Finite Group Databasebeta

← #10799083

Subgroups — #10799083

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×#909C211yesmaximal, characteristic
2C2×C3×#428C211yesmaximal, characteristic, complemented
3C2×#5122C211yesmaximal, characteristic, complemented
4#82230C211yesmaximal, characteristic, complemented
5C2×#5002C211yesmaximal, characteristic
6#82186C211yesmaximal, characteristic
7C22×#810C211yesmaximal, characteristic, complemented
8#70629C212yesmaximal, complemented
9#70627C212yesmaximal, complemented
10#70575C212yesmaximal, complemented
11#70574C212yesmaximal, complemented
12#21234—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C2×#909—24no—
21C2×#909—24no—
22C2×#909C418yescomplemented
23C22×Dic12C2212yes—
24C22×#204C2212yes—
25C3×#428—24no—
26C2×C3×#107C2212yes—
27C2×C3×#106C2212yes—
28#5213—22no—
29#5122—24no—
30C2×#815C2212yes—
31C2×C8×Dic3C2212yes—
32#5002—24no—
33#4988C2212yes—
34#4987—22no—
35#4973C2212yes—
36#4973C2212yes—
37#4973C2212yes—
38C2×#810—24no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2212yes—
42C2×#810C2212yes—
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4943—22no—
46#4929—22no—
47C22×#137—31no—
48C2×#428D311yescharacteristic, complemented
49C2×#416—31no—
50C22×#103—31no—
51C2×#405—31no—
52#2311—31no—
53#2310—31no—
54#1973—32no—
55#1971—32no—
56#1969—32no—
57#1968—32no—
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C23×Dic3—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235C2×C414yes—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C22×Dic6—22no—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—21no—
75C22×Dic6—22no—
76C22×Dic6C2×C414yes—
77C22×Dic6C2312yes—
78C22×Dic6C2311yescharacteristic
79C2×C3×M16C2×C414yes—
80C2×C3×M16D414yes—
81C2×C3×M16D414yes—
82C22×C3×C8C2312yes—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×#217D412yes—
94C2×(C2×D3)⋊C4—24no—
95#926—22no—
96#926—22no—
97C2×C4⋊Dic3D412yes—
98C2×C4×Dic3C2312yes—
99C22×C3⋊C8—22no—
100C22×C3⋊C8—22no—
101C22×C3⋊C8D414yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104#909—232no—
105C2×Dic12—24no—
106C2×Dic12—24no—
107C2×Dic12C2×C418yescomplemented
108C2×#204—24no—
109C2×#204—24no—
110C2×#204C2×C418yescomplemented
111C3×#107—24no—
112C3×#106—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116#815—24no—
117C8×Dic3—24no—
118#810—24no—
119#810—24no—
120#810—22no—
121#810—22no—
122#809—22no—
123#809—22no—
124C22×#41—31no—
125C23×Q8—31no—
126C2×#137—38no—
127C2×#137—64no—
128C2×#137—64no—
129C22×Q16—32no—
130C22×SD16—32no—
131C22×M16C2×D311yescharacteristic
132C23×C8—31no—
133C2×#117—31no—
134C2×#117C2×D311yescharacteristic
135C22×C4⋊C4C2×D311yescharacteristic
136#428—24no—
137C2×#107C2×D312yes—
138C2×#106C2×D312yes—
139#416—64no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—64no—
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—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—42no—
164C22×C3×C4—82no—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4112yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—42no—
175C2×#67—42no—
176C22×Dic3—24no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—42no—
180C22×Dic3—42no—
181C22×Dic3C2×D414yes—
182#235—28no—
183#235—28no—
184#235—44no—
185#235—44no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12C22×C412yes—
192C2×C4×D3—24no—
193C2×C4×D3—42no—
194C2×C4×D3—42no—
195C2×Dic6—28no—
196C2×Dic6—28no—
197C2×Dic6—22no—
198C2×Dic6—21no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—22no—
204C2×Dic6—21no—
205C2×Dic6—24no—
206C2×Dic6—22no—
207C2×Dic6—21no—
208C2×Dic6—44no—
209C2×Dic6—42no—
210C2×Dic6—44no—
211C2×Dic6—42no—
212C2×Dic6C22×C414yes—
213C2×Dic6C22×C412yes—
214C3×M16C22⋊C4116yescomplemented
215C2×C3×C8C22×C414yes—
216C2×C3×C8C2×D414yes—
217C2×C3×C8C2×D414yes—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C42—42no—
227#217—24no—
228(C2×D3)⋊C4—84no—
229C4⋊Dic3—24no—
230C4×Dic3—22no—
231C4×Dic3—22no—
232C2×C3⋊C8—28no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8C2×D412yes—
238C2×C3⋊C8C2×D412yes—
239Dic12—216no—
240#204—216no—
241C2×#41—34no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—62no—
247C22×Q8—34no—
248C22×Q8—32no—
249C22×Q8—31no—
250C22×Q8—61no—
251C22×Q8—62no—
252C22×Q8—62no—
253C22×Q8—62no—
254C22×Q8—62no—
255C22×Q8—62no—
256C22×D4—31no—
257C22×D4—61no—
258C23×C4—21no—
259C23×C4—61no—
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×M16C4×D314yes—
270C2×M16D1214yes—
271C2×M16#6714yes—
272C22×C8—34no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—62no—
276C22×C8—62no—
277C22×C8C22×D312yes—
278#117—22no—
279#117—22no—
280#117—62no—
281#117—62no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4C22×D312yes—
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—32no—
291C2×C22⋊C4—64no—
292C2×C42—21no—
293C2×C42—32no—
294#107—24no—
295#106—24no—
296#103—64no—
297#103—64no—
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—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—84no—
325C2×C3×C4C2×C22⋊C414yes—
326C2×C3×C4C4×D414yes—
327C2×C3×C4C4×D412yes—
328C2×C3×C4#12114yes—
329C2×C3×C4#12112yes—
330C2×C3×C4#12414yes—
331C2×C3×C4#12712yes—
332C2×C3×C4C23×C411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338#67—48no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—48no—
349D12—22no—
350D12—22no—
351D12—41no—
352D12—41no—
353C4×D3—48no—
354Dic6—24no—
355Dic6—22no—
356Dic6—24no—
357Dic6—22no—
358Dic6—42no—
359Dic6—41no—
360Dic6—42no—
361Dic6—41no—
362Dic6—48no—
363C3×C8C2×C22⋊C418yes—
364C3⋊C8—22no—
365C3⋊C8—22no—
366C24—61no—
367C24D3×D411yescharacteristic
368#41—68no—
369#41—68no—
370#41—124no—
371#41—124no—
372C2×Q8—34no—
373C2×Q8—32no—
374C2×Q8—64no—
375C2×Q8—62no—
376C2×Q8—61no—
377C2×Q8—68no—
378C2×Q8—68no—
379C2×Q8—62no—
380C2×Q8—61no—
381C2×Q8—62no—
382C2×Q8—61no—
383C2×Q8—62no—
384C2×Q8—61no—
385C2×Q8—62no—
386C2×Q8—61no—
387C2×Q8—124no—
388C2×Q8—122no—
389C2×Q8—122no—
390C2×Q8—124no—
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—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—34no—
404C22×C4—42no—
405C22×C4—64no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—64no—
409C22×C4—82no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4C2×C4×D312yes—
415C22×C4C2×D1212yes—
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D412yes—
418C22×C4#23912yes—
419C22×C4#23912yes—
420C22×C4C2×#6712yes—
421C22×C4C23×D311yescharacteristic
422Q16—616no—
423SD16—616no—
424M16(C2×D3)⋊C4116yescomplemented
425C2×C8—32no—
426C2×C8—32no—
427C2×C8—68no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8C2×C4×D314yes—
433C2×C8C2×D1214yes—
434C2×C8C2×#6714yes—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—64no—
443C22⋊C4—42no—
444C22⋊C4—64no—
445C22⋊C4—244no—
446C42—42no—
447C42—62no—
448C42—62no—
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—81no—
465C3×C4—82no—
466C3×C4—81no—
467C3×C4#39014yes—
468C3×C4C22×C22⋊C411yescharacteristic
469C3×C4C2×C4×D411yescharacteristic
470C3×C4C2×#12111yescharacteristic
471C3×C4C2×#12411yescharacteristic
472Dic3—44no—
473Dic3—42no—
474Dic3—44no—
475C23—22no—
476C23—22no—
477C23—62no—
478C23—121no—
479C23—122no—
480C23—122no—
481C23—121no—
482C23#89312yes—
483C23#89512yes—
484C23#94012yes—
485C23C2×D3×D411yescharacteristic
486Q8—64no—
487Q8—62no—
488Q8—64no—
489Q8—62no—
490Q8—128no—
491Q8—122no—
492Q8—121no—
493Q8—122no—
494Q8—121no—
495D4—62no—
496D4—62no—
497D4—128no—
498D4—121no—
499D4—121no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—84no—
512C2×C4—128no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—128no—
518C2×C4#89312yes—
519C2×C4#89314yes—
520C2×C4#89512yes—
521C2×C4#89514yes—
522C2×C4C2×(C2×D3)⋊C414yes—
523C2×C4#94012yes—
524C2×C4#94714yes—
525C2×C4C22×C4×D311yescharacteristic
526C2×C4C22×D1211yescharacteristic
527C2×C4C2×D3×D411yescharacteristic
528C2×C4C2×#23911yescharacteristic
529C2×C4C2×#23911yescharacteristic
530C2×C4C22×#6711yescharacteristic
531C8—62no—
532C8—62no—
533C8C2×(C2×D3)⋊C418yes—
534C2×C3—24no—
535C2×C3C2×#39011yescharacteristic
536C2×C3#294712yes—
537C2×C3#306312yes—
538C2×C3#315411yescharacteristic
539C2×C3#315611yescharacteristic
540D3—82no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—124no—
546C22—122no—
547C22—241no—
548C22—242no—
549C22—241no—
550C22#498814yes—
551C22C2×#89311yescharacteristic
552C22C2×#89511yescharacteristic
553C22#577512yes—
554C22#577912yes—
555C22C2×#94011yescharacteristic
556C4—81no—
557C4—82no—
558C4—81no—
559C4—124no—
560C4—122no—
561C4—124no—
562C4#498814yes—
563C4C2×#89311yescharacteristic
564C4C2×#89511yescharacteristic
565C4C22×(C2×D3)⋊C411yescharacteristic
566C4C2×#94711yescharacteristic
567C3#2123411yescharacteristic, complemented, Sylow
568C2—24no—
569C2—242no—
570C2#7920212yes—
571C2#7921412yes—
572C2C2×#498811yescharacteristic
573C2#8393911yescharacteristic
574C2#8397511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#428C2
socleC23×C3C22×D4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#428D3
hypocenterC3#21234
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C3×#428C2semidirect
3. C2×#5122C2semidirect
4. #82230C2semidirect
7. C22×#810C2semidirect
8. #70629C2semidirect
9. #70627C2semidirect
10. #70575C2semidirect
11. #70574C2semidirect
22. C2×#909C4semidirect
48. C2×#428D3semidirect
107. C2×Dic12C2×C4semidirect
110. C2×#204C2×C4semidirect
214. C3×M16C22⋊C4semidirect
424. M16(C2×D3)⋊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×#9092
C2×C3×#4282
C2×#51222
#822302
C2×#50022
#821862
C22×#8102
#706292
#706272
#705752
#705742
#212343