Finite Group Databasebeta

← #10702910

Subgroups — #10702910

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5006C211yesmaximal, characteristic
2#82213C211yesmaximal, characteristic
3C2×#5002C211yesmaximal, characteristic
4C3×#1740C211yesmaximal, characteristic, complemented
5#71580C211yesmaximal, characteristic, complemented
6#71579C211yesmaximal, characteristic, complemented
7C2×#4531C211yesmaximal, characteristic, complemented
8#11176—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16#5006—22no—
17#5006—22no—
18#5004—21no—
19#5002—22no—
20#5002—22no—
21#5002C414yescomplemented
22#5001—21no—
23C2×#810—21no—
24#4966—22no—
25C2×#809—21no—
26C2×#806—22no—
27C3×#342—22no—
28#4531—22no—
29#4529—22no—
30#4524—22no—
31#2896—31no—
32C2×#416—31no—
33C2×#410—31no—
34#1740D311yescharacteristic, complemented
35#1735—31no—
36C2×H(3,ℤ/4)—31no—
37#1729—31no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235C2×C412yes—
51C2×#235D412yes—
52C22×D12D411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6D411yescharacteristic
55C3×#117—21no—
56C3×#117—21no—
57C3×#117—21no—
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117C2×C412yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×#217—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926D412yes—
81C2×C4⋊Dic3—21no—
82C2×C4×Dic3—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C22×C3⋊C8—21no—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8C2×C412yes—
96C22×C3⋊C8D412yes—
97#810—24no—
98#810—24no—
99#809—24no—
100#809—24no—
101#806—28no—
102C22×#41—31no—
103C22×M16—31no—
104C23×C8—31no—
105C2×#117—31no—
106C2×#117C2×D311yescharacteristic
107C2×#117C2×D311yescharacteristic
108C2×#117C2×D311yescharacteristic
109#419—61no—
110#418—61no—
111#417—62no—
112#416—34no—
113#416—62no—
114#416—62no—
115C2×#103—61no—
116C2×#102—61no—
117#410—62no—
118#410—62no—
119C2×#99—62no—
120#342—22no—
121#340—62no—
122H(3,ℤ/4)—62no—
123#336—62no—
124C24×C3C2×D411yescharacteristic
125C23×D3—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—22no—
144C2×#67—22no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C22×Dic3—21no—
150C22×Dic3—41no—
151#235—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C22⋊C412yes—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6C22⋊C412yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—41no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—41no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—41no—
195C3×C22⋊C4—41no—
196C3×C22⋊C4—41no—
197C3×C42—22no—
198C3×C42—41no—
199C3×C42—41no—
200C3×C42—41no—
201C3×C42—41no—
202C3×C42—41no—
203#217—22no—
204#217—41no—
205C4⋊Dic3—22no—
206C4⋊Dic3—41no—
207C4×Dic3—22no—
208C4×Dic3—41no—
209#208—41no—
210#208—41no—
211#208—41no—
212#208—41no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—41no—
220C2×C3⋊C8—41no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×#41—32no—
224C2×#41—32no—
225C2×#41—61no—
226C2×#41—61no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C22×Q8—31no—
236C22×D4—31no—
237C22×D4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—61no—
241C2×M16—61no—
242C2×M16—61no—
243C2×M16—61no—
244C2×M16—61no—
245C2×M16—61no—
246C22×C8—32no—
247C22×C8—32no—
248C22×C8—61no—
249C22×C8—61no—
250C22×C8—61no—
251C22×C8—61no—
252C22×C8—61no—
253C22×C8—61no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—32no—
266#117—61no—
267#117—61no—
268#117—61no—
269#117C4×D312yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4D1212yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4#6712yes—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—61no—
282#103—64no—
283#103—64no—
284#102—64no—
285#102—64no—
286#99—68no—
287C23×C3—42no—
288C23×C3C4×D411yescharacteristic
289C23×C3#12011yescharacteristic
290C23×C3#12111yescharacteristic
291C23×C3#12111yescharacteristic
292C23×C3#12111yescharacteristic
293C23×C3#12311yescharacteristic
294C23×C3#12411yescharacteristic
295C22×D3—22no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—82no—
299C2×C3×C4—44no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—82no—
304C2×C3×C4—82no—
305C2×C3×C4C2×C22⋊C411yescharacteristic
306C2×C3×C4#11711yescharacteristic
307C2×C3×C4#11711yescharacteristic
308C2×C3×C4C4×D411yescharacteristic
309C2×C3×C4C4×D411yescharacteristic
310C2×C3×C4C4×D411yescharacteristic
311C2×C3×C4#12011yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12311yescharacteristic
318C2×C3×C4#12311yescharacteristic
319C2×C3×C4#12311yescharacteristic
320C2×C3×C4#12311yescharacteristic
321C2×C3×C4#12311yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12411yescharacteristic
327C2×C3×C4#12411yescharacteristic
328C2×C3×C4#12411yescharacteristic
329C2×C3×C4#12611yescharacteristic
330C2×C3×C4#12611yescharacteristic
331C2×C3×C4#12611yescharacteristic
332C2×C3×C4#12611yescharacteristic
333#67—42no—
334#67—42no—
335#67—42no—
336#67—42no—
337C2×Dic3—22no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—42no—
341C2×Dic3—82no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350Dic6—22no—
351Dic6—22no—
352Dic6—41no—
353Dic6—41no—
354C3⋊C8—42no—
355C3⋊C8—42no—
356C24—61no—
357C24C2×#6711yescharacteristic
358#41—62no—
359#41—62no—
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—121no—
367#41—121no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374C2×Q8—32no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×D4—32no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—121no—
389C2×D4—121no—
390C2×D4—121no—
391C2×D4—121no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—121no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4C2×C4×D311yescharacteristic
404C22×C4C2×D1211yescharacteristic
405C22×C4#23511yescharacteristic
406C22×C4#23511yescharacteristic
407C22×C4#23511yescharacteristic
408C22×C4#23511yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4#23711yescharacteristic
414C22×C4#23711yescharacteristic
415C22×C4#23911yescharacteristic
416C22×C4#23911yescharacteristic
417M16—121no—
418M16—121no—
419M16—121no—
420M16—121no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—121no—
428C2×C8—121no—
429C2×C8—121no—
430C2×C8—121no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—62no—
438C4⋊C4—121no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—62no—
446C22⋊C4—121no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—62no—
454C42—121no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3#38611yescharacteristic
463C22×C3#38811yescharacteristic
464C22×C3#38811yescharacteristic
465C22×C3#39411yescharacteristic
466C22×C3#39411yescharacteristic
467C22×C3#39411yescharacteristic
468C22×C3#39711yescharacteristic
469C2×D3—42no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4—82no—
474C3×C4—82no—
475C3×C4—82no—
476C3×C4#38111yescharacteristic
477C3×C4#38611yescharacteristic
478C3×C4#38811yescharacteristic
479C3×C4#38811yescharacteristic
480C3×C4#39411yescharacteristic
481C3×C4#39711yescharacteristic
482C3×C4#39711yescharacteristic
483C3×C4#39711yescharacteristic
484Dic3—42no—
485Dic3—82no—
486C23—42no—
487C23—62no—
488C23—121no—
489C23—121no—
490C23—242no—
491C23C4×#6711yescharacteristic
492C23#92911yescharacteristic
493C23#93011yescharacteristic
494C23#93711yescharacteristic
495C23#93811yescharacteristic
496C23#93911yescharacteristic
497C23#94711yescharacteristic
498Q8—62no—
499Q8—62no—
500Q8—121no—
501Q8—121no—
502D4—62no—
503D4—62no—
504D4—122no—
505D4—122no—
506D4—122no—
507D4—122no—
508D4—121no—
509D4—121no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—62no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—242no—
525C2×C4#87211yescharacteristic
526C2×C4C4×D1211yescharacteristic
527C2×C4#87511yescharacteristic
528C2×C4#87611yescharacteristic
529C2×C4D3×C22⋊C411yescharacteristic
530C2×C4#67⋊C411yescharacteristic
531C2×C4#88211yescharacteristic
532C2×C4#88311yescharacteristic
533C2×C4#88311yescharacteristic
534C2×C4#88311yescharacteristic
535C2×C4#88411yescharacteristic
536C2×C4#88411yescharacteristic
537C2×C4#88411yescharacteristic
538C2×C4#88511yescharacteristic
539C2×C4#88511yescharacteristic
540C2×C4#88611yescharacteristic
541C2×C4#89211yescharacteristic
542C2×C4#89311yescharacteristic
543C2×C4#89411yescharacteristic
544C2×C4#89411yescharacteristic
545C2×C4#89411yescharacteristic
546C2×C4#89411yescharacteristic
547C2×C4#89411yescharacteristic
548C2×C4#89511yescharacteristic
549C2×C4#89511yescharacteristic
550C2×C4#89811yescharacteristic
551C2×C4#89811yescharacteristic
552C2×C4#89811yescharacteristic
553C8—122no—
554C8—122no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3#144011yescharacteristic
562C2×C3#205312yes—
563D3—82no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—122no—
572C22—242no—
573C22—242no—
574C22—242no—
575C22#494011yescharacteristic
576C22#495611yescharacteristic
577C22#495911yescharacteristic
578C22#496111yescharacteristic
579C22#499011yescharacteristic
580C22#499111yescharacteristic
581C22#499211yescharacteristic
582C4—82no—
583C4—82no—
584C4—82no—
585C4—122no—
586C4—242no—
587C4#466411yescharacteristic
588C4#466511yescharacteristic
589C4#466911yescharacteristic
590C4#467011yescharacteristic
591C4#467211yescharacteristic
592C4#467411yescharacteristic
593C4#467511yescharacteristic
594C4#467511yescharacteristic
595C3#1117611yescharacteristic, complemented, Sylow
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—242no—
603C2#6942911yescharacteristic
604C2#7004312yes—
subgroupG/H
centerC2×C4#885
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1740C2
socleC22×C3#394
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1740D3
hypocenterC3#11176
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C3×#1740C2semidirect
5. #71580C2semidirect
6. #71579C2semidirect
7. C2×#4531C2semidirect
21. #5002C4semidirect
34. #1740D3semidirect
595. C3#11176semidirect

Maximal subgroups

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

subgroupindex
C2×#50062
#822132
C2×#50022
C3×#17402
#715802
#715792
C2×#45312
#111763