Finite Group Databasebeta

← #10691767

Subgroups — #10691767

Order 768.

564 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2368C211yesmaximal, characteristic, complemented
2#82736C211yesmaximal, characteristic
3#71690C211yesmaximal, characteristic, complemented
4#69456C211yesmaximal, characteristic
5C2×#4451C211yesmaximal, characteristic
6#67617C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9278—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#391—21no—
12C3×#387—21no—
13C3×#387—21no—
14C3×#385—21no—
15C3×C4×C4⋊C4—21no—
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×C3×#95C2211yescharacteristic
19#5227—21no—
20#5212C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5048—22no—
24#5046C412yescomplemented
25#5043C412yescomplemented
26#4968—21no—
27C2×#806—21no—
28C2×#806—21no—
29#4538—22no—
30#4451—21no—
31#4451—21no—
32#3187—31no—
33#2368D311yescharacteristic, complemented
34#1964—31no—
35#1520—31no—
36C2×#327—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8C2×C412yes—
47C3×C22⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C22⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#215—22no—
60#930—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2×C411yescharacteristic
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77#835—22no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C412yes—
83#809—22no—
84#808—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121—31no—
89#483—32no—
90#481—61no—
91#465—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98#391—21no—
99#387—21no—
100#387—21no—
101#385—21no—
102#380—31no—
103C4×C4⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4C2×D311yescharacteristic
106C2×#95C2×D311yescharacteristic
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—81no—
125C22×C3×C4—81no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—41no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—21no—
139C22×Dic3—41no—
140C2×D12—21no—
141C2×D12—21no—
142C2×D12—81no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—81no—
148C3×C4⋊C4C22⋊C412yes—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4SD1612yescomplemented
155C3×C42—42no—
156C3×C42—81no—
157C3×C42—81no—
158#217—41no—
159#217—41no—
160#215—82no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—81no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—81no—
169C2×C3⋊C8—81no—
170C22×Q8#6711yescharacteristic
171C22×D4—31no—
172C22×D4—61no—
173C23×C4—21no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C22×C8—61no—
177C22×C8—61no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8C4×D312yes—
181C22⋊Q8D1212yes—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4#6711yescharacteristic
196C2×C22⋊C4#6711yescharacteristic
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200#106—62no—
201#103—62no—
202#102—62no—
203#99—122no—
204#99—122no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—61no—
216#95#6711yescharacteristic
217#95#6711yescharacteristic
218C23×C3—41no—
219C23×C3—41no—
220C23×C3C4×D411yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12111yescharacteristic
223C23×C3C2×SD1611yescharacteristic
224C23×C3#13511yescharacteristic
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C23×C3#13711yescharacteristic
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—42no—
233C3×Q8—81no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—82no—
259C2×C3×C4—82no—
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12311yescharacteristic
264#67—81no—
265#67—81no—
266#67—81no—
267#67—81no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274D12—42no—
275D12—81no—
276C3⋊C8—42no—
277C3⋊C8—81no—
278C24—61no—
279C24C2×#6711yescharacteristic
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8(C2×D3)⋊C412yes—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—241no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—61no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—121no—
305C22×C4C2×C4×D311yescharacteristic
306C22×C4C2×D1211yescharacteristic
307C22×C4#23511yescharacteristic
308C22×C4#23511yescharacteristic
309C22×C4C2×#6711yescharacteristic
310C22×C4C2×#6711yescharacteristic
311SD16—242no—
312C2×C8—62no—
313C2×C8—121no—
314C2×C8—241no—
315C2×C8—241no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—81no—
323C4⋊C4—241no—
324C4⋊C4(C2×D3)⋊C412yes—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4#21512yescomplemented
335C42—42no—
336C42—81no—
337C42—81no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3#38611yescharacteristic
349C22×C3C4×SD1611yescharacteristic
350C22×C3#44011yescharacteristic
351C22×C3#44811yescharacteristic
352C22×C3#44911yescharacteristic
353C22×C3#46011yescharacteristic
354C22×C3#46411yescharacteristic
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—81no—
358C2×D3—161no—
359C2×D3—161no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—81no—
369C3×C4—81no—
370Dic3—81no—
371Dic3—161no—
372C23—41no—
373C23—41no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—241no—
378C23—241no—
379C23C4×#6711yescharacteristic
380C23#93011yescharacteristic
381C23C2×#21511yescharacteristic
382C23#94211yescharacteristic
383C23#94911yescharacteristic
384C23#95011yescharacteristic
385C23#95311yescharacteristic
386Q8—42no—
387Q8—81no—
388D4—122no—
389D4—241no—
390D4—241no—
391D4—241no—
392D4—241no—
393D4—241no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4—241no—
426C2×C4C2×(C2×D3)⋊C411yescharacteristic
427C2×C4C4×#6711yescharacteristic
428C2×C4#92911yescharacteristic
429C2×C4#93011yescharacteristic
430C8—122no—
431C8—241no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3#188011yescharacteristic
439C2×C3#190611yescharacteristic
440C2×C3#191311yescharacteristic
441D3—161no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—241no—
453C22—241no—
454C22—241no—
455C22—481no—
456C22—481no—
457C22C4×#21511yescharacteristic
458C22#502611yescharacteristic
459C22#502711yescharacteristic
460C22#502811yescharacteristic
461C22#521311yescharacteristic
462C22#522711yescharacteristic
463C22#523811yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—241no—
475C4—481no—
476C3#927811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—481no—
484C2#7163311yescharacteristic
485C2#7164511yescharacteristic
486C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2368C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2368D3
hypocenterC3#9278
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2368

Complements

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

NH ≅ G/Ntype
1. C3×#2368C2semidirect
3. #71690C2semidirect
6. #67617C2semidirect
7. #67612C2semidirect
24. #5046C4semidirect
25. #5043C4semidirect
33. #2368D3semidirect
154. C3×C22⋊C4SD16semidirect
334. C22⋊C4#215semidirect
476. C3#9278semidirect

Maximal subgroups

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

subgroupindex
C3×#23682
#827362
#716902
#694562
C2×#44512
#676172
#676122
#92783