Finite Group Databasebeta

← #10804473

Subgroups — #10804473

Order 768.

1310 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
1C2×#5668C211yesmaximal, characteristic
2#79038C211yesmaximal, characteristic, complemented
3#78517C211yesmaximal, characteristic, complemented
4C3×#1967C211yesmaximal, characteristic, complemented
5#71727C211yesmaximal, characteristic, complemented
6#70523C211yesmaximal, characteristic, complemented
7#70114C212yesmaximal, complemented
8#70098C212yesmaximal, complemented
9#70080C212yesmaximal, complemented
10#70064C212yesmaximal, complemented
11#69878C211yesmaximal, characteristic, complemented
12#21800—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×#237C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×#944C2211yescharacteristic
17C2×#940C2211yescharacteristic
18#5668—22no—
19#5668—22no—
20#5668—22no—
21#5668—22no—
22C2×#884C414yes—
23C2×#67⋊C4C414yes—
24C2×#875C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×#101C2212yes—
28C2×C3×#100C2212yes—
29C3×C4×M16C2211yescharacteristic
30C2×#836C2212yes—
31C2×#833C2212yes—
32C2×#824C2212yes—
33C2×#824C2212yes—
34C2×#823C2212yes—
35C2×#823C2212yes—
36#5126C2212yes—
37#5126C2212yes—
38#5118C2212yes—
39Dic3×M16C2212yes—
40#4924C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4835—22no—
43#4833—22no—
44#4705—22no—
45#4685—22no—
46C2×#532—31no—
47#2920—31no—
48#2920—31no—
49#1967—31no—
50#1967—31no—
51#1967—31no—
52#1967D311yescharacteristic, complemented
53#1966—32no—
54#1965—32no—
55#1785—32no—
56#1785—32no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C22×#67C2×C412yes—
60C22×#67C2×C414yes—
61C23×Dic3C2×C412yes—
62C2×#239—22no—
63C2×#239—22no—
64C2×#239—42no—
65C2×#239—42no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—42no—
69C2×#237—42no—
70C22×D12C2×C412yes—
71C22×D12C2311yescharacteristic
72C22×C4×D3C2×C412yes—
73C22×C4×D3C2×C412yes—
74C22×C4×D3C2311yescharacteristic
75C22×Dic6C2311yescharacteristic
76C2×C3×M16C2312yes—
77C2×C3×M16C2312yes—
78C3×#124—21no—
79C3×#124—21no—
80C3×#124—22no—
81C2×C3×C22⋊C4C2×C414yes—
82C2×C3×C42C2311yescharacteristic
83#944—21no—
84#944—22no—
85#944—21no—
86#940—21no—
87#940—22no—
88#940—21no—
89C2×(C2×D3)⋊C4C2×C414yes—
90C2×Dic3⋊C4C2×C414yes—
91C2×C4×Dic3C2×C412yes—
92C2×C4×Dic3C2311yescharacteristic
93C2×#208C2312yes—
94C2×#208C2312yes—
95#884—24no—
96#884—24no—
97#884—24no—
98#884—24no—
99#67⋊C4—24no—
100#67⋊C4—24no—
101#67⋊C4—24no—
102#67⋊C4—24no—
103#875—21no—
104#875—21no—
105#875—22no—
106C42×D3D412yes—
107C42×D3D412yes—
108C42×D3D412yes—
109C42×D3D412yes—
110C3×C4⋊C8—22no—
111C3×#101—42no—
112C3×#100—42no—
113C3×#97—21no—
114C3×C4×C8—21no—
115#836—42no—
116#833—42no—
117#824—42no—
118#824—42no—
119#823—42no—
120#823—42no—
121(C2×D3)⋊C8—24no—
122(C2×D3)⋊C8—24no—
123#815—22no—
124Dic3⋊C8—24no—
125C8×Dic3—22no—
126C4⋊(C3⋊C8)—22no—
127#803—21no—
128C4×C3⋊C8—21no—
129C22×#41—31no—
130C22×#41—31no—
131#532—62no—
132#532—62no—
133#532—62no—
134#532—62no—
135C2×C4⋊Q8—31no—
136C2×#127—31no—
137C2×#124—31no—
138C2×#124C2×D311yescharacteristic
139C2×#121—34no—
140C2×C4×D4—34no—
141C22×C42—31no—
142#432—62no—
143#432—62no—
144#431—62no—
145#431—62no—
146#423—32no—
147#423—31no—
148#423C2×D311yescharacteristic
149C2×#101—32no—
150C2×#101—32no—
151C2×#101—32no—
152C2×#101C2×D312yes—
153C2×#100—32no—
154C2×#100—32no—
155C2×#100—32no—
156C2×#100C2×D312yes—
157#407—32no—
158#407—32no—
159C4×M16—32no—
160C4×M16—31no—
161C4×M16C2×D311yescharacteristic
162C24×C3C22×C412yes—
163C23×D3C22×C412yes—
164C23×D3C22×C411yescharacteristic
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C22×C3×C4C22×C412yes—
174C22×C3×C4C22×C412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—24no—
179C2×#67—44no—
180C2×#67—48no—
181C22×Dic3—24no—
182C22×Dic3—44no—
183C22×Dic3C22×C412yes—
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C412yes—
186#239—44no—
187#239—44no—
188#239—48no—
189#237—44no—
190#237—44no—
191#237—48no—
192C2×D12—24no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—22no—
196C2×D12—22no—
197C2×D12—44no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—24no—
201C2×C4×D3—44no—
202C2×C4×D3—44no—
203C2×C4×D3C22⋊C412yes—
204C2×C4×D3C22⋊C412yes—
205C2×C4×D3C22⋊C412yes—
206C2×C4×D3C22⋊C412yes—
207C2×C4×D3C2×D412yes—
208C2×C4×D3C2×D412yes—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C3×M16—22no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D412yes—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228C4×Dic3C22⋊C412yes—
229C4×Dic3C22⋊C412yes—
230C4×Dic3C22⋊C412yes—
231C4×Dic3C22⋊C412yes—
232C4×Dic3C2×D411yescharacteristic
233C4×Dic3C2×D412yes—
234C4×Dic3C2×D411yescharacteristic
235#208—22no—
236#208—22no—
237#208—22no—
238#208—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×#41—62no—
242C2×#41—62no—
243C2×#41—62no—
244C2×#41—62no—
245C2×#41—122no—
246C2×#41—122no—
247C2×#41—122no—
248C2×#41—122no—
249C22×Q8—31no—
250C22×Q8C22×D311yescharacteristic
251C22×D4—34no—
252C22×D4—32no—
253C22×D4—32no—
254C22×D4—31no—
255C22×D4C22×D311yescharacteristic
256C23×C4—31no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—32no—
260C2×M16—32no—
261C2×M16—32no—
262C2×M16C22×D312yes—
263C2×M16C22×D312yes—
264C4⋊Q8—61no—
265C4⋊Q8—61no—
266C4⋊Q8—62no—
267#127—61no—
268#127—61no—
269#127—62no—
270#124—21no—
271#124—21no—
272#124—22no—
273#124—61no—
274#124—61no—
275#124—62no—
276#121—64no—
277#121—64no—
278#121—64no—
279#121—64no—
280C4×D4—64no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—64no—
284C2×C4⋊C4—34no—
285C2×C22⋊C4—34no—
286C2×C22⋊C4C4×D314yes—
287C2×C42—32no—
288C2×C42—31no—
289C2×C42—32no—
290C2×C42—32no—
291C2×C42—32no—
292C2×C42—32no—
293C2×C42C22×D311yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—64no—
296C4⋊C8—62no—
297#101—42no—
298#101—122no—
299#101—122no—
300#101—122no—
301#100—42no—
302#100—122no—
303#100—122no—
304#100—122no—
305C22⋊C8—64no—
306C22⋊C8—64no—
307#97—21no—
308#97—62no—
309#97—61no—
310C4×C8—21no—
311C4×C8—62no—
312C4×C8—61no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—44no—
316C23×C3C23×C411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—24no—
320C22×D3—42no—
321C22×D3—44no—
322C22×D3C2×C22⋊C412yes—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—42no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—42no—
332C2×C3×C4C2×C22⋊C412yes—
333C2×C3×C4C2×C22⋊C412yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4H(5,𝔽2)12yes—
337C2×C3×C4H(5,𝔽2)12yes—
338#67—48no—
339#67—48no—
340#67—48no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—22no—
344C2×Dic3—24no—
345C2×Dic3—48no—
346C2×Dic3C2×C22⋊C412yes—
347C2×Dic3C2×C22⋊C412yes—
348C2×Dic3C2×C22⋊C412yes—
349D12—48no—
350D12—41no—
351D12—41no—
352D12—42no—
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—24no—
356C4×D3—24no—
357C4×D3—24no—
358C4×D3—42no—
359C4×D3—42no—
360C4×D3—48no—
361Dic6—41no—
362Dic6—41no—
363Dic6—42no—
364C3×C8—42no—
365C3×C8—42no—
366C3⋊C8—42no—
367C3⋊C8—42no—
368C24—31no—
369C24—32no—
370C24C2×C4×D312yes—
371#41—124no—
372#41—124no—
373#41—128no—
374#41—124no—
375#41—124no—
376#41—128no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—62no—
396C2×D4—62no—
397C2×D4—124no—
398C2×D4—124no—
399C2×D4—128no—
400C22×C4—32no—
401C22×C4—31no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4C2×C4×D312yes—
417C22×C4C2×C4×D312yes—
418C22×C4C23×D311yescharacteristic
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—62no—
424M16—62no—
425M16—62no—
426M16—62no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—62no—
430C2×C8—62no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—64no—
436C22⋊C4—64no—
437C42—32no—
438C42—32no—
439C42—32no—
440C42—32no—
441C42—31no—
442C42—31no—
443C42—32no—
444C42D3×D411yescharacteristic
445C42D3×D411yescharacteristic
446C42D3×D412yes—
447C22×C3—22no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—84no—
451C22×C3C22×C22⋊C411yescharacteristic
452C22×C3#51812yes—
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—44no—
457C2×D3—42no—
458C2×D3—44no—
459C2×D3—84no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—42no—
463C3×C4#53511yescharacteristic
464C3×C4#53511yescharacteristic
465C3×C4#53512yes—
466Dic3—22no—
467Dic3—22no—
468Dic3—42no—
469Dic3#41314yes—
470C23—22no—
471C23—24no—
472C23—32no—
473C23—44no—
474C23—62no—
475C23—62no—
476C23—64no—
477C23—122no—
478C23—124no—
479C23C22×C4×D311yescharacteristic
480Q8—41no—
481Q8—41no—
482Q8—42no—
483Q8—121no—
484Q8—121no—
485Q8—122no—
486D4—41no—
487D4—41no—
488D4—42no—
489D4—128no—
490D4—128no—
491D4—128no—
492D4—128no—
493D4—121no—
494D4—121no—
495D4—122no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—32no—
501C2×C4—42no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—64no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—128no—
514C2×C4—128no—
515C2×C4D3×C22⋊C412yes—
516C2×C4D3×C22⋊C412yes—
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4#100412yes—
520C2×C4#100912yes—
521C8—42no—
522C8—42no—
523C8—122no—
524C8—122no—
525C2×C3—22no—
526C2×C3—82no—
527C2×C3#230911yescharacteristic
528C2×C3C2×#41312yes—
529D3—42no—
530D3—82no—
531C22—22no—
532C22—44no—
533C22—42no—
534C22—62no—
535C22—62no—
536C22—62no—
537C22—84no—
538C22—124no—
539C22—124no—
540C22—122no—
541C22—244no—
542C22C2×D3×C22⋊C411yescharacteristic
543C22#554512yes—
544C4—22no—
545C4—22no—
546C4—34no—
547C4—42no—
548C4—62no—
549C4—62no—
550C4—122no—
551C4#560712yes—
552C4#567411yescharacteristic
553C4#571511yescharacteristic
554C3#2180011yescharacteristic, complemented, Sylow
555C2—22no—
556C2—82no—
557C2—122no—
558C2—242no—
559C2#7642811yescharacteristic
560C2D3×#41312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1967C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5668C2
cosocleC22×C3×C4C24
hypercenter#1967D3
hypocenterC3#21800
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1967

Complements

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

NH ≅ G/Ntype
2. #79038C2semidirect
3. #78517C2semidirect
4. C3×#1967C2semidirect
5. #71727C2semidirect
6. #70523C2semidirect
7. #70114C2semidirect
8. #70098C2semidirect
9. #70080C2semidirect
10. #70064C2semidirect
11. #69878C2semidirect
52. #1967D3semidirect
554. C3#21800semidirect

Maximal subgroups

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

subgroupindex
C2×#56682
#790382
#785172
C3×#19672
#717272
#705232
#701142
#700982
#700802
#700642
#698782
#218003