Finite Group Databasebeta

← #10691440

Subgroups — #10691440

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3101C211yesmaximal, characteristic
2C4×D4×Dic3C211yesmaximal, characteristic
3#71839C211yesmaximal, characteristic, complemented
4#71835C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9231—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C22×C4×Dic3—21no—
13C3×#483—21no—
14C3×#476C412yescomplemented
15C3×#461C412yescomplemented
16C3×#450—21no—
17C3×C4×SD16—22no—
18C3×#431C2211yescharacteristic
19C4×#217—21no—
20Dic3×C4⋊C4—21no—
21Dic3×C22⋊C4—22no—
22C4×C4⋊Dic3C2211yescharacteristic
23C42×Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4538—22no—
27D4⋊(C3⋊C8)—22no—
28#4532—21no—
29#4532—21no—
30#3101D311yescharacteristic
31C42×D4—31no—
32#1994—31no—
33#1503—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×C3×SD16—22no—
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—28no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—42no—
64C2×C4×Dic3—42no—
65C2×C4×Dic3D411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C2×#208—21no—
68C3×#106—22no—
69C3×C4⋊C8C2×C412yescomplemented
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#830—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4×C3⋊C8—21no—
81#533C2×D311yescharacteristic
82C2×C4×D4—61no—
83C2×C4×D4C2×D311yescharacteristic
84C22×C42—61no—
85#483—21no—
86#476Dic312yescomplemented
87#461Dic312yescomplemented
88#450—21no—
89C4×SD16—22no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423—31no—
93C4×C4⋊C4—31no—
94C4×C4⋊C4—61no—
95C4×C22⋊C4—62no—
96C4×C22⋊C4—61no—
97C43—31no—
98#339—61no—
99#339—61no—
100#328—62no—
101D4⋊C8—62no—
102C24×C3—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—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
118C22×Dic3—21no—
119C22×Dic3—28no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—44no—
123C22×Dic3—44no—
124C3×SD16—44no—
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—28no—
143C4⋊Dic3—24no—
144C4×Dic3—24no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147C4×Dic3—21no—
148C4×Dic3—42no—
149C4×Dic3—44no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3SD1612yescomplemented
153#208—41no—
154#208—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C22×D4#6711yescharacteristic
158C23×C4—21no—
159C23×C4—62no—
160C2×SD16—22no—
161C2×M16—61no—
162C4⋊Q8C2×Dic311yescharacteristic, complemented
163C4⋊Q8C2×Dic311yescharacteristic, complemented
164#125—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—68no—
172C4×D4C2×Dic312yes—
173C4×D4#6712yes—
174#117—21no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C42—31no—
180C2×C42—31no—
181C2×C42—61no—
182C2×C42—62no—
183C2×C42—61no—
184C2×C42—61no—
185C2×C42—122no—
186C2×C42—122no—
187C2×C42C22×D311yescharacteristic
188#106—22no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8C2×Dic312yescomplemented
193#103—22no—
194#102—22no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197C4×C8—21no—
198C4×C8—61no—
199#95—32no—
200#95—62no—
201C23×C3—24no—
202C23×C3—22no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C3×D4—24no—
210C3×D4—22no—
211C3×D4#10414yes—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—24no—
238C2×Dic3—416no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C3×C8—42no—
244C3⋊C8—81no—
245C3⋊C8—81no—
246C24—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4#21712yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—68no—
261C22×C4—62no—
262C22×C4—61no—
263C22×C4—124no—
264C22×C4—124no—
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4C2×#6711yescharacteristic
268SD16—44no—
269M16—121no—
270M16—121no—
271C2×C8—22no—
272C2×C8—121no—
273C2×C8—121no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—64no—
282C4⋊C4#21712yes—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—68no—
287C42—32no—
288C42—41no—
289C42—64no—
290C42—61no—
291C42—61no—
292C42—61no—
293C42—124no—
294C42—122no—
295C42—121no—
296C42—121no—
297C42#23711yescharacteristic
298C42#23711yescharacteristic
299C42C22×Dic311yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—24no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3#38611yescharacteristic
309C22×C3#45011yescharacteristic
310C22×C3#45211yescharacteristic
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—42no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4C2×#10412yes—
318C3×C4C4×SD1611yescharacteristic
319C3×C4#44011yescharacteristic
320C3×C4#46111yescharacteristic
321C3×C4#46311yescharacteristic
322Dic3—24no—
323Dic3—42no—
324Dic3—44no—
325C23—24no—
326C23—22no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23#93711yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—24no—
334D4—22no—
335D4#83714yes—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—64no—
352C2×C4—1216no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4D3×SD1611yescharacteristic
358C2×C4#91411yescharacteristic
359C2×C4#91511yescharacteristic
360C2×C4#91611yescharacteristic
361C2×C4D4×Dic311yescharacteristic
362C2×C4D4×Dic311yescharacteristic
363C2×C4#93511yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4C2×#21711yescharacteristic
367C8—42no—
368C8—241no—
369C8—241no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—44no—
373C2×C3#187711yescharacteristic
374C2×C3#189611yescharacteristic
375C2×C3#190611yescharacteristic
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22#516911yescharacteristic
384C22#517311yescharacteristic
385C22#522011yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—64no—
391C4—81no—
392C4—81no—
393C4—124no—
394C4—122no—
395C4Dic3×SD1611yescharacteristic
396C4#516211yescharacteristic
397C4#516411yescharacteristic
398C4#516511yescharacteristic
399C4C2×#83712yes—
400C3#923111yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—44no—
404C2#7171111yescharacteristic
405C2#7204711yescharacteristic
406C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3101C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3101D3
hypocenterC3#9231
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
3. #71839C2semidirect
4. #71835C2semidirect
6. #67746C2semidirect
7. #67736C2semidirect
14. C3×#476C4semidirect
15. C3×#461C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
69. C3×C4⋊C8C2×C4semidirect
86. #476Dic3semidirect
87. #461Dic3semidirect
152. C4×Dic3SD16semidirect
162. C4⋊Q8C2×Dic3semidirect
163. C4⋊Q8C2×Dic3semidirect
192. C4⋊C8C2×Dic3semidirect
400. C3#9231semidirect

Maximal subgroups

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

subgroupindex
C3×#31012
C4×D4×Dic32
#718392
#718352
#715982
#677462
#677362
#92313