Finite Group Databasebeta

← #10691096

Subgroups — #10691096

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3077C211yesmaximal, characteristic
2#82333C211yesmaximal, characteristic
3#71838C211yesmaximal, characteristic, complemented
4#71835C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic
6#67715C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9175—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#478—22no—
13C3×#476C412yescomplemented
14C3×#474C412yescomplemented
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#418—21no—
18#5212—21no—
19C2×C3⋊#95—21no—
20#4984—21no—
21#4950—22no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—21no—
27#4538—21no—
28#4538—22no—
29D4⋊(C3⋊C8)—22no—
30#3077D311yescharacteristic
31#2349—31no—
32#1518—31no—
33#1503—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×C3×M16—21no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4C2×C412yes—
51C3×C4×D4D412yes—
52C3×#117—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57D4×Dic3—42no—
58D4×Dic3—42no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—22no—
62C2×C4×Dic3—21no—
63C3×#107—22no—
64C3×#106—22no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C412yescomplemented
67C3×#102—22no—
68C3×#102—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80#533C2×D311yescharacteristic
81C2×C4×D4—61no—
82C2×C4×D4C2×D311yescharacteristic
83#478—22no—
84#476Dic312yescomplemented
85#474Dic312yescomplemented
86#431—31no—
87#431—31no—
88#428—21no—
89#423C2×D311yescharacteristic
90#418—21no—
91#389—61no—
92#385—62no—
93#382—31no—
94#380—61no—
95C4×C4⋊C4—31no—
96C2×#95—61no—
97#328—61no—
98#328—61no—
99#328—62no—
100D4⋊C8—62no—
101C24×C3—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C2×C3×D4SD1612yes—
109C2×C3×D4Q1612yes—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—82no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—22no—
119C22×Dic3—21no—
120C22×Dic3—44no—
121C22×Dic3—44no—
122C22×Dic3—82no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4C22⋊C412yes—
134C3×C22⋊C4—22no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142#217—44no—
143C4⋊Dic3—42no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—82no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C22×D4#6711yescharacteristic
152C23×C4—21no—
153C23×C4—62no—
154C2×M16—21no—
155C4⋊Q8C2×Dic311yescharacteristic, complemented
156C4⋊Q8C2×Dic311yescharacteristic, complemented
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—122no—
165C4×D4—122no—
166C4×D4C2×Dic312yes—
167C4×D4#6712yes—
168#117—21no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—62no—
175C2×C42—61no—
176C2×C42C22×D311yescharacteristic
177#107—22no—
178#106—22no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8C2×Dic312yescomplemented
183#102—22no—
184#102—22no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187C4×C8—61no—
188C4×C8—61no—
189#95—31no—
190#95—31no—
191#95—32no—
192#95—62no—
193#95—61no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3#12011yescharacteristic
200C3×Q8—81no—
201C3×Q8—81no—
202C3×D4—22no—
203C3×D4—22no—
204C3×D4—22no—
205C3×D4#10314yes—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4C2×Q1611yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—22no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—88no—
234C3×C8—42no—
235C3⋊C8—81no—
236C3⋊C8—81no—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4#21512yes—
244C2×D4#21612yes—
245C2×D4#21712yes—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—61no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4—82no—
253C22×C4—124no—
254C22×C4—124no—
255C22×C4—242no—
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4C2×#6711yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—121no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—122no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C4⋊C4#21712yes—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C22⋊C4—124no—
280C42—41no—
281C42—121no—
282C42—121no—
283C42#23711yescharacteristic
284C42#23711yescharacteristic
285C42C22×Dic311yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—24no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3#38611yescharacteristic
295C22×C3#45011yescharacteristic
296C22×C3#45111yescharacteristic
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4C2×#10311yescharacteristic
305C3×C4#41711yescharacteristic
306C3×C4#48311yescharacteristic
307C3×C4#48411yescharacteristic
308Dic3—42no—
309Dic3—81no—
310Dic3—82no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23#93711yescharacteristic
317Q8—81no—
318Q8—81no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4#83514yes—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—62no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—248no—
341C2×C4D4×Dic311yescharacteristic
342C2×C4D4×Dic311yescharacteristic
343C2×C4#93511yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4C2×#21511yescharacteristic
347C2×C4C2×#21611yescharacteristic
348C2×C4#94911yescharacteristic
349C2×C4#95111yescharacteristic
350C2×C4C2×#21711yescharacteristic
351C8—42no—
352C8—241no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—44no—
357C2×C3#187211yescharacteristic
358C2×C3#190411yescharacteristic
359C2×C3#190611yescharacteristic
360C22—21no—
361C22—21no—
362C22—24no—
363C22—44no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22#504611yescharacteristic
368C22#505011yescharacteristic
369C22#522011yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—122no—
378C4—242no—
379C4—241no—
380C4#504311yescharacteristic
381C4#504411yescharacteristic
382C4C2×#83511yescharacteristic
383C4#523211yescharacteristic
384C3#917511yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—44no—
388C2#7169211yescharacteristic
389C2#7208911yescharacteristic
390C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3077D3
hypocenterC3#9175
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
3. #71838C2semidirect
4. #71835C2semidirect
6. #67715C2semidirect
7. #67713C2semidirect
13. C3×#476C4semidirect
14. C3×#474C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
66. C3×C4⋊C8C2×C4semidirect
84. #476Dic3semidirect
85. #474Dic3semidirect
155. C4⋊Q8C2×Dic3semidirect
156. C4⋊Q8C2×Dic3semidirect
182. C4⋊C8C2×Dic3semidirect
384. C3#9175semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#823332
#718382
#718352
#715972
#677152
#677132
#91753