Finite Group Databasebeta

← #10691452

Subgroups — #10691452

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
1C3×C42×D4C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3#71598C211yesmaximal, characteristic, complemented
4#69469C211yesmaximal, characteristic, complemented
5#69395C211yesmaximal, characteristic, complemented
6#67620C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9233—31nomaximal, Sylow
9C2×C3×C4×D4—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C22×C3×C42—21no—
12#5477C2211yescharacteristic
13C3×C4×C4⋊C4—21no—
14C3×C4×C4⋊C4C2211yescharacteristic
15C3×C4×C22⋊C4—21no—
16C3×C4×C22⋊C4—22no—
17C3×C43C2211yescharacteristic
18#5217—21no—
19#5015C412yescomplemented
20C4×#213—22no—
21#5007C412yescomplemented
22#4967—21no—
23C2×#805—21no—
24C2×#805—21no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28D4⋊(C3⋊C8)—22no—
29#4451—22no—
30#3100—31no—
31C42×D4D311yescharacteristic, complemented
32#1962—31no—
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C2×#235—21no—
41C3×C4×D4—28no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4C2×C412yes—
45C3×C4×D4D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42—22no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42—42no—
55C2×C3×C42—42no—
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×#213—22no—
60#930—21no—
61#930—21no—
62C2×#208—21no—
63#875—21no—
64#874C2×C411yescharacteristic, complemented
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic, complemented
68C3×#95—22no—
69C3×#95D412yes—
70#832—22no—
71#830—21no—
72#830—21no—
73#809—22no—
74#807—22no—
75#805—42no—
76#805—42no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yescomplemented
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4×D4—21no—
85C2×C4×D4C2×D311yescharacteristic
86C22×C42—21no—
87#480—61no—
88#474—32no—
89#459—32no—
90#447—61no—
91C4×D8—62no—
92#431—31no—
93#431—31no—
94#423—31no—
95C2×#104—61no—
96C2×#104—61no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×D311yescharacteristic
99C4×C22⋊C4—21no—
100C4×C22⋊C4—22no—
101C43C2×D311yescharacteristic
102#327—62no—
103D4⋊C8—62no—
104C24×C3—21no—
105C2×C3×D4—24no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C22×C3×C4—21no—
109C22×C3×C4—28no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—44no—
116C22×C3×C4—44no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C2×#67—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—28no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C42—24no—
141C3×C42—21no—
142C3×C42—21no—
143C3×C42—21no—
144C3×C42—42no—
145C3×C42—44no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42D812yescomplemented
149C3×C42C22×C411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152C3×C42#4111yescharacteristic
153#213—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×#41—61no—
164C22×D4#6711yescharacteristic
165C23×C4—22no—
166C23×C4—21no—
167C2×D8—62no—
168C2×M16—61no—
169C4⋊Q8—31no—
170#127—31no—
171#124—61no—
172#121—61no—
173#121—61no—
174C4×D4—28no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4C4×D312yes—
180C4×D4D1212yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4#6711yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42#6711yescharacteristic
192C2×C42#6711yescharacteristic
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199#104—122no—
200#104—122no—
201#102—62no—
202#102—62no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C4×C8—61no—
206C4×C8—61no—
207#95—22no—
208#95#6712yes—
209C23×C3—24no—
210C23×C3—22no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3#12011yescharacteristic
215C22×D3—41no—
216C3×D4—24no—
217C3×D4—22no—
218C3×D4#10414yescomplemented
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—416no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12311yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248#67—81no—
249#67—81no—
250C2×Dic3—41no—
251D12—42no—
252D12—41no—
253D12—41no—
254D12—81no—
255C4×D3—81no—
256C4×D3—81no—
257Dic6—41no—
258Dic6—41no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C3⋊C8—81no—
262C24—21no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—61no—
268C2×D4—24no—
269C2×D4—21no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—121no—
274C2×D4(C2×D3)⋊C412yes—
275C22×C4—21no—
276C22×C4—28no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—121no—
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292D8—124no—
293M16—121no—
294M16—121no—
295C2×C8—62no—
296C2×C8—121no—
297C2×C8—121no—
298C4⋊C4—24no—
299C4⋊C4—21no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4(C2×D3)⋊C412yes—
303C22⋊C4—28no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—24no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—44no—
313C42—42no—
314C42—41no—
315C42—41no—
316C42#21312yescomplemented
317C42C2×C4×D311yescharacteristic
318C42C2×D1211yescharacteristic
319C42#23511yescharacteristic
320C42#23511yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—24no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3#38611yescharacteristic
329C22×C3#44711yescharacteristic
330C22×C3#44911yescharacteristic
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—24no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—44no—
340C3×C4—42no—
341C3×C4C2×#10412yes—
342C3×C4C4×D811yescharacteristic
343C3×C4#44211yescharacteristic
344C3×C4#45911yescharacteristic
345C3×C4#46311yescharacteristic
346Dic3—81no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—121no—
353C23#95311yescharacteristic
354Q8—121no—
355Q8—121no—
356D4—24no—
357D4—22no—
358D4—122no—
359D4—121no—
360D4—121no—
361D4—241no—
362D4—241no—
363D4—241no—
364D4#80514yescomplemented
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—416no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4C2×(C2×D3)⋊C411yescharacteristic
388C2×C4C4×#6711yescharacteristic
389C2×C4C4×#6711yescharacteristic
390C2×C4#92911yescharacteristic
391C2×C4#93011yescharacteristic
392C2×C4#93011yescharacteristic
393C2×C4C2×#21311yescharacteristic
394C2×C4#93211yescharacteristic
395C2×C4#94911yescharacteristic
396C2×C4#95011yescharacteristic
397C8—122no—
398C8—241no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—44no—
403C2×C3#187911yescharacteristic
404C2×C3#189611yescharacteristic
405C2×C3#190611yescharacteristic
406D3—161no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—44no—
411C22—44no—
412C22—42no—
413C22—42no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22#521311yescharacteristic
418C22#521711yescharacteristic
419C22#523811yescharacteristic
420C4—24no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—42no—
426C4—42no—
427C4—241no—
428C4C2×#80512yes—
429C4C4×#21311yescharacteristic
430C4#501411yescharacteristic
431C4#501511yescharacteristic
432C4#501611yescharacteristic
433C3#923311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—44no—
437C2—481no—
438C2#7153211yescharacteristic
439C2#7161311yescharacteristic
440C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C42×D4C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenterC42×D4D3
hypocenterC3#9233
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
1. C3×C42×D4C2semidirect
3. #71598C2semidirect
4. #69469C2semidirect
5. #69395C2semidirect
6. #67620C2semidirect
7. #67611C2semidirect
19. #5015C4semidirect
21. #5007C4semidirect
31. C42×D4D3semidirect
64. #874C2×C4semidirect
67. #869C2×C4semidirect
80. C4⋊(C3⋊C8)C2×C4semidirect
148. C3×C42D8semidirect
218. C3×D4#104semidirect
316. C42#213semidirect
364. D4#805semidirect
433. C3#9233semidirect

Maximal subgroups

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

subgroupindex
C3×C42×D42
#823082
#715982
#694692
#693952
#676202
#676112
#92333