Finite Group Databasebeta

← #10691746

Subgroups — #10691746

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2364C211yesmaximal, characteristic
2#82630C211yesmaximal, characteristic
3#71675C211yesmaximal, characteristic
4#69457C211yesmaximal, characteristic, complemented
5C2×#4452C211yesmaximal, characteristic, complemented
6#67617C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9275—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×C4×D4—21no—
11C2×#924C2211yescharacteristic
12C3×#388—21no—
13C3×#387—21no—
14C3×#386—21no—
15C3×#386—21no—
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×C3×#95C2211yescharacteristic
19#5218—21no—
20#5212C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5041—22no—
24#5040C412yescomplemented
25#5034C412yescomplemented
26#4971—21no—
27#4537—22no—
28#4527—21no—
29#4527—21no—
30#4452—21no—
31#4452—21no—
32#3187—31no—
33#2364D311yescharacteristic
34#1996—31no—
35#1519—31no—
36C2×#328—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4—21no—
43C23×C3×C4C2311yescharacteristic
44C22×Dic6C2×C411yescharacteristic, complemented
45C3×#121—21no—
46C3×#121—21no—
47C3×#121C2×C412yes—
48C3×#121D412yes—
49C3×C4×D4—42no—
50C3×C4×D4—42no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4D411yescharacteristic
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#214—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic, complemented
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95D411yescharacteristic
78C3×#95D411yescharacteristic
79#832—22no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830C2×C412yescomplemented
85#810—22no—
86#808—22no—
87C2×C22⋊Q8—31no—
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—21no—
90#483—61no—
91#481—32no—
92#465—62no—
93#450—61no—
94#448—32no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97#388—21no—
98#387—21no—
99#386—21no—
100#386—21no—
101#380—31no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4C2×D311yescharacteristic
104C2×#95C2×D311yescharacteristic
105#342—61no—
106#342—61no—
107#328—61no—
108#328—61no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—81no—
120C2×C3×D4C22⋊C412yes—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—81no—
132C22×C3×C4—81no—
133C22×C3×C4—81no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—41no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—81no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4C22⋊C412yes—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—81no—
157C3×C22⋊C4SD1612yescomplemented
158C3×C42—42no—
159C3×C42—81no—
160#217—41no—
161#217—41no—
162#217—41no—
163#217—41no—
164#214—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—81no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—81no—
173C2×C3⋊C8—81no—
174C22×Q8—31no—
175C22×D4—21no—
176C22×D4#6711yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C22×D311yescharacteristic
180C2×SD16—62no—
181C22×C8—61no—
182C22×C8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C22⋊Q8—61no—
186C22⋊Q8—61no—
187#121—21no—
188#121—21no—
189#121C4×D312yes—
190#121D1212yes—
191C4×D4—42no—
192C4×D4—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4#6711yescharacteristic
205C2×C22⋊C4#6711yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208#106—62no—
209#103—62no—
210#102—62no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—61no—
220#95#6711yescharacteristic
221#95#6711yescharacteristic
222C23×C3—22no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3C4×D411yescharacteristic
232C23×C3#12011yescharacteristic
233C23×C3#12111yescharacteristic
234C23×C3C2×SD1611yescharacteristic
235C23×C3#13511yescharacteristic
236C23×C3AGL(1,ℤ/8)11yescharacteristic
237C23×C3#13711yescharacteristic
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—81no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
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—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—84no—
266C2×C3×C4—82no—
267C2×C3×C4—82no—
268C2×C3×C4C2×C22⋊C411yescharacteristic
269C2×C3×C4C4×D411yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12311yescharacteristic
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278C2×Dic3—81no—
279Dic6—42no—
280Dic6—81no—
281C3⋊C8—42no—
282C3⋊C8—81no—
283C24—21no—
284C24C2×#6711yescharacteristic
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—241no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—81no—
296C2×D4(C2×D3)⋊C412yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—121no—
313C22×C4C2×C4×D311yescharacteristic
314C22×C4C2×D1211yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4#23511yescharacteristic
317C22×C4C2×#6711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319SD16—242no—
320C2×C8—62no—
321C2×C8—121no—
322C2×C8—241no—
323C2×C8—241no—
324C4⋊C4—21no—
325C4⋊C4—42no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C4⋊C4—241no—
331C4⋊C4(C2×D3)⋊C412yes—
332C22⋊C4—21no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—81no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—121no—
344C22⋊C4#21412yescomplemented
345C42—42no—
346C42—81no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3#38611yescharacteristic
365C22×C3C4×SD1611yescharacteristic
366C22×C3#44011yescharacteristic
367C22×C3#45011yescharacteristic
368C22×C3#45211yescharacteristic
369C22×C3#46111yescharacteristic
370C22×C3#46311yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4—81no—
380Dic3—81no—
381Dic3—81no—
382Dic3—161no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23C4×#6711yescharacteristic
393C23#93011yescharacteristic
394C23#93211yescharacteristic
395C23C2×#21411yescharacteristic
396C23#95011yescharacteristic
397C23#95111yescharacteristic
398C23#95311yescharacteristic
399Q8—122no—
400Q8—241no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—81no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
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—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4C2×(C2×D3)⋊C411yescharacteristic
439C2×C4C4×#6711yescharacteristic
440C2×C4#92911yescharacteristic
441C2×C4#93011yescharacteristic
442C8—122no—
443C8—241no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—82no—
451C2×C3#187711yescharacteristic
452C2×C3#190611yescharacteristic
453C2×C3#191311yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22#501611yescharacteristic
472C22C4×#21411yescharacteristic
473C22#501811yescharacteristic
474C22#501911yescharacteristic
475C22#521311yescharacteristic
476C22#521811yescharacteristic
477C22#523911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—241no—
488C4—241no—
489C4—481no—
490C3#927511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#7161211yescharacteristic
499C2#7164511yescharacteristic
500C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2364C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2364D3
hypocenterC3#9275
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2364

Complements

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

NH ≅ G/Ntype
4. #69457C2semidirect
5. C2×#4452C2semidirect
6. #67617C2semidirect
7. #67612C2semidirect
24. #5040C4semidirect
25. #5034C4semidirect
44. C22×Dic6C2×C4semidirect
65. C2×C4⋊Dic3C2×C4semidirect
84. #830C2×C4semidirect
157. C3×C22⋊C4SD16semidirect
344. C22⋊C4#214semidirect
490. C3#9275semidirect

Maximal subgroups

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

subgroupindex
C3×#23642
#826302
#716752
#694572
C2×#44522
#676172
#676122
#92753