Finite Group Databasebeta

← #10691770

Subgroups — #10691770

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
1#82630C211yesmaximal, characteristic
2#73473C211yesmaximal, characteristic, complemented
3C3×#1964C211yesmaximal, characteristic
4C2×#4537C211yesmaximal, characteristic
5#69510C211yesmaximal, characteristic, complemented
6#67624C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9278—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C2×C3×#99—21no—
12C2×C3×#99—21no—
13C3×#380C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5218C412yescomplemented
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5041—22no—
20#5040—21no—
21#5034—21no—
22#4971C412yescomplemented
23#4946—21no—
24Dic3×C22⋊C4—21no—
25#4933—21no—
26C4×Dic3⋊C4—21no—
27#4654—21no—
28#4646—21no—
29#4537—21no—
30#4537—21no—
31#4452—22no—
32#3187—31no—
33#2368—31no—
34#1964D311yescharacteristic
35#1520—31no—
36C2×#327—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C22×Dic6D411yescharacteristic
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×#217—21no—
56C2×#214—22no—
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60#924C2×C412yes—
61#924D412yes—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#99—42no—
70C3×#99—42no—
71C3×#95—21no—
72C3×#95—21no—
73#832—22no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D411yescharacteristic
80C3⋊#95D411yescharacteristic
81#830—21no—
82#830—21no—
83#830—21no—
84#830C2×C412yes—
85#810—22no—
86#808—22no—
87C2×C22⋊Q8—31no—
88C2×#121C2×D311yescharacteristic
89#483—32no—
90#481—61no—
91#465—62no—
92#450—32no—
93#448—61no—
94C2×#99—21no—
95C2×#99—21no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98#391—61no—
99#387—61no—
100#387—61no—
101#385—61no—
102#380C2×D311yescharacteristic
103C4×C4⋊C4—61no—
104C4×C22⋊C4—61no—
105C4×C22⋊C4C2×D311yescharacteristic
106C2×#95—31no—
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—81no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
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
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—42no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C22×Dic3—81no—
140C2×Dic6—21no—
141C2×Dic6—81no—
142C2×Dic6C22⋊C412yes—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—81no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C22⋊C4—21no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4SD1612yescomplemented
154C3×C42—81no—
155#217—22no—
156#217—22no—
157#214—82no—
158C4⋊Dic3—21no—
159C4⋊Dic3C22⋊C412yes—
160Dic3⋊C4—22no—
161Dic3⋊C4—22no—
162Dic3⋊C4—42no—
163Dic3⋊C4—81no—
164C4×Dic3—42no—
165C4×Dic3—81no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—81no—
169C2×C3⋊C8—81no—
170C22×Q8—31no—
171C22×D4—21no—
172C22×D4C4×D311yescharacteristic
173C23×C4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C22×C8—61no—
177C22×C8—61no—
178C22⋊Q8—32no—
179C22⋊Q8—32no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4C4×D311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4#6711yescharacteristic
196C2×C22⋊C4#6711yescharacteristic
197C2×C42—21no—
198C2×C42—61no—
199C2×C42—61no—
200#106—62no—
201#103—62no—
202#102—62no—
203#99—42no—
204#99—42no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209#95—21no—
210#95—21no—
211#95—31no—
212#95—31no—
213#95—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217#95—61no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3C4×D411yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3#13511yescharacteristic
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C23×C3#13711yescharacteristic
232C3×D4—42no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—22no—
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—81no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12311yescharacteristic
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—22no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—82no—
273C2×Dic3—82no—
274Dic6—42no—
275Dic6—81no—
276C3⋊C8—42no—
277C3⋊C8—81no—
278C24—21no—
279C24C2×#6711yescharacteristic
280C2×Q8—32no—
281C2×Q8—61no—
282C2×Q8—241no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—81no—
301C22×C4—122no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—241no—
305C22×C4C2×C4×D311yescharacteristic
306C22×C4#23511yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4#23711yescharacteristic
311SD16—242no—
312C2×C8—62no—
313C2×C8—121no—
314C2×C8—241no—
315C2×C8—241no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—32no—
319C4⋊C4—62no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—81no—
323C4⋊C4—122no—
324C4⋊C4—241no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4#21412yescomplemented
335C42—81no—
336C42—122no—
337C42—241no—
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—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—161no—
352C22×C3—161no—
353C22×C3#38611yescharacteristic
354C22×C3C4×SD1611yescharacteristic
355C22×C3#44011yescharacteristic
356C22×C3#44811yescharacteristic
357C22×C3#44911yescharacteristic
358C22×C3#46011yescharacteristic
359C22×C3#46411yescharacteristic
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—42no—
365C3×C4—81no—
366C3×C4—81no—
367C3×C4—161no—
368Dic3—42no—
369Dic3—42no—
370Dic3—42no—
371Dic3—81no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—81no—
378C23—81no—
379C23C4×#6711yescharacteristic
380C23#93211yescharacteristic
381C23C2×#21411yescharacteristic
382C23#93711yescharacteristic
383C23#93811yescharacteristic
384C23#95011yescharacteristic
385C23#95111yescharacteristic
386Q8—122no—
387Q8—241no—
388D4—42no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—81no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
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—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—242no—
425C2×C4—242no—
426C2×C4D3×C22⋊C411yescharacteristic
427C2×C4#67⋊C411yescharacteristic
428C2×C4#88311yescharacteristic
429C2×C4#88411yescharacteristic
430C8—122no—
431C8—241no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—161no—
439C2×C3#188011yescharacteristic
440C2×C3#190611yescharacteristic
441C2×C3#191311yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—161no—
456C22—161no—
457C22#495611yescharacteristic
458C22C4×#21411yescharacteristic
459C22#501811yescharacteristic
460C22#503711yescharacteristic
461C22#504011yescharacteristic
462C22#505411yescharacteristic
463C22#507511yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—81no—
470C4—81no—
471C4—122no—
472C4—122no—
473C4—122no—
474C4—161no—
475C4—241no—
476C3#927811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—161no—
484C2#6966011yescharacteristic
485C2#7003411yescharacteristic
486C2#7005511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1964D3
hypocenterC3#9278
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #73473C2semidirect
5. #69510C2semidirect
6. #67624C2semidirect
7. #67617C2semidirect
15. #5218C4semidirect
22. #4971C4semidirect
153. C3×C22⋊C4SD16semidirect
334. C22⋊C4#214semidirect
476. C3#9278semidirect

Maximal subgroups

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

subgroupindex
#826302
#734732
C3×#19642
C2×#45372
#695102
#676242
#676172
#92783