31no—
32#1962D311yescharacteristic, complemented
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—22no—
40C22×C4×D3—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#127C2×C411yescharacteristic, complemented
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42D411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×C3×C42C2311yescharacteristic
54C2×#213—22no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4×Dic3—22no—
60C2×C4×Dic3—21no—
61C4×D12—28no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C42×D3—42no—
67C42×D3—42no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#104—42no—
71C3×#104—42no—
72#832—22no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#809—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yescomplemented
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#532C2×D311yescharacteristic
84C2×C4×D4—31no—
85C2×C4×D4—61no—
86C22×C42—61no—
87#480—61no—
88#474—32no—
89#459—32no—
90#447—61no—
91C4×D8—62no—
92#431—31no—
93#431—31no—
94#423C2×D311yescharacteristic
95C2×#104—21no—
96C2×#104—21no—
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99C4×C22⋊C4—62no—
100C4×C22⋊C4—61no—
101C43C2×D311yescharacteristic
102#327—62no—
103D4⋊C8—62no—
104C23×D3—21no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C2×C3×Q8—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—22no—
122C2×D12—21no—
123C2×D12—24no—
124C2×D12C22⋊C412yes—
125C2×C4×D3—28no—
126C2×C4×D3—22no—
127C2×C4×D3—22no—
128C2×C4×D3—44no—
129C2×C4×D3—44no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—24no—
142C3×C42—21no—
143C3×C42—21no—
144C3×C42—21no—
145C3×C42—42no—
146C3×C42D812yescomplemented
147C3×C42C22×C411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150C3×C42#4111yescharacteristic
151#213—44no—
152(C2×D3)⋊C4—28no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155C4⋊Dic3—24no—
156C4⋊Dic3—21no—
157C4⋊Dic3C22⋊C412yes—
158C4×Dic3—44no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C22×D4—31no—
165C23×C4—62no—
166C23×C4—61no—
167C2×D8—62no—
168C2×M16—21no—
169C4⋊Q8C4×D311yescharacteristic, complemented
170#127C4×D311yescharacteristic, complemented
171#124—21no—
172#121—21no—
173#121—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—32no—
177C4×D4—32no—
178C4×D4—68no—
179C4×D4—61no—
180C4×D4—61no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—62no—
188C2×C42—61no—
189C2×C42—122no—
190C2×C42—122no—
191C2×C42#6711yescharacteristic
192C2×C42#6711yescharacteristic
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199#104—42no—
200#104—42no—
201#102—62no—
202#102—62no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C4×C8—61no—
206C4×C8—61no—
207#95—32no—
208#95—62no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C22×D3—24no—
212C22×D3—22no—
213C22×D3—42no—
214C22×D3—41no—
215C22×D3—41no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
245C2×Dic3—24no—
246C2×Dic3—22no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251D12—24no—
252D12—22no—
253D12#10414yescomplemented
254C4×D3—416no—
255C4×D3—42no—
256C4×D3—42no—
257C4×D3—42no—
258C4×D3—42no—
259C3×C8—81no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—61no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—32no—
272C2×D4—41no—
273C2×D4—64no—
274C2×D4—61no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—68no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—124no—
284C22×C4—124no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292D8—124no—
293M16—41no—
294M16—41no—
295C2×C8—41no—
296C2×C8—62no—
297C2×C8—121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—32no—
301C4⋊C4—64no—
302C4⋊C4—61no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—68no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C42—24no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—42no—
313C42—124no—
314C42—121no—
315C42—121no—
316C42#21312yescomplemented
317C42C2×C4×D311yescharacteristic
318C42#23511yescharacteristic
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3#38611yescharacteristic
327C22×C3#44711yescharacteristic
328C22×C3#44911yescharacteristic
329C2×D3—24no—
330C2×D3—44no—
331C2×D3—44no—
332C2×D3—42no—
333C2×D3—42no—
334C3×C4—24no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—81no—
340C3×C4C2×#10412yes—
341C3×C4C4×D811yescharacteristic
342C3×C4#44211yescharacteristic
343C3×C4#45911yescharacteristic
344C3×C4#46311yescharacteristic
345Dic3—44no—
346Dic3—42no—
347C23—41no—
348C23—64no—
349C23—62no—
350C23—122no—
351C23—121no—
352C23—121no—
353C23#93711yescharacteristic
354Q8—41no—
355Q8—41no—
356D4—34no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—64no—
361D4—62no—
362D4—81no—
363D4—81no—
364D4—81no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—64no—
375C2×C4—62no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—1216no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4D3×C22⋊C411yescharacteristic
388C2×C4#67⋊C411yescharacteristic
389C2×C4#88311yescharacteristic
390C2×C4#88411yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4C2×#21311yescharacteristic
393C2×C4#93211yescharacteristic
394C2×C4#93811yescharacteristic
395C2×C4#94911yescharacteristic
396C2×C4#95011yescharacteristic
397C8—81no—
398C8—122no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—161no—
403C2×C3#187911yescharacteristic
404C2×C3#189611yescharacteristic
405C2×C3#190611yescharacteristic
406D3—44no—
407C22—21no—
408C22—21no—
409C22—64no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—124no—
414C22—124no—
415C22—122no—
416C22—122no—
417C22#495611yescharacteristic
418C22#503611yescharacteristic
419C22#503711yescharacteristic
420C4—24no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—124no—
427C4—122no—
428C4D3×#10412yes—
429C4C4×#21311yescharacteristic
430C4#501411yescharacteristic
431C4#505711yescharacteristic
432C4#507211yescharacteristic
433C3#923311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—124no—
437C2—161no—
438C2#6965711yescharacteristic
439C2#7038211yescharacteristic
440C2#7039111yescharacteristic
subgroupG/H
centerC2×C4#938
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#386
prime radical#83298C2
cosocleC2×C3×C42C23
hypercenter#1962D3
hypocenterC3#9233
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. C42×D12C2semidirect
3. C3×#1962C2semidirect
4. #71811C2semidirect
5. #70496C2semidirect
6. #67715C2semidirect
7. #67611C2semidirect
17. #5087C4semidirect
18. #5072C4semidirect
32. #1962D3semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×#127C2×C4semidirect
80. C4⋊(C3⋊C8)C2×C4semidirect
146. C3×C42D8semidirect
169. C4⋊Q8C4×D3semidirect
170. #127C4×D3semidirect
253. D12#104semidirect
316. C42#213semidirect
433. C3#9233semidirect

Maximal subgroups

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

subgroupindex
#832982
C42×D122
C3×#19622
#718112
#704962
#677152
#676112
#92333