Finite Group Databasebeta

← #10698530

Subgroups — #10698530

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73379C211yesmaximal, characteristic
2C3×#1444C211yesmaximal, characteristic, complemented
3#71560C211yesmaximal, characteristic
4#71559C211yesmaximal, characteristic, complemented
5C2×#4452C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7#67613C211yesmaximal, characteristic, complemented
8#10405—31nomaximal, Sylow
9C22×C3×C4⋊C4—21no—
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C2×C3×#95—21no—
14C2×C3×#95C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5213—21no—
17#5213—21no—
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#810—21no—
21C2×#809—21no—
22#4674—21no—
23#4673—21no—
24#4672—21no—
25#4659—21no—
26#4529—21no—
27#4529—21no—
28#4452—22no—
29#4451—22no—
30#2639—31no—
31#1874—31no—
32#1872—31no—
33C2×#328—31no—
34C2×#327—31no—
35#1444D311yescharacteristic, complemented
36#1280—31no—
37C23×C3×C4—21no—
38C23×C3×C4—21no—
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×Dic6D411yescharacteristic
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—42no—
48C2×C3×C4⋊C4—42no—
49C2×C3×C4⋊C4—42no—
50C2×C3×C4⋊C4—42no—
51C2×#217—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—42no—
74C3×#95D412yes—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#810—42no—
84#809—42no—
85C2×C22⋊Q8—31no—
86C2×#121—31no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×D311yescharacteristic
89C2×#103—61no—
90C2×#102—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93#400—61no—
94#397—61no—
95#394—61no—
96#393—61no—
97#386—61no—
98#386—61no—
99C2×#95—21no—
100C2×#95C2×D311yescharacteristic
101C2×#95C2×D311yescharacteristic
102#340—61no—
103#340—61no—
104#328—62no—
105#327—62no—
106C24×C3C2×D411yescharacteristic
107C23×D3—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—41no—
152C2×D12—41no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—48no—
162#217—41no—
163#217—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—81no—
167(C2×D3)⋊C4—81no—
168(C2×D3)⋊C4—81no—
169(C2×D3)⋊C4—81no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174Dic3⋊C4—41no—
175Dic3⋊C4—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C22×Q8—31no—
183C22×D4—31no—
184C22×D4—61no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4#6711yescharacteristic
188C23×C4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—61no—
191C22×C8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215#103—122no—
216#102—122no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—42no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95—61no—
229#95D1212yes—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3#12011yescharacteristic
235C23×C3#12111yescharacteristic
236C23×C3#12411yescharacteristic
237C23×C3#13511yescharacteristic
238C23×C3#13511yescharacteristic
239C23×C3AGL(1,ℤ/8)11yescharacteristic
240C23×C3#13711yescharacteristic
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4—81no—
279C2×C3×C4—81no—
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12311yescharacteristic
283C2×C3×C4#12311yescharacteristic
284#67—81no—
285#67—81no—
286#67—81no—
287#67—81no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297D12—42no—
298D12—81no—
299Dic6—42no—
300Dic6—81no—
301C3⋊C8—81no—
302C3⋊C8—81no—
303C24—61no—
304C24C2×#6711yescharacteristic
305C2×Q8—61no—
306C2×Q8—61no—
307C2×Q8—121no—
308C2×Q8—121no—
309C2×D4—61no—
310C2×D4—61no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4C2×D1211yescharacteristic
349C22×C4#23511yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4D3×D411yescharacteristic
352C22×C4#23911yescharacteristic
353C22×C4#23911yescharacteristic
354C2×C8—121no—
355C2×C8—121no—
356C2×C8—121no—
357C2×C8—121no—
358C2×C8—121no—
359C2×C8—121no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—48no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—241no—
376C22⋊C4—241no—
377C22⋊C4—241no—
378C22⋊C4—241no—
379C22×C3—21no—
380C22×C3—21no—
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#39411yescharacteristic
392C22×C3#44911yescharacteristic
393C22×C3#45211yescharacteristic
394C22×C3#45311yescharacteristic
395C22×C3#45611yescharacteristic
396C22×C3B(4,𝔽2)11yescharacteristic
397C22×C3#45811yescharacteristic
398C22×C3#46311yescharacteristic
399C22×C3#46411yescharacteristic
400C22×C3#48811yescharacteristic
401C22×C3#48911yescharacteristic
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—44no—
414C3×C4—81no—
415C3×C4—82no—
416C3×C4—81no—
417Dic3—81no—
418Dic3—81no—
419Dic3—81no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23#93011yescharacteristic
432C23#93211yescharacteristic
433C23#93711yescharacteristic
434C23#94211yescharacteristic
435C23#94711yescharacteristic
436C23#95011yescharacteristic
437C23#95011yescharacteristic
438Q8—122no—
439Q8—241no—
440D4—122no—
441D4—241no—
442D4—241no—
443D4—241no—
444D4—241no—
445D4—241no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4#89411yescharacteristic
488C2×C4#89411yescharacteristic
489C2×C4#89511yescharacteristic
490C2×C4#89511yescharacteristic
491C8—241no—
492C8—241no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3#160011yescharacteristic
498C2×C3#160311yescharacteristic
499C2×C3#160911yescharacteristic
500C2×C3#161811yescharacteristic
501C2×C3#203911yescharacteristic
502C2×C3#204211yescharacteristic
503C2×C3#204411yescharacteristic
504D3—161no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22#499111yescharacteristic
525C22#501611yescharacteristic
526C22#502811yescharacteristic
527C22#503211yescharacteristic
528C22#503711yescharacteristic
529C22#504711yescharacteristic
530C22#505311yescharacteristic
531C22#506811yescharacteristic
532C22#507711yescharacteristic
533C22#508911yescharacteristic
534C22#509511yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—44no—
540C4—81no—
541C4—82no—
542C4—81no—
543C4—241no—
544C4—241no—
545C4—241no—
546C3#1040511yescharacteristic, complemented, Sylow
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—481no—
552C2#6959311yescharacteristic
553C2#6959611yescharacteristic
554C2#6962811yescharacteristic
555C2#6969711yescharacteristic
556C2#6970511yescharacteristic
557C2#6971111yescharacteristic
558C2#6971511yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1444C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenter#1444D3
hypocenterC3#10405
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C3×#1444C2semidirect
4. #71559C2semidirect
5. C2×#4452C2semidirect
7. #67613C2semidirect
35. #1444D3semidirect
546. C3#10405semidirect

Maximal subgroups

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

subgroupindex
#733792
C3×#14442
#715602
#715592
C2×#44522
C2×#44512
#676132
#104053