Finite Group Databasebeta

← #10811963

Subgroups — #10811963

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5559C211yesmaximal, characteristic
2#79046C212yesmaximal, complemented
3C3×#1800C211yesmaximal, characteristic, complemented
4C4×#833C211yesmaximal, characteristic, complemented
5#70523C212yesmaximal, complemented
6#70065C212yesmaximal, complemented
7#70064C212yesmaximal
8#68997C211yesmaximal, characteristic, complemented
9#68994C211yesmaximal, characteristic, complemented
10#68973C211yesmaximal, characteristic, complemented
11#68972C211yesmaximal, characteristic, complemented
12#22539—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#940C2211yescharacteristic
15C2×#936C2211yescharacteristic
16C2×#929C412yes—
17C2×C4×#67C412yes—
18#5559—22no—
19#5559—22no—
20#5559—22no—
21#5559—22no—
22C2×#893C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×#885C412yes—
25C2×#883C412yes—
26C2×#875C2211yescharacteristic
27C2×C4×Dic6C2211yescharacteristic
28C3×#423C2212yes—
29C2×C3×#100C2212yes—
30C2×#833C2212yes—
31C2×#824C2212yes—
32C2×#824C2212yes—
33C2×#823C2212yes—
34C2×#823C2212yes—
35#5118C2212yes—
36#5118C2212yes—
37Dic3×M16C2212yes—
38Dic3×M16C2212yes—
39C4×#208C2212yes—
40#4831—22no—
41#4830—22no—
42#4828—22no—
43#4828—22no—
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46#4530C2211yescharacteristic
47#4530C2211yescharacteristic
48C2×#559—31no—
49Q8×M16—32no—
50#1967—32no—
51#1967—32no—
52#1965—32no—
53#1801—31no—
54#1800—31no—
55#1800D311yescharacteristic, complemented
56C4×#101—31no—
57C4×#100—31no—
58C4×#100—31no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4C2×C412yes—
61C22×#67C2×C412yes—
62C22×#67C2×C412yes—
63C22×D12C2311yescharacteristic
64C22×C4×D3C2×C412yes—
65C22×Dic6C2×C411yescharacteristic
66C22×Dic6C2311yescharacteristic
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4C2×C412yes—
73C2×C3×C42C2311yescharacteristic
74C2×#217C2×C412yes—
75C2×#217C2×C412yes—
76#940—22no—
77#940—22no—
78#936—22no—
79#936—22no—
80#929—24no—
81#929—24no—
82C4×#67—22no—
83C4×#67—22no—
84C4×#67—22no—
85C4×#67—22no—
86C2×(C2×D3)⋊C4C2×C412yes—
87C2×(C2×D3)⋊C4C2×C412yes—
88C2×(C2×D3)⋊C4C2×C412yes—
89C2×C4⋊Dic3C2×C411yescharacteristic
90C2×Dic3⋊C4C2×C412yes—
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×C4×Dic3C2×C412yes—
94C2×C4×Dic3C2311yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C2×#208C2312yes—
97C2×#208C2312yes—
98#893—22no—
99#893—22no—
100#887—22no—
101#887—22no—
102#885—24no—
103#885—24no—
104#883—24no—
105#883—24no—
106#875—22no—
107#875—22no—
108C4×Dic6D412yes—
109C4×Dic6D412yes—
110C4×Dic6D412yes—
111C4×Dic6D412yes—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—22no—
114C3×#100—42no—
115#833—24no—
116#824—24no—
117#824—42no—
118#823—24no—
119#823—42no—
120#815—22no—
121#815—22no—
122Dic3⋊C8—22no—
123Dic3⋊C8—22no—
124Dic3⋊C8—22no—
125Dic3⋊C8—22no—
126C8×Dic3—22no—
127C8×Dic3—22no—
128#803—22no—
129C4×C3⋊C8—22no—
130#559—62no—
131#559—62no—
132#559—62no—
133#559—62no—
134C2×#127—31no—
135C2×#124—31no—
136C2×#124—32no—
137C2×#124—31no—
138C2×#123—32no—
139C2×#123—32no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—32no—
143C2×C4×D4—31no—
144C2×C4×D4C2×D311yescharacteristic
145#446—62no—
146#446—62no—
147#446—62no—
148C8×Q8—62no—
149#423—32no—
150#423—32no—
151#423C2×D312yes—
152C2×#101—32no—
153C2×#101—32no—
154C2×#100—32no—
155C2×#100—32no—
156C2×#100—32no—
157C2×#100C2×D312yes—
158C4×M16—32no—
159C4×M16—32no—
160C4×M16—32no—
161#343—31no—
162#343—31no—
163#343C2×D311yescharacteristic
164#343C2×D311yescharacteristic
165C24×C3C22×C412yes—
166C23×D3C22×C412yes—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—82no—
174C22×C3×C4C22×C412yes—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C2×#67—22no—
179C2×#67—22no—
180C2×#67—24no—
181C2×#67—44no—
182C2×#67—82no—
183C22×Dic3C22×C412yes—
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C411yescharacteristic
187C22×Dic3C22×C411yescharacteristic
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—21no—
191C2×D12—21no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—82no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—42no—
200C2×Dic6C22⋊C412yes—
201C2×Dic6C22⋊C412yes—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C42C2×D412yes—
211C3×C42C2×D412yes—
212#217—22no—
213#217—22no—
214#217—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—24no—
219C4⋊Dic3C22⋊C412yes—
220C4⋊Dic3C22⋊C412yes—
221Dic3⋊C4—24no—
222Dic3⋊C4C22⋊C411yescharacteristic
223Dic3⋊C4C22⋊C411yescharacteristic
224Dic3⋊C4C22⋊C411yescharacteristic
225Dic3⋊C4C22⋊C411yescharacteristic
226Dic3⋊C4C22⋊C412yes—
227Dic3⋊C4C22⋊C412yes—
228C4×Dic3—22no—
229C4×Dic3—22no—
230C4×Dic3C2×D412yes—
231C4×Dic3C2×D412yes—
232C4×Dic3C2×D412yes—
233C4×Dic3C2×D412yes—
234#208—22no—
235#208—22no—
236#208—22no—
237#208#4114yes—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C22×Q8—31no—
241C22×Q8—31no—
242C22×D4—32no—
243C22×D4—32no—
244C22×D4—31no—
245C22×D4C22×D311yescharacteristic
246C23×C4—32no—
247C23×C4C4×D312yes—
248C2×M16—32no—
249C2×M16—32no—
250C2×M16C22×D312yes—
251C2×M16C22×D312yes—
252#127—62no—
253#127—62no—
254#124—62no—
255#124—62no—
256#124—64no—
257#124—64no—
258#124—62no—
259#124—62no—
260#123—64no—
261#123—64no—
262#123—64no—
263#123—64no—
264C4×Q8—32no—
265C4×Q8—32no—
266C4×Q8—32no—
267C4×Q8—32no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—62no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—32no—
284C2×C22⋊C4—32no—
285C2×C22⋊C4—32no—
286C2×C22⋊C4—32no—
287C2×C22⋊C4—32no—
288C2×C22⋊C4C4×D312yes—
289C2×C42—31no—
290C2×C42—31no—
291C2×C42—32no—
292C2×C42C22×D311yescharacteristic
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—62no—
296C4⋊C8—62no—
297C4⋊C8—62no—
298C4⋊C8—62no—
299#101—64no—
300#101—122no—
301#100—42no—
302#100—64no—
303#100—64no—
304#100—122no—
305#97—62no—
306#97—62no—
307#97—62no—
308C4×C8—62no—
309C4×C8—62no—
310C4×C8—62no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C23×C411yescharacteristic
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—42no—
321C22×D3—42no—
322C3×D4—42no—
323C3×D4—42no—
324C2×C3×C4—22no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—82no—
329C2×C3×C4—82no—
330C2×C3×C4C2×C22⋊C411yescharacteristic
331C2×C3×C4C2×C22⋊C411yescharacteristic
332C2×C3×C4C2×C22⋊C412yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C2×#4112yes—
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336C2×C3×C42−1+411yescharacteristic
337#67—44no—
338#67—44no—
339#67—84no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3C2×C22⋊C411yescharacteristic
348C2×Dic3C2×C22⋊C412yes—
349C2×Dic3C2×C22⋊C411yescharacteristic
350C2×Dic3C2×C22⋊C411yescharacteristic
351C2×Dic3C2×C22⋊C412yes—
352C2×Dic3C2×C22⋊C411yescharacteristic
353D12—42no—
354D12—42no—
355C4×D3—82no—
356C4×D3—82no—
357Dic6—24no—
358Dic6—42no—
359Dic6—42no—
360Dic6—42no—
361C3×C8—42no—
362C3×C8—42no—
363C3⋊C8—24no—
364C3⋊C8—42no—
365C24—32no—
366C24C2×C4×D312yes—
367C2×Q8—32no—
368C2×Q8—32no—
369C2×Q8—62no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—62no—
373C2×Q8—122no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—64no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—124no—
386C2×D4—242no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—32no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—82no—
397C22×C4—242no—
398C22×C4C2×C4×D312yes—
399C22×C4C2×C4×D311yescharacteristic
400C22×C4C2×C4×D311yescharacteristic
401C22×C4C23×D311yescharacteristic
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—34no—
407M16—62no—
408M16—62no—
409M16—62no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—62no—
413C2×C8—62no—
414C4⋊C4—22no—
415C4⋊C4—31no—
416C4⋊C4—31no—
417C4⋊C4—31no—
418C4⋊C4—31no—
419C4⋊C4—32no—
420C4⋊C4—32no—
421C4⋊C4—32no—
422C4⋊C4—32no—
423C4⋊C4—64no—
424C22⋊C4—24no—
425C22⋊C4—64no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—64no—
429C22⋊C4—62no—
430C22⋊C4—62no—
431C22⋊C4—64no—
432C42—32no—
433C42—32no—
434C42—32no—
435C42—32no—
436C42—62no—
437C42—62no—
438C42D3×D412yes—
439C42D3×D412yes—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3C22×C22⋊C411yescharacteristic
448C22×C3C4×#4111yescharacteristic
449C22×C3#52011yescharacteristic
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—82no—
454C2×D3—82no—
455C3×C4—22no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4#54812yes—
460C3×C4#55012yes—
461Dic3—22no—
462Dic3—21no—
463Dic3—21no—
464Dic3—21no—
465Dic3—22no—
466Dic3—42no—
467Dic3#41312yes—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—122no—
477C23—122no—
478C23C22×C4×D311yescharacteristic
479Q8—64no—
480Q8—122no—
481Q8—122no—
482Q8—122no—
483D4—42no—
484D4—42no—
485D4—124no—
486D4—124no—
487D4—122no—
488D4—122no—
489D4—244no—
490C2×C4—22no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—32no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—32no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—242no—
510C2×C4—242no—
511C2×C4D3×C22⋊C412yes—
512C2×C4D3×C22⋊C411yescharacteristic
513C2×C4D3×C22⋊C411yescharacteristic
514C2×C4C2×D3×D412yes—
515C2×C4#100411yescharacteristic
516C2×C4#100711yescharacteristic
517C2×C4D3×#4112yes—
518C8—42no—
519C8—42no—
520C8—64no—
521C8—122no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—82no—
525C2×C3#234111yescharacteristic
526C2×C3C2×#41311yescharacteristic
527C2×C3#288811yescharacteristic
528D3—82no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22—242no—
540C22—242no—
541C22C2×D3×C22⋊C411yescharacteristic
542C22#550411yescharacteristic
543C22#552211yescharacteristic
544C4—21no—
545C4—21no—
546C4—22no—
547C4—32no—
548C4—42no—
549C4—61no—
550C4—61no—
551C4—62no—
552C4—61no—
553C4—62no—
554C4—122no—
555C4#559912yes—
556C4#563312yes—
557C3#2253911yescharacteristic, complemented, Sylow
558C2—21no—
559C2—21no—
560C2—82no—
561C2—242no—
562C2#7387711yescharacteristic
563C2#7869711yescharacteristic
564C2D3×#41311yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1800C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#940C22
cosocleC22×C3×C4C24
hypercenter#1800D3
hypocenterC3#22539
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #79046C2semidirect
3. C3×#1800C2semidirect
4. C4×#833C2semidirect
5. #70523C2semidirect
6. #70065C2semidirect
8. #68997C2semidirect
9. #68994C2semidirect
10. #68973C2semidirect
11. #68972C2semidirect
55. #1800D3semidirect
557. C3#22539semidirect

Maximal subgroups

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

subgroupindex
C2×#55592
#790462
C3×#18002
C4×#8332
#705232
#700652
#700642
#689972
#689942
#689732
#689722
#225393