Finite Group Databasebeta

← #10799081

Subgroups — #10799081

Order 768.

1518 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C22×C3×#103C211yesmaximal, characteristic, complemented
3#84062C211yesmaximal, characteristic, complemented
4C2×#5113C211yesmaximal, characteristic
5C2×#5101C211yesmaximal, characteristic, complemented
6#82186C211yesmaximal, characteristic
7C2×#4964C211yesmaximal, characteristic, complemented
8#70260C212yesmaximal, complemented
9#70259C212yesmaximal, complemented
10#70258C212yesmaximal, complemented
11#70257C212yesmaximal, complemented
12#21234—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×#216C2212yes—
18C2×#942—24no—
19C2×#942—24no—
20C2×#942C418yescomplemented
21C22×#215C2212yes—
22C2×#926C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C22×#208C2211yescharacteristic
25C2×C3×#103—24no—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28#5229—22no—
29#5229C2212yes—
30#5229C2212yes—
31#5213—22no—
32#5211—22no—
33#5113—24no—
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#5101—24no—
39C2×#815C2212yes—
40C2×C8×Dic3C2212yes—
41#4988C2212yes—
42#4987—22no—
43#4973C2212yes—
44#4964—24no—
45C2×#808C2212yes—
46C2×#807C2212yes—
47C22×#137—31no—
48C2×#428—31no—
49C2×#416—31no—
50C22×#103D311yescharacteristic, complemented
51C2×#405—31no—
52#2311—31no—
53#2310—31no—
54#1973—32no—
55#1971—32no—
56#1969—32no—
57#1968—32no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8C2×C414yes—
65C22×C3×Q8C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C2×#235—22no—
74C2×#235—22no—
75C2×#235C2×C414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8D414yes—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2312yes—
88C2×#217—21no—
89C2×#217D412yes—
90C2×#216—24no—
91C2×#216—24no—
92C2×#216C2×C418yescomplemented
93#942—232no—
94C2×#215—24no—
95C2×#215—24no—
96C2×#215C2×C418yescomplemented
97C2×(C2×D3)⋊C4—24no—
98#926—22no—
99#926—22no—
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3D412yes—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2312yes—
107C2×#208C2×C414yes—
108C2×#208D414yes—
109C2×#208D414yes—
110C22×C3⋊C8C2312yes—
111C3×#103—24no—
112C3×#103—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116#821—22no—
117#821—22no—
118#816—22no—
119#816—22no—
120#815—24no—
121C8×Dic3—24no—
122#808—24no—
123#807—24no—
124C22×#41—31no—
125C23×Q8C2×D311yescharacteristic
126C2×#137—38no—
127C2×#137—64no—
128C2×#137—64no—
129C22×Q16—32no—
130C22×SD16—32no—
131C22×M16—31no—
132C23×C8C2×D311yescharacteristic
133C2×#117—31no—
134C2×#117—31no—
135C22×C4⋊C4C2×D311yescharacteristic
136#428—64no—
137C2×#107—32no—
138C2×#106—32no—
139#416—64no—
140C2×#103—24no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103C2×D312yes—
144C2×#103C2×D312yes—
145C2×#102—31no—
146C2×#102—31no—
147#405—64no—
148C2×#97—32no—
149C2×C4×C8—32no—
150#391—32no—
151#391—32no—
152#391—32no—
153#391—62no—
154#390—32no—
155#386—62no—
156#381—62no—
157#380—62no—
158C24×C3C2×D411yescharacteristic
159C23×D3—21no—
160C2×C3×Q8—24no—
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—28no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—42no—
169C2×C3×Q8—44no—
170C2×C3×Q8—44no—
171C2×C3×Q8—42no—
172C2×C3×Q8C22×C414yes—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—42no—
177C22×C3×C4—42no—
178C22×C3×C4—42no—
179C22×C3×C4—82no—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—24no—
189C2×#67—42no—
190C2×#67—42no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D414yes—
195#235—28no—
196#235—28no—
197#235—44no—
198#235—44no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12C22×C412yes—
205C2×C4×D3—24no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6C22×C412yes—
214C2×C3×C8—28no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8C2×D412yes—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224#217—24no—
225#217—42no—
226#216—216no—
227#215—216no—
228(C2×D3)⋊C4—84no—
229C4⋊Dic3—24no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4×Dic3—22no—
235C4×Dic3—22no—
236C4×Dic3—42no—
237#208C22⋊C4116yescomplemented
238C2×C3⋊C8C22×C414yes—
239C2×C3⋊C8C2×D414yes—
240C2×C3⋊C8C2×D414yes—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—31no—
254C22×Q8C4×D314yes—
255C22×Q8C22×D312yes—
256C22×D4—31no—
257C22×D4—61no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—61no—
261C23×C4C22×D311yescharacteristic
262#137—632no—
263C2×Q16—38no—
264C2×Q16—64no—
265C2×Q16—64no—
266C2×SD16—38no—
267C2×SD16—64no—
268C2×SD16—64no—
269C2×M16—34no—
270C2×M16—34no—
271C2×M16—34no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—32no—
275C22×C8#6714yes—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278#117—62no—
279#117—62no—
280#117—62no—
281#117—62no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4C22×D312yes—
289C2×C22⋊C4—32no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—64no—
292C2×C42—32no—
293C2×C42—61no—
294#107—64no—
295#106—64no—
296#103—24no—
297#103—24no—
298#103—62no—
299#103—62no—
300#102—62no—
301#102—62no—
302#97—64no—
303C4×C8—64no—
304#95—64no—
305#95—64no—
306#95—64no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3C4×D412yes—
310C23×C3#12112yes—
311C23×C3#12712yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—42no—
317C22×D3—42no—
318C3×Q8—24no—
319C3×Q8—24no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—48no—
323C2×C3×C4—24no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—48no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—84no—
330C2×C3×C4C2×C22⋊C414yes—
331C2×C3×C4C4×D412yes—
332C2×C3×C4C4×D414yes—
333C2×C3×C4#12112yes—
334C2×C3×C4#12114yes—
335C2×C3×C4#12414yes—
336C2×C3×C4#12712yes—
337C2×C3×C4C23×C411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343#67—48no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354D12—22no—
355D12—22no—
356D12—41no—
357D12—41no—
358C4×D3—48no—
359Dic6—22no—
360Dic6—22no—
361Dic6—41no—
362Dic6—41no—
363C3×C8—22no—
364C3×C8—22no—
365C3⋊C8C2×C22⋊C418yes—
366C24—61no—
367C24D3×D411yescharacteristic
368#41—68no—
369#41—68no—
370#41—124no—
371#41—124no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—32no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8C2×C4×D314yes—
391C2×D4—32no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—122no—
399C2×D4—122no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—34no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—61no—
408C22×C4—64no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—82no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4C2×C4×D312yes—
415C22×C4#23512yes—
416C22×C4D3×D412yes—
417C22×C4D3×D412yes—
418C22×C4D3×D411yescharacteristic
419C22×C4#23712yes—
420C22×C4C2×#6712yes—
421C22×C4C23×D311yescharacteristic
422Q16—616no—
423SD16—616no—
424M16—316no—
425C2×C8—28no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—34no—
431C2×C8—34no—
432C2×C8—34no—
433C2×C8C2×#6712yes—
434C2×C8C2×#6712yes—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—64no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C22⋊C4—64no—
444C22⋊C4—122no—
445C22⋊C4—244no—
446C42—62no—
447C42—62no—
448C42—122no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3#39014yes—
454C22×C3C2×C4×D411yescharacteristic
455C22×C3C2×#12111yescharacteristic
456C22×C3C2×#12711yescharacteristic
457C22×C3#57712yes—
458C22×C3#57812yes—
459C2×D3—44no—
460C2×D3—42no—
461C2×D3—81no—
462C2×D3—81no—
463C2×D3—82no—
464C3×C4—44no—
465C3×C4—82no—
466C3×C4#39014yes—
467C3×C4C22×C22⋊C411yescharacteristic
468C3×C4C2×C4×D411yescharacteristic
469C3×C4C2×#12111yescharacteristic
470C3×C4C2×#12411yescharacteristic
471Dic3—42no—
472Dic3—44no—
473Dic3—81no—
474Dic3—81no—
475C23—22no—
476C23—22no—
477C23—62no—
478C23—121no—
479C23—122no—
480C23—122no—
481C23—121no—
482C23#67⋊C412yes—
483C23#88412yes—
484C23#94012yes—
485C23C2×D3×D411yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491Q8—62no—
492Q8—62no—
493Q8—121no—
494Q8—121no—
495D4—62no—
496D4—62no—
497D4—128no—
498D4—121no—
499D4—121no—
500C2×C4—24no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—84no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—128no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4D3×C22⋊C414yes—
519C2×C4#67⋊C412yes—
520C2×C4#88412yes—
521C2×C4#88514yes—
522C2×C4C4×#6714yes—
523C2×C4#93914yes—
524C2×C4#94012yes—
525C2×C4C22×C4×D311yescharacteristic
526C2×C4C2×#23511yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×#23711yescharacteristic
530C2×C4C22×#6711yescharacteristic
531C8—22no—
532C8—22no—
533C8—38no—
534C2×C3—24no—
535C2×C3C2×#39011yescharacteristic
536C2×C3#294712yes—
537C2×C3#306312yes—
538C2×C3#315411yescharacteristic
539C2×C3#315611yescharacteristic
540D3—82no—
541C22—22no—
542C22—22no—
543C22—24no—
544C22—24no—
545C22—122no—
546C22—124no—
547C22—241no—
548C22—241no—
549C22—242no—
550C22#495714yes—
551C22C2×#67⋊C411yescharacteristic
552C22C2×#88411yescharacteristic
553C22#576212yes—
554C22#576612yes—
555C22C2×#94011yescharacteristic
556C4—44no—
557C4—82no—
558C4—122no—
559C4—124no—
560C4—241no—
561C4—241no—
562C4#495714yes—
563C4C2×D3×C22⋊C411yescharacteristic
564C4C2×#88511yescharacteristic
565C4C2×C4×#6711yescharacteristic
566C4C2×#93911yescharacteristic
567C3#2123411yescharacteristic, complemented, Sylow
568C2—24no—
569C2—242no—
570C2#7877412yes—
571C2#7880012yes—
572C2C2×#495711yescharacteristic
573C2#8385411yescharacteristic
574C2#8388111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#103C2
socleC23×C3C22×D4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC22×#103D3
hypocenterC3#21234
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
2. C22×C3×#103C2semidirect
3. #84062C2semidirect
5. C2×#5101C2semidirect
7. C2×#4964C2semidirect
8. #70260C2semidirect
9. #70259C2semidirect
10. #70258C2semidirect
11. #70257C2semidirect
20. C2×#942C4semidirect
50. C22×#103D3semidirect
92. C2×#216C2×C4semidirect
96. C2×#215C2×C4semidirect
237. #208C22⋊C4semidirect
567. C3#21234semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C22×C3×#1032
#840622
C2×#51132
C2×#51012
#821862
C2×#49642
#702602
#702592
#702582
#702572
#212343