Finite Group Databasebeta

← #10848659

Subgroups — #10848659

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic, complemented
2#83336C211yesmaximal, characteristic, complemented
3#82979C211yesmaximal, characteristic, complemented
4#82526C211yesmaximal, characteristic, complemented
5#71867C212yesmaximal, complemented
6#71856C212yesmaximal, complemented
7#71840C211yesmaximal, characteristic, complemented
8#71727C211yesmaximal, characteristic, complemented
9#71661C212yesmaximal, complemented
10#71650C212yesmaximal, complemented
11#71643C211yesmaximal, characteristic, complemented
12#25595—31nomaximal, Sylow
13C3×#559—22no—
14C3×#556—22no—
15C3×#551—24no—
16C3×#542—22no—
17C3×#536C2211yescharacteristic
18C3×#535C2211yescharacteristic
19C3×#533C2211yescharacteristic
20C3×#529—22no—
21C2×C3×#124C2211yescharacteristic
22C3×#519C2211yescharacteristic
23C3×#518C2211yescharacteristic
24C3×C4×#41C2211yescharacteristic
25C2×#949C2212yes—
26C2×#947C2212yes—
27C2×#940C2212yes—
28C2×#874C2211yescharacteristic
29C2×#837C2212yes—
30C2×#836C2212yes—
31C2×#833C2212yes—
32#5088—22no—
33#5067—22no—
34#5027—22no—
35#5015—22no—
36#5001C2212yes—
37#5001C2212yes—
38#5001C2212yes—
39#4960—24no—
40#4932C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4532C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#3510D311yescharacteristic, complemented
47#3138—31no—
48#3038—31no—
49#2657—31no—
50#2003—32no—
51#2002—32no—
52#1995—31no—
53#1967—31no—
54#1888—32no—
55#1887—32no—
56#1759—31no—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C22×#67—24no—
61C22×D12—22no—
62C22×D12C2312yes—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D412yescomplemented
65C3×#127—21no—
66C3×#126—24no—
67C3×#126—24no—
68C3×#125D412yescomplemented
69C3×#124—24no—
70C3×#124—21no—
71C3×#124—21no—
72C3×#124—22no—
73C3×#124—21no—
74C3×#124—21no—
75C3×#123—24no—
76C3×#123—24no—
77C3×#123—22no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×#121—22no—
82C3×#120—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8D412yescomplemented
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4D412yescomplemented
92C3×#117—24no—
93C3×#117—22no—
94C3×#117—21no—
95C3×#117C2311yescharacteristic
96C3×#117C2311yescharacteristic
97C3×#117C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4D414yes—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102#949—42no—
103#949—42no—
104#947—24no—
105#947—42no—
106C2×#215—22no—
107#940—24no—
108#940—42no—
109C2×#213—22no—
110C2×(C2×D3)⋊C4—24no—
111C2×C4×Dic3C2312yes—
112C2×#208C2312yes—
113C2×#208C2312yes—
114#874—22no—
115#874—21no—
116#874—21no—
117#837—24no—
118#836—42no—
119#833—42no—
120C3⋊#95—22no—
121#809—24no—
122#809—24no—
123#809—24no—
124C4⋊(C3⋊C8)—22no—
125#803—21no—
126C4×C3⋊C8—21no—
127C2×AGL(1,ℤ/8)—32no—
128#559—22no—
129#556—22no—
130#551—24no—
131#542—22no—
132#536C2×D311yescharacteristic
133#535C2×D311yescharacteristic
134#533C2×D311yescharacteristic
135C2×#127—32no—
136C2×#127—31no—
137#529—22no—
138C2×#124—32no—
139C2×#124C2×D311yescharacteristic
140#519C2×D311yescharacteristic
141#518C2×D311yescharacteristic
142C4×#41C2×D311yescharacteristic
143#490—62no—
144#486—62no—
145#460—62no—
146#459—62no—
147#423—31no—
148C2×#104—32no—
149#418—32no—
150#418—32no—
151#418—32no—
152C2×#101—32no—
153C2×#100—32no—
154C4×M16—31no—
155#392—64no—
156#376—31no—
157#339—32no—
158#339—32no—
159#339—32no—
160C24×C3C2×D412yes—
161C23×D3—22no—
162C3×#41—22no—
163C3×#41—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4C2×D411yescharacteristic
177C22×C3×C4—24no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C2×#67—48no—
184C2×#67—88no—
185C22×Dic3—22no—
186C2×D12—22no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—42no—
193C2×D12—42no—
194C2×D12—84no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4C2×D412yes—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—22no—
213C3×C42—24no—
214C3×C42—22no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D412yes—
221#215—44no—
222#213—44no—
223(C2×D3)⋊C4—48no—
224C4×Dic3—22no—
225C4×Dic3#4114yes—
226#208—22no—
227#208—22no—
228#208—22no—
229#208—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×#41C22×D311yescharacteristic
233C22×Q8C22×D311yescharacteristic
234C22×D4—32no—
235C22×D4—64no—
236C22×D4—62no—
237C22×D4C22×D311yescharacteristic
238AGL(1,ℤ/8)—122no—
239AGL(1,ℤ/8)—122no—
240C2×SD16—62no—
241C2×D8—62no—
242C2×M16—32no—
243C2×M16—32no—
244C4⋊Q8—21no—
245C4⋊Q8#6712yescomplemented
246#127—21no—
247#127—64no—
248#127—62no—
249#127—61no—
250#127—61no—
251#127—122no—
252#126—24no—
253#126—24no—
254#125#6712yescomplemented
255#124—24no—
256#124—21no—
257#124—21no—
258#124—21no—
259#124—21no—
260#124—22no—
261#124—64no—
262#124—122no—
263#123—24no—
264#123—24no—
265#123—22no—
266C22⋊Q8—24no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—22no—
270#120—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8#6712yescomplemented
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4#6712yescomplemented
280#117—24no—
281#117—22no—
282#117—21no—
283#117C22×D311yescharacteristic
284#117C22×D311yescharacteristic
285#117C22×D311yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4#6714yes—
289C2×C42—21no—
290C2×C42—32no—
291C2×C42C22×D311yescharacteristic
292C4⋊C8—62no—
293#104—64no—
294#102—64no—
295#102—64no—
296#102—64no—
297#101—122no—
298#100—122no—
299#97—61no—
300C4×C8—61no—
301#95—62no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—24no—
305C23×C3—44no—
306C23×C3C22×D411yescharacteristic
307C22×D3—48no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—41no—
314C3×Q8—41no—
315C3×Q8—42no—
316C3×Q8—82no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—42no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—42no—
323C3×D4—84no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—82no—
333C2×C3×C4—84no—
334C2×C3×C4#12012yes—
335C2×C3×C4#12012yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4112yes—
339C2×C3×C4H(5,𝔽2)12yes—
340#67—816no—
341C2×Dic3—24no—
342C2×Dic3—44no—
343D12—42no—
344D12—42no—
345D12—42no—
346D12—42no—
347D12—84no—
348D12—84no—
349C3⋊C8—42no—
350C3⋊C8—42no—
351C24—62no—
352C24C2×#6712yes—
353#41—22no—
354#41—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8C2×#6711yescharacteristic
361C2×D4—21no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—128no—
374C2×D4—122no—
375C2×D4—122no—
376C2×D4—244no—
377C2×D4—248no—
378C2×D4C2×#6711yescharacteristic
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—62no—
383C22×C4C2×#6712yes—
384C22×C4C2×#6712yes—
385C22×C4C23×D311yescharacteristic
386SD16—124no—
387D8—124no—
388M16—62no—
389M16—62no—
390M16—62no—
391M16—62no—
392C2×C8—62no—
393C2×C8—62no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4C2×#6712yes—
401C4⋊C4C2×#6712yes—
402C4⋊C4C2×#6712yes—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—128no—
413C42—24no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—34no—
419C42—62no—
420C42C2×#6711yescharacteristic
421C42C2×#6711yescharacteristic
422C42C2×#6712yes—
423C22×C3—22no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3—84no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#54612yes—
431C2×D3—84no—
432C2×D3—84no—
433C2×D3—82no—
434C2×D3—84no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4#53211yescharacteristic
443C3×C4#53511yescharacteristic
444C3×C4#55012yes—
445Dic3—44no—
446C23—22no—
447C23—21no—
448C23—24no—
449C23—44no—
450C23—128no—
451C23—122no—
452C23—122no—
453C23—122no—
454C23C22×#6711yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—82no—
461D4—21no—
462D4—21no—
463D4—42no—
464D4—41no—
465D4—41no—
466D4—42no—
467D4—84no—
468D4—122no—
469D4—122no—
470D4—122no—
471D4—122no—
472D4—244no—
473D4—244no—
474D4—2416no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—64no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—124no—
487C2×C4#95312yes—
488C2×C4#95312yes—
489C2×C4D3×#4112yes—
490C2×C4#100912yes—
491C2×C4C22×#6711yescharacteristic
492C2×C4C22×#6711yescharacteristic
493C8—122no—
494C8—122no—
495C2×C3—22no—
496C2×C3—81no—
497C2×C3—82no—
498C2×C3#243111yescharacteristic
499C2×C3#302112yes—
500D3—162no—
501C22—22no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—84no—
507C22—244no—
508C22—244no—
509C22—242no—
510C22—244no—
511C22#582912yes—
512C22C2×#95311yescharacteristic
513C4—22no—
514C4—22no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—124no—
521C4#583011yescharacteristic
522C4#583211yescharacteristic
523C4#583412yes—
524C3#2559511yescharacteristic, complemented, Sylow
525C2—22no—
526C2—82no—
527C2—81no—
528C2—482no—
529C2#8409211yescharacteristic
530C2#8412212yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3510C2
socleC22×C3C2×#120
hypercenter#3510D3
hypocenterC3#25595
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3510

Complements

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

NH ≅ G/Ntype
1. C3×#3510C2semidirect
2. #83336C2semidirect
3. #82979C2semidirect
4. #82526C2semidirect
5. #71867C2semidirect
6. #71856C2semidirect
7. #71840C2semidirect
8. #71727C2semidirect
9. #71661C2semidirect
10. #71650C2semidirect
11. #71643C2semidirect
46. #3510D3semidirect
64. C3×C4⋊Q8D4semidirect
68. C3×#125D4semidirect
86. C3×C4×Q8D4semidirect
91. C3×C4×D4D4semidirect
245. C4⋊Q8#67semidirect
254. #125#67semidirect
274. C4×Q8#67semidirect
279. C4×D4#67semidirect
524. C3#25595semidirect

Maximal subgroups

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

subgroupindex
C3×#35102
#833362
#829792
#825262
#718672
#718562
#718402
#717272
#716612
#716502
#716432
#255953