Finite Group Databasebeta

← #10698466

Subgroups — #10698466

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2618C211yesmaximal, characteristic
2#71677C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic
4#71565C211yesmaximal, characteristic, complemented
5C2×#4528C211yesmaximal, characteristic, complemented
6#69505C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#10396—31nomaximal, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#398—21no—
14C3×#396—21no—
15C3×#396—21no—
16C3×#394—21no—
17C3×#386—21no—
18C3×#380C2211yescharacteristic
19C2×#832—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#808—21no—
26#4537—22no—
27#4529—21no—
28#4529—21no—
29#4528—22no—
30#2618D311yescharacteristic
31#1874—31no—
32#1853—31no—
33C2×#327—31no—
34C2×#324—31no—
35#1441—31no—
36#1294—31no—
37C22×C3×D4—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3—21no—
44C3×#123—42no—
45C3×#123—42no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—42no—
70C22×C3⋊C8—21no—
71C22×C3⋊C8—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75#832—42no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—42no—
79C3⋊#95—42no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#808—42no—
85C2×#123—21no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4—31no—
88C22×C4⋊C4C2×D311yescharacteristic
89C2×#106—61no—
90C2×#102—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93#398—21no—
94#396—21no—
95#396—21no—
96#394—21no—
97#386—21no—
98#380C2×D311yescharacteristic
99C2×#95—31no—
100C2×#95—61no—
101C2×#95—61no—
102#340—61no—
103#340—61no—
104#327—62no—
105#324—62no—
106C24×C3—21no—
107C24×C3C2×D411yescharacteristic
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4SD1612yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C22⋊C4—42no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—81no—
172C3×C22⋊C4—81no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C22×D4—21no—
184C22×D4#6711yescharacteristic
185C23×C4—61no—
186C23×C4—61no—
187C23×C4#6711yescharacteristic
188C23×C4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—61no—
191C22×C8—61no—
192#123—42no—
193#123—42no—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—122no—
209C2×C4⋊C4#6711yescharacteristic
210C2×C4⋊C4#6711yescharacteristic
211C2×C4⋊C4#6711yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218#106—122no—
219#102—122no—
220C22⋊C8—32no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—61no—
228#95—61no—
229#95—122no—
230#95—122no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3#12011yescharacteristic
243C23×C3#12111yescharacteristic
244C23×C3#12311yescharacteristic
245C23×C3C2×SD1611yescharacteristic
246C23×C3#13511yescharacteristic
247C23×C3AGL(1,ℤ/8)11yescharacteristic
248C23×C3AGL(1,ℤ/8)11yescharacteristic
249C3×D4—42no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C3×D4—81no—
254C3×D4—81no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
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—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—81no—
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12311yescharacteristic
287C2×C3×C4#12411yescharacteristic
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—81no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299C2×Dic3—81no—
300C2×Dic3—81no—
301C2×Dic3—81no—
302C3⋊C8—42no—
303C3⋊C8—81no—
304C24—21no—
305C24C2×#6711yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
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—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4#21412yes—
347C22×C4D3×D411yescharacteristic
348C22×C4#23711yescharacteristic
349C22×C4#23711yescharacteristic
350C22×C4#23711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6711yescharacteristic
353C2×C8—62no—
354C2×C8—121no—
355C2×C8—121no—
356C2×C8—121no—
357C2×C8—121no—
358C2×C8—121no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3#39411yescharacteristic
401C22×C3#44911yescharacteristic
402C22×C3#45011yescharacteristic
403C22×C3#45411yescharacteristic
404C22×C3#45611yescharacteristic
405C22×C3B(4,𝔽2)11yescharacteristic
406C22×C3#45811yescharacteristic
407C22×C3#46511yescharacteristic
408C22×C3#46911yescharacteristic
409C22×C3#48111yescharacteristic
410C22×C3#48211yescharacteristic
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—42no—
416C3×C4—81no—
417C3×C4—81no—
418C3×C4—81no—
419C3×C4—81no—
420Dic3—82no—
421Dic3—82no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23#93211yescharacteristic
434C23C2×#21411yescharacteristic
435C23#93511yescharacteristic
436C23#93911yescharacteristic
437C23#94911yescharacteristic
438C23#95011yescharacteristic
439C23#95311yescharacteristic
440D4—42no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—241no—
482C2×C4—241no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4#93511yescharacteristic
490C2×C4#93611yescharacteristic
491C2×C4#93811yescharacteristic
492C2×C4#93911yescharacteristic
493C8—122no—
494C8—241no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—161no—
500C2×C3#159711yescharacteristic
501C2×C3#160411yescharacteristic
502C2×C3#160911yescharacteristic
503C2×C3#161611yescharacteristic
504C2×C3#203411yescharacteristic
505C2×C3#203811yescharacteristic
506C2×C3#204411yescharacteristic
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22#503411yescharacteristic
527C22#503511yescharacteristic
528C22#503911yescharacteristic
529C22#504111yescharacteristic
530C22#521811yescharacteristic
531C22#522211yescharacteristic
532C22#523811yescharacteristic
533C22#524311yescharacteristic
534C22#524411yescharacteristic
535C22#524511yescharacteristic
536C22#524611yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—42no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—242no—
547C4—242no—
548C3#1039611yescharacteristic, complemented, Sylow
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—161no—
554C2#7168011yescharacteristic
555C2#7185011yescharacteristic
556C2#7186311yescharacteristic
557C2#7187511yescharacteristic
558C2#7188011yescharacteristic
559C2#7188611yescharacteristic
560C2#7188811yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2618C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2618D3
hypocenterC3#10396
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
4. #71565C2semidirect
5. C2×#4528C2semidirect
6. #69505C2semidirect
7. #67624C2semidirect
548. C3#10396semidirect

Maximal subgroups

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

subgroupindex
C3×#26182
#716772
C2×#45372
#715652
C2×#45282
#695052
#676242
#103963