Finite Group Databasebeta

← #10691450

Subgroups — #10691450

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
1C4×D4×Dic3C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3C3×#1486C211yesmaximal, characteristic, complemented
4#70789C211yesmaximal, characteristic, complemented
5#70512C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9233—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×D4×Dic3—21no—
11C22×C4×Dic3—21no—
12#5477C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5217—21no—
16C4×#217—21no—
17C2×#825—21no—
18C2×#825—21no—
19#5015C412yescomplemented
20C4×#213—22no—
21#5007C412yescomplemented
22Dic3×C4⋊C4—21no—
23#4967—21no—
24Dic3×C22⋊C4—22no—
25C4×C4⋊Dic3C2211yescharacteristic
26C42×Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C3×D4⋊C8—22no—
29#4470—22no—
30#3100—31no—
31C42×D4—31no—
32#1962—31no—
33#1502—31no—
34#1486D311yescharacteristic, complemented
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×#235—21no—
41C2×C3×M16—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4C2×C412yes—
45C3×C4×D4D412yes—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—22no—
50D4×Dic3—28no—
51C2×#213—22no—
52#930—21no—
53#930—21no—
54C2×C4⋊Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—42no—
60C2×C4×Dic3—42no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63#875—21no—
64#874C2×C411yescharacteristic, complemented
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic, complemented
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#832—22no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#830—21no—
77#825—42no—
78#825—42no—
79#809—22no—
80#807—22no—
81C4⋊(C3⋊C8)C2×C412yescomplemented
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4×D4—61no—
85C2×C4×D4C2×D311yescharacteristic
86C22×C42—61no—
87#480—61no—
88#474—32no—
89#459—32no—
90#447—61no—
91C4×D8—62no—
92#431—31no—
93#431C2×D311yescharacteristic
94#423C2×D311yescharacteristic
95C2×#104—61no—
96C2×#104—61no—
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99C4×C22⋊C4—62no—
100C4×C22⋊C4—61no—
101C43—31no—
102#327—62no—
103D4⋊C8—22no—
104C24×C3—21no—
105C2×C3×D4—24no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—41no—
119C22×Dic3—21no—
120C22×Dic3—28no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—44no—
124C22×Dic3—44no—
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×M16—41no—
135C3×M16—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4C22⋊C412yes—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#217—28no—
147#213—44no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—24no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4×Dic3—24no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—42no—
158C4×Dic3—44no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3D812yescomplemented
162C2×C3⋊C8—22no—
163C2×#41—61no—
164C22×D4D1211yescharacteristic
165C23×C4—21no—
166C23×C4—62no—
167C2×D8—62no—
168C2×M16—21no—
169C4⋊Q8—31no—
170#127—31no—
171#124—61no—
172#121—61no—
173#121—61no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—68no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4C4×D312yes—
180C4×D4#6712yes—
181C2×C4⋊C4—61no—
182C2×C4⋊C4D1211yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—62no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188C2×C42—62no—
189C2×C42—61no—
190C2×C42—61no—
191C2×C42—122no—
192C2×C42—122no—
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199#104—122no—
200#104—122no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C4×C8—21no—
206C4×C8—61no—
207#95—32no—
208#95—62no—
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—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238#67—81no—
239#67—81no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—24no—
245C2×Dic3—416no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251D12—42no—
252D12—41no—
253D12—41no—
254D12—81no—
255C4×D3—81no—
256C4×D3—81no—
257Dic6—41no—
258Dic6—41no—
259C3×C8—81no—
260C3×C8—81no—
261C3⋊C8—42no—
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—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—61no—
283C22×C4—68no—
284C22×C4—62no—
285C22×C4—61no—
286C22×C4—124no—
287C22×C4—124no—
288C22×C4—121no—
289C22×C4C2×D1211yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292D8—124no—
293M16—41no—
294M16—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—62no—
298C4⋊C4—21no—
299C4⋊C4—64no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4(C2×D3)⋊C412yes—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—68no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—32no—
309C42—64no—
310C42—61no—
311C42—61no—
312C42—61no—
313C42—124no—
314C42—122no—
315C42—121no—
316C42—121no—
317C42C2×C4×D311yescharacteristic
318C42#23711yescharacteristic
319C42#23711yescharacteristic
320C42C2×#6711yescharacteristic
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—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4C2×#10412yes—
339C3×C4C4×D811yescharacteristic
340C3×C4#44211yescharacteristic
341C3×C4#45911yescharacteristic
342C3×C4#46311yescharacteristic
343Dic3—24no—
344Dic3—42no—
345Dic3—44no—
346Dic3—81no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—121no—
355Q8—121no—
356D4—24no—
357D4—22no—
358D4—122no—
359D4—121no—
360D4—121no—
361D4—241no—
362D4—241no—
363D4—241no—
364D4#82514yescomplemented
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—64no—
379C2×C4—1216no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4#67⋊C411yescharacteristic
388C2×C4#67⋊C411yescharacteristic
389C2×C4#88611yescharacteristic
390C2×C4D3×D811yescharacteristic
391C2×C4#91111yescharacteristic
392C2×C4#91411yescharacteristic
393C2×C4#91611yescharacteristic
394C2×C4C2×(C2×D3)⋊C411yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#93911yescharacteristic
397C8—81no—
398C8—81no—
399C8—122no—
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#477311yescharacteristic
418C22#478111yescharacteristic
419C22#495811yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—64no—
425C4—124no—
426C4—122no—
427C4—241no—
428C4#475611yescharacteristic
429C4#478511yescharacteristic
430C4C2×#82512yes—
431C4#515011yescharacteristic
432C4#516511yescharacteristic
433C3#923311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—44no—
437C2—481no—
438C2#6996511yescharacteristic
439C2#7017211yescharacteristic
440C2#7039511yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9233
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. C4×D4×Dic3C2semidirect
3. C3×#1486C2semidirect
4. #70789C2semidirect
5. #70512C2semidirect
6. #67736C2semidirect
7. #67715C2semidirect
19. #5015C4semidirect
21. #5007C4semidirect
34. #1486D3semidirect
64. #874C2×C4semidirect
67. #869C2×C4semidirect
81. C4⋊(C3⋊C8)C2×C4semidirect
161. C4×Dic3D8semidirect
218. C3×D4#104semidirect
364. D4#825semidirect
433. C3#9233semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Dic32
#823082
C3×#14862
#707892
#705122
#677362
#677152
#92333