Finite Group Databasebeta

← #10691449

Subgroups — #10691449

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
2C4×#893C211yesmaximal, characteristic, complemented
3C3×#1502C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9233—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#893—21no—
11C2×C4×D12C2211yescharacteristic
12C2×C42×D3—21no—
13C3×#431C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#837—21no—
16C2×#837—21no—
17#5087C412yescomplemented
18#5072C412yescomplemented
19#5036—21no—
20#5033—21no—
21C4×#213—22no—
22Dic3×C4⋊C4—21no—
23C4×(C2×D3)⋊C4—21no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4666—22no—
28C3×#327—22no—
29#4459—22no—
30#3100—31no—
31C42×D4—31no—
32#1962—31no—
33#1502D311yescharacteristic, complemented
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—22no—
40C22×C4×D3—21no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#127C2×C411yescharacteristic, complemented
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#213—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×#208—21no—
61#893—28no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12C2×C412yes—
65C4×D12D412yes—
66C42×D3—42no—
67C42×D3—42no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95D412yes—
72#837—42no—
73#837—42no—
74#832—22no—
75C3⋊#95—22no—
76#830—21no—
77#809—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)C2×C412yescomplemented
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#431C2×D311yescharacteristic
94#423—31no—
95C2×#104—61no—
96C2×#104—61no—
97C4×C4⋊C4—61no—
98C4×C4⋊C4C2×D311yescharacteristic
99C4×C22⋊C4—61no—
100C4×C22⋊C4—62no—
101C43—31no—
102#327—22no—
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—22no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
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—
134C2×C3×C8—41no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145C3×C42#4111yescharacteristic
146#213—44no—
147(C2×D3)⋊C4—28no—
148(C2×D3)⋊C4—22no—
149(C2×D3)⋊C4—22no—
150C4⋊Dic3—21no—
151C4⋊Dic3C22⋊C412yes—
152C4×Dic3—24no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—44no—
157C4×Dic3—42no—
158C4×Dic3D812yescomplemented
159#208—41no—
160#208—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—61no—
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—21no—
182C2×C4⋊C4—31no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—61no—
185C2×C42—21no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—61no—
189C2×C42—62no—
190C2×C42—61no—
191C2×C42—122no—
192C2×C42—122no—
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
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—62no—
208#95#6712yes—
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—24no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4C2×C22⋊C411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12311yescharacteristic
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
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—22no—
276C22×C4—41no—
277C22×C4—68no—
278C22×C4—61no—
279C22×C4—61no—
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—121no—
294M16—121no—
295C2×C8—41no—
296C2×C8—62no—
297C2×C8—121no—
298C4⋊C4—24no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—32no—
302C4⋊C4—61no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—68no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C42—32no—
309C42—41no—
310C42—41no—
311C42—64no—
312C42—61no—
313C42—61no—
314C42—61no—
315C42—124no—
316C42—122no—
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—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—44no—
338C3×C4—81no—
339C3×C4C2×#10412yes—
340C3×C4C4×D811yescharacteristic
341C3×C4#44211yescharacteristic
342C3×C4#45911yescharacteristic
343C3×C4#46311yescharacteristic
344Dic3—24no—
345Dic3—42no—
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—24no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—1216no—
379C2×C4—121no—
380C2×C4—121no—
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×C4D3×D811yescharacteristic
392C2×C4#91111yescharacteristic
393C2×C4#91411yescharacteristic
394C2×C4#91611yescharacteristic
395C2×C4C4×#6711yescharacteristic
396C2×C4#93811yescharacteristic
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#515611yescharacteristic
419C22#517211yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—44no—
424C4—64no—
425C4—81no—
426C4—122no—
427C4—122no—
428C4#475611yescharacteristic
429C4#478511yescharacteristic
430C4#478611yescharacteristic
431C4#478711yescharacteristic
432C4D3×#10412yes—
433C3#923311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—124no—
437C2—161no—
438C2#7005511yescharacteristic
439C2#7020811yescharacteristic
440C2#7038211yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#386
prime radical#83298C2
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9233
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. C4×#893C2semidirect
3. C3×#1502C2semidirect
4. #71815C2semidirect
5. #70762C2semidirect
6. #67736C2semidirect
7. #67620C2semidirect
17. #5087C4semidirect
18. #5072C4semidirect
33. #1502D3semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×#127C2×C4semidirect
81. C4⋊(C3⋊C8)C2×C4semidirect
158. C4×Dic3D8semidirect
169. C4⋊Q8C4×D3semidirect
170. #127C4×D3semidirect
253. D12#104semidirect
433. C3#9233semidirect

Maximal subgroups

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

subgroupindex
#832982
C4×#8932
C3×#15022
#718152
#707622
#677362
#676202
#92333