Finite Group Databasebeta

← #10690475

Subgroups — #10690475

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4693C211yesmaximal, characteristic
2#73656C211yesmaximal, characteristic, complemented
3#70457C211yesmaximal, characteristic
4C2×#4481C211yesmaximal, characteristic, complemented
5#69487C211yesmaximal, characteristic
6C3×#1279C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#9063—31nomaximal, Sylow
9C22×#204—22no—
10C2×#874C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#821—21no—
16C2×#821—21no—
17Dic3×C4⋊C4—21no—
18#4972—21no—
19C2×#809—21no—
20C2×#809—21no—
21C4×C4⋊Dic3—21no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4928—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4693—22no—
26#4693C414yescomplemented
27#4646—22no—
28#4481—22no—
29C2×#492—31no—
30#2369—31no—
31#1982—31no—
32#1982—31no—
33C2×#333—31no—
34#1279—31no—
35#1279D311yescharacteristic, complemented
36C22×D12—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×Dic6D412yes—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—22no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3—22no—
53C22×C3⋊C8—21no—
54C2×#204—28no—
55C2×#204—44no—
56C2×#204—44no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61#869—21no—
62#869—21no—
63#869C2×C412yes—
64#869D412yes—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4×C8—21no—
68C3×C4×C8C2×C412yes—
69C3×#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73#821—42no—
74#821—42no—
75#809—42no—
76#809—42no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C22×SD16—62no—
80C2×C4⋊Q8—31no—
81C2×#127—31no—
82#492—34no—
83#492—62no—
84C2×C4⋊C8—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#102—61no—
87C2×#102—61no—
88C2×#102—61no—
89C2×#102—61no—
90C2×C4×C8C2×D311yescharacteristic
91#391—62no—
92#389—61no—
93#389—61no—
94C4×C4⋊C4—31no—
95C4×C4⋊C4—61no—
96C4×C4⋊C4—61no—
97C4×C4⋊C4C2×D311yescharacteristic
98#333—62no—
99C23×D3—41no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—22no—
107C22×Dic3—21no—
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—42no—
112C2×D12—42no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×Dic6—22no—
125C2×Dic6—42no—
126C2×Dic6—42no—
127C2×Dic6C22⋊C414yes—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4SD1612yescomplemented
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3SD1612yescomplemented
152Dic3⋊C4—82no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C4×Dic3—82no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159#204—416no—
160C22×Q8—32no—
161C22×D4—61no—
162C22×D4—61no—
163C22×D4—61no—
164C2×SD16—68no—
165C2×SD16—124no—
166C2×SD16—124no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—61no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—61no—
173C4⋊Q8—61no—
174#127—32no—
175#127—61no—
176#127—61no—
177#127—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4D1211yescharacteristic
184C2×C4⋊C4D1211yescharacteristic
185C2×C42—21no—
186C2×C42—61no—
187C2×C42—62no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#102—122no—
194#102—122no—
195#102—122no—
196#102—122no—
197C4×C8—21no—
198C4×C8C4×D312yes—
199#95—21no—
200#95—62no—
201#95—62no—
202#95—61no—
203C23×C3#12011yescharacteristic
204C22×D3—42no—
205C22×D3—81no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—81no—
209C22×D3—81no—
210C22×D3—81no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4C2×SD1611yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—24no—
234C2×Dic3—22no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242D12—44no—
243D12—42no—
244D12—42no—
245D12—42no—
246D12—42no—
247D12—82no—
248D12—81no—
249D12—81no—
250D12—81no—
251D12—81no—
252Dic6—44no—
253Dic6—82no—
254C3×C8—24no—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—121no—
258C2×Q8—34no—
259C2×Q8—62no—
260C2×Q8—122no—
261C2×Q8—122no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—122no—
266C2×D4—122no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—61no—
283C22×C4C2×D1211yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286SD16—1216no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—121no—
294C2×C8—121no—
295C2×C8—121no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—32no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—242no—
308C4⋊C4#20412yescomplemented
309C42—41no—
310C42—41no—
311C42—121no—
312C42—121no—
313C42—242no—
314C42C2×C4×D311yescharacteristic
315C42#23511yescharacteristic
316C42D3×D411yescharacteristic
317C42#23911yescharacteristic
318C22×C3#38611yescharacteristic
319C22×C3#44811yescharacteristic
320C22×C3#44811yescharacteristic
321C22×C3#44911yescharacteristic
322C22×C3#44911yescharacteristic
323C22×C3#45311yescharacteristic
324C22×C3#45311yescharacteristic
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—161no—
329C2×D3—161no—
330C2×D3—161no—
331C2×D3—161no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—81no—
336C3×C4C4×SD1611yescharacteristic
337C3×C4C4×SD1611yescharacteristic
338C3×C4#44011yescharacteristic
339C3×C4#44011yescharacteristic
340C3×C4#46011yescharacteristic
341C3×C4#46011yescharacteristic
342C3×C4#46411yescharacteristic
343C3×C4#46411yescharacteristic
344Dic3—42no—
345Dic3—44no—
346Dic3—81no—
347C23—122no—
348C23—241no—
349C23—241no—
350C23—241no—
351C23—241no—
352C23—241no—
353C23—241no—
354C23#88211yescharacteristic
355Q8—124no—
356Q8—242no—
357D4—124no—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—242no—
363D4—241no—
364D4—241no—
365D4—241no—
366D4—241no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—64no—
376C2×C4—62no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4C4×D1211yescharacteristic
385C2×C4D3×C22⋊C411yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#89311yescharacteristic
388C2×C4#89411yescharacteristic
389C2×C4#89511yescharacteristic
390C2×C4C2×#20411yescharacteristic
391C2×C4#90411yescharacteristic
392C2×C4#90811yescharacteristic
393C2×C4#90911yescharacteristic
394C2×C4#91111yescharacteristic
395C2×C4D3×SD1611yescharacteristic
396C2×C4#91611yescharacteristic
397C2×C4#91811yescharacteristic
398C8—24no—
399C8—81no—
400C8—241no—
401C2×C3#162211yescharacteristic
402C2×C3#162211yescharacteristic
403C2×C3#162311yescharacteristic
404C2×C3#162411yescharacteristic
405C2×C3#188011yescharacteristic
406C2×C3#188011yescharacteristic
407C2×C3#188711yescharacteristic
408D3—162no—
409C22—242no—
410C22—242no—
411C22—242no—
412C22—481no—
413C22—481no—
414C22—481no—
415C22—481no—
416C22#466511yescharacteristic
417C22#473411yescharacteristic
418C22#473811yescharacteristic
419C22#478111yescharacteristic
420C22#480611yescharacteristic
421C22#482211yescharacteristic
422C22#482211yescharacteristic
423C4—21no—
424C4—21no—
425C4—42no—
426C4—81no—
427C4—122no—
428C4—124no—
429C4—241no—
430C4C4×#20411yescharacteristic
431C4#471011yescharacteristic
432C4#476211yescharacteristic
433C4#479811yescharacteristic
434C4#484511yescharacteristic
435C4#484811yescharacteristic
436C4#485211yescharacteristic
437C4#488911yescharacteristic
438C3#906311yescharacteristic, complemented, Sylow
439C2—482no—
440C2#6819811yescharacteristic
441C2#6819911yescharacteristic
442C2#6824411yescharacteristic
443C2#6826411yescharacteristic
444C2#6868611yescharacteristic
445C2#6876011yescharacteristic
446C2#6890911yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9063
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #73656C2semidirect
4. C2×#4481C2semidirect
6. C3×#1279C2semidirect
7. #67655C2semidirect
26. #4693C4semidirect
35. #1279D3semidirect
137. C3×C4⋊C4SD16semidirect
151. C4⋊Dic3SD16semidirect
308. C4⋊C4#204semidirect
438. C3#9063semidirect

Maximal subgroups

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

subgroupindex
C2×#46932
#736562
#704572
C2×#44812
#694872
C3×#12792
#676552
#90633