Finite Group Databasebeta

← #10691762

Subgroups — #10691762

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3186C211yesmaximal, characteristic
2#82638C211yesmaximal, characteristic, complemented
3#71686C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic, complemented
6#67678C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9277—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#480—21no—
13C3×#480C412yescomplemented
14C3×#466—22no—
15C3×#447—21no—
16C3×#447C412yescomplemented
17C2×C3×C22⋊C8C2211yescharacteristic
18#5220—21no—
19#5220—21no—
20C2×#834—21no—
21C2×#834—21no—
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4980—21no—
27#4951—21no—
28Dic3×C22⋊C4—21no—
29#4537—22no—
30#4537—21no—
31#4537—21no—
32#3186D311yescharacteristic
33#2364—31no—
34#1964—31no—
35#1519—31no—
36C2×#327—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C411yescharacteristic, complemented
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C2×C3×D8—22no—
47C22×C3×C8—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121C2×C412yes—
55C3×#121D412yes—
56C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#217D411yescharacteristic
65C2×#217D411yescharacteristic
66D4×Dic3—42no—
67D4×Dic3—42no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4×Dic3—21no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8—21no—
73C3×#107—22no—
74C3×#102—22no—
75C3×#102—22no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8C2×C412yescomplemented
78#834—42no—
79#834—42no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95D411yescharacteristic
86C3⋊#95D411yescharacteristic
87#830—21no—
88#830—21no—
89C2×#121C2×D311yescharacteristic
90C2×#121C2×D311yescharacteristic
91C2×C4×D4—61no—
92#480—21no—
93#480Dic312yescomplemented
94#466—22no—
95#447—21no—
96#447Dic312yescomplemented
97C2×#99—61no—
98C2×#99—61no—
99C2×C22⋊C8—31no—
100C2×C22⋊C8C2×D311yescharacteristic
101#388—61no—
102#387—61no—
103#386—61no—
104#386—61no—
105#380—31no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4—61no—
108C2×#95—31no—
109#327—61no—
110#327—61no—
111#327—62no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—81no—
131C2×C3×D4—81no—
132C2×C3×D4C22⋊C412yes—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—81no—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—42no—
147C22×Dic3—42no—
148C22×Dic3—42no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—81no—
153C22×Dic3—81no—
154C3×D8—82no—
155C2×C3×C8—22no—
156C2×C3×C8—81no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4C22⋊C412yes—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165#217—21no—
166#217—21no—
167#217—21no—
168#217—42no—
169#217—42no—
170#217—41no—
171#217—41no—
172#217—81no—
173#217D812yescomplemented
174C4⋊Dic3—42no—
175C4⋊Dic3—81no—
176C4×Dic3—42no—
177C4×Dic3—81no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—81no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C2×Dic311yescharacteristic, complemented
183C22×D4#6711yescharacteristic
184C23×C4—61no—
185C23×C4—61no—
186C23×C4C22×D311yescharacteristic
187C2×D8—22no—
188C22×C8—21no—
189C22×C8—61no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196#121C2×Dic312yes—
197#121#6712yes—
198C4×D4—122no—
199C4×D4—122no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4C2×Dic311yescharacteristic, complemented
203C2×C4⋊C4#6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C42—61no—
213C2×C42—61no—
214#107—22no—
215#102—22no—
216#102—22no—
217#99—122no—
218#99—122no—
219C22⋊C8—21no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C22⋊C8C2×Dic312yescomplemented
223#95—31no—
224#95—31no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229#95—61no—
230C23×C3—22no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—81no—
243C23×C3—81no—
244C23×C3C4×D411yescharacteristic
245C23×C3#12011yescharacteristic
246C23×C3#12111yescharacteristic
247C23×C3C2×D811yescharacteristic
248C23×C3#13511yescharacteristic
249C23×C3AGL(1,ℤ/8)11yescharacteristic
250C23×C3AGL(1,ℤ/8)11yescharacteristic
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—81no—
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4C2×C22⋊C411yescharacteristic
276C2×C3×C4C4×D411yescharacteristic
277C2×C3×C4#12111yescharacteristic
278C2×C3×C4#12311yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—84no—
292C2×Dic3—82no—
293C2×Dic3—82no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C3×C8—42no—
298C3⋊C8—81no—
299C24—21no—
300C24—21no—
301C24D3×D411yescharacteristic
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4#21712yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—61no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—81no—
327C22×C4—122no—
328C22×C4—122no—
329C22×C4—122no—
330C22×C4—121no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—241no—
334C22×C4—241no—
335C22×C4D3×D411yescharacteristic
336C22×C4#23711yescharacteristic
337C22×C4#23711yescharacteristic
338C22×C4C22×Dic311yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341D8—82no—
342C2×C8—22no—
343C2×C8—81no—
344C2×C8—121no—
345C2×C8—241no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—122no—
350C4⋊C4—241no—
351C4⋊C4#21712yes—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—32no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—61no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—241no—
365C42—122no—
366C42—241no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—161no—
388C22×C3—161no—
389C22×C3#38611yescharacteristic
390C22×C3C4×D811yescharacteristic
391C22×C3#44211yescharacteristic
392C22×C3#44711yescharacteristic
393C22×C3#44911yescharacteristic
394C22×C3#45911yescharacteristic
395C22×C3#46311yescharacteristic
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—42no—
401C3×C4—81no—
402Dic3—42no—
403Dic3—42no—
404Dic3—81no—
405Dic3—81no—
406Dic3—161no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—81no—
420C23—81no—
421C23D3×D811yescharacteristic
422C23#91111yescharacteristic
423C23#91111yescharacteristic
424C23#91211yescharacteristic
425C23D4×Dic311yescharacteristic
426C23#93711yescharacteristic
427C23#93911yescharacteristic
428D4—42no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—81no—
439D4—81no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—244no—
465C2×C4—242no—
466C2×C4—242no—
467C2×C4—241no—
468C2×C4—241no—
469C2×C4—241no—
470C2×C4D4×Dic311yescharacteristic
471C2×C4#93511yescharacteristic
472C2×C4#93911yescharacteristic
473C2×C4C2×#21711yescharacteristic
474C8—42no—
475C8—241no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—82no—
483C2×C3—161no—
484C2×C3#187911yescharacteristic
485C2×C3#190611yescharacteristic
486C2×C3#191311yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—161no—
508C22—161no—
509C22Dic3×D811yescharacteristic
510C22#515011yescharacteristic
511C22#515211yescharacteristic
512C22#515311yescharacteristic
513C22#515611yescharacteristic
514C22#515811yescharacteristic
515C22#522011yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—122no—
523C4—122no—
524C4—241no—
525C4—241no—
526C4—481no—
527C3#927711yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—82no—
535C2—161no—
536C2#7184811yescharacteristic
537C2#7203711yescharacteristic
538C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3186C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#3186D3
hypocenterC3#9277
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
2. #82638C2semidirect
3. #71686C2semidirect
4. #71675C2semidirect
5. C2×#4537C2semidirect
6. #67678C2semidirect
7. #67675C2semidirect
13. C3×#480C4semidirect
16. C3×#447C4semidirect
41. C22×C3×D4C2×C4semidirect
56. C2×C3×C4⋊C4C2×C4semidirect
77. C3×C22⋊C8C2×C4semidirect
93. #480Dic3semidirect
96. #447Dic3semidirect
173. #217D8semidirect
182. C22×D4C2×Dic3semidirect
202. C2×C4⋊C4C2×Dic3semidirect
222. C22⋊C8C2×Dic3semidirect
527. C3#9277semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#826382
#716862
#716752
C2×#45372
#676782
#676752
#92773