Finite Group Databasebeta

← #10691124

Subgroups — #10691124

Order 768.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3036C211yesmaximal, characteristic
2#82333C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9179—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×D4×Dic3—21no—
13C3×#463—22no—
14C3×#461C412yescomplemented
15C3×#459C412yescomplemented
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5212—21no—
19C2×C3⋊#95—21no—
20#4984—21no—
21#4950—22no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4537—21no—
27#4537—21no—
28#4537—22no—
29D4⋊(C3⋊C8)—22no—
30#3036D311yescharacteristic
31#2349—31no—
32#1517—31no—
33#1502—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C2×C3×#41—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×Dic3—22no—
41C3×AGL(1,ℤ/8)—42no—
42C3×AGL(1,ℤ/8)—42no—
43C2×C3×SD16—22no—
44C2×C3×D8—22no—
45C2×C3×M16—21no—
46C3×C4⋊Q8C2×C411yescharacteristic, complemented
47C3×#127C2×C411yescharacteristic, complemented
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4C2×C412yes—
56C3×C4×D4D412yes—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61D4×Dic3—42no—
62D4×Dic3—42no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—22no—
66C2×C4×Dic3—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yescomplemented
69C3×#103—22no—
70C3×#102—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73C3⋊#95D411yescharacteristic
74C3⋊#95D411yescharacteristic
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C2×AGL(1,ℤ/8)—21no—
83#532C2×D311yescharacteristic
84C2×C4×D4—61no—
85C2×C4×D4C2×D311yescharacteristic
86#463—22no—
87#461Dic312yescomplemented
88#459Dic312yescomplemented
89#431—31no—
90#431—31no—
91#423C2×D311yescharacteristic
92#417—21no—
93#389—61no—
94#385—62no—
95#382—31no—
96#380—61no—
97C4×C4⋊C4—31no—
98C2×#95—61no—
99#327—61no—
100#327—61no—
101#327—62no—
102D4⋊C8—62no—
103C24×C3—21no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4C22⋊C412yes—
117C2×C3×D4D812yes—
118C2×C3×D4SD1612yes—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—41no—
123C22×C3×C4—82no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—22no—
129C22×Dic3—21no—
130C22×Dic3—44no—
131C22×Dic3—44no—
132C22×Dic3—82no—
133C3×SD16—44no—
134C3×D8—44no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4C22⋊C412yes—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149C3×C42#4111yescharacteristic
150#217—44no—
151C4⋊Dic3—42no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C4⋊Dic3—82no—
155C4×Dic3—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×#41—21no—
160C22×D4#6711yescharacteristic
161C23×C4—21no—
162C23×C4—62no—
163AGL(1,ℤ/8)—42no—
164AGL(1,ℤ/8)—42no—
165C2×SD16—22no—
166C2×D8—22no—
167C2×M16—21no—
168C4⋊Q8C2×Dic311yescharacteristic, complemented
169#127C2×Dic311yescharacteristic, complemented
170#124—21no—
171#121—21no—
172#121—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—122no—
178C4×D4—122no—
179C4×D4C2×Dic312yes—
180C4×D4#6712yes—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4#6711yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—62no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8C2×Dic312yescomplemented
193#103—22no—
194#102—22no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C4×C8—61no—
198C4×C8—61no—
199#95—31no—
200#95—31no—
201#95—32no—
202#95—62no—
203#95—61no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—41no—
212C3×Q8—41no—
213C3×D4—22no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—41no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4#10214yes—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—82no—
229C2×C3×C4—82no—
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×C4C2×SD1611yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241C2×C3×C4#13711yescharacteristic
242C2×Dic3—22no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—88no—
252C3×C8—42no—
253C3⋊C8—81no—
254C3⋊C8—81no—
255C24—21no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4#21312yes—
269C2×D4#21412yes—
270C2×D4#21712yes—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—61no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4—82no—
279C22×C4—124no—
280C22×C4—124no—
281C22×C4—242no—
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4C2×#6711yescharacteristic
285SD16—44no—
286D8—44no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—121no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—122no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—242no—
299C4⋊C4#21712yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—124no—
305C42—121no—
306C42—121no—
307C42#23711yescharacteristic
308C42#23711yescharacteristic
309C42C22×Dic311yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—24no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3#38611yescharacteristic
322C22×C3#44711yescharacteristic
323C22×C3#44811yescharacteristic
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—81no—
330C3×C4C2×#10211yescharacteristic
331C3×C4#41711yescharacteristic
332C3×C4#48011yescharacteristic
333C3×C4#48111yescharacteristic
334Dic3—42no—
335Dic3—81no—
336Dic3—82no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23#93711yescharacteristic
344Q8—41no—
345Q8—41no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—41no—
350D4—41no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4#83214yes—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—62no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—122no—
374C2×C4—248no—
375C2×C4C2×#21311yescharacteristic
376C2×C4C2×#21411yescharacteristic
377C2×C4D4×Dic311yescharacteristic
378C2×C4D4×Dic311yescharacteristic
379C2×C4#93511yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#93911yescharacteristic
382C2×C4#94911yescharacteristic
383C2×C4#95111yescharacteristic
384C2×C4C2×#21711yescharacteristic
385C8—42no—
386C8—241no—
387C8—241no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—44no—
391C2×C3—161no—
392C2×C3#187411yescharacteristic
393C2×C3#190411yescharacteristic
394C2×C3#190611yescharacteristic
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#503611yescharacteristic
406C22#504011yescharacteristic
407C22#522011yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—81no—
414C4—122no—
415C4—242no—
416C4—241no—
417C4#503311yescharacteristic
418C4#503411yescharacteristic
419C4C2×#83211yescharacteristic
420C4#523211yescharacteristic
421C3#917911yescharacteristic, complemented, Sylow
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2#7167711yescharacteristic
427C2#7208911yescharacteristic
428C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3036D3
hypocenterC3#9179
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #82333C2semidirect
3. #71814C2semidirect
4. #71811C2semidirect
5. #71597C2semidirect
6. #67715C2semidirect
7. #67713C2semidirect
14. C3×#461C4semidirect
15. C3×#459C4semidirect
46. C3×C4⋊Q8C2×C4semidirect
47. C3×#127C2×C4semidirect
68. C3×C4⋊C8C2×C4semidirect
87. #461Dic3semidirect
88. #459Dic3semidirect
168. C4⋊Q8C2×Dic3semidirect
169. #127C2×Dic3semidirect
192. C4⋊C8C2×Dic3semidirect
421. C3#9179semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#823332
#718142
#718112
#715972
#677152
#677132
#91793