Finite Group Databasebeta

← #10883536

Subgroups — #10883536

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2957C211yesmaximal, characteristic, complemented
2#83928C211yesmaximal, characteristic, complemented
3#83914C211yesmaximal, characteristic, complemented
4#78553C211yesmaximal, characteristic, complemented
5#72029C212yesmaximal, complemented
6#72020C212yesmaximal, complemented
7#70719C211yesmaximal, characteristic, complemented
8#70711C211yesmaximal, characteristic, complemented
9#70406C212yesmaximal, complemented
10#70404C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#28180—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C3×#568C2212yes—
15#5828C2212yes—
16#5822C2212yes—
17C2×#901C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19#5524C2211yescharacteristic
20C3×#444—22no—
21C3×#443—22no—
22C2×C3×#108C2211yescharacteristic
23C3×#421C2212yes—
24C3×#405C2211yescharacteristic
25#5237C2212yes—
26#5233C2212yes—
27#5192—21no—
28#5188—21no—
29#5177—22no—
30#5171—22no—
31#5160—21no—
32#5157—21no—
33#5139C2212yes—
34#5129C2211yescharacteristic
35#5126C2211yescharacteristic
36#5126C2212yes—
37C2×#819C2211yescharacteristic
38#5106C2211yescharacteristic
39#4999C2211yescharacteristic
40#4706—22no—
41C8×D12—22no—
42#4558C2212yes—
43#4556C2212yes—
44#4556C2212yes—
45#4531C2212yes—
46#3067—31no—
47#3050—31no—
48#2957D311yescharacteristic, complemented
49#2929—31no—
50#2080—32no—
51#2010—32no—
52#1891—32no—
53#1881—32no—
54#1833—31no—
55#1814—31no—
56#1810—31no—
57C2×C3×#41C2312yes—
58C2×#235C2311yescharacteristic
59C22×C4×D3C2311yescharacteristic
60C3×#135—21no—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135—21no—
64C2×C3×Q16D411yescharacteristic, complemented
65C2×C3×SD16D412yescomplemented
66C2×C3×D8D411yescharacteristic, complemented
67C3×#131—24no—
68C3×#131—42no—
69C2×C3×M16—42no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#117C2311yescharacteristic
74#951—22no—
75#950—42no—
76#949—22no—
77#946—22no—
78#942—22no—
79#938—22no—
80#937—22no—
81#935—22no—
82#932—22no—
83C4×#67—42no—
84#926C2312yes—
85C2×#208C2311yescharacteristic
86C2×#208C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88#901—21no—
89#901—21no—
90#901—21no—
91#901—21no—
92C2×ΓL(1,𝔽25)—21no—
93C2×D3×C8—21no—
94C2×D3×C8—21no—
95C2×D3×C8—21no—
96C2×D3×C8—21no—
97C2×D3×C8—21no—
98C2×D3×C8—21no—
99C2×D3×C8—21no—
100#897—21no—
101#895—21no—
102#886—21no—
103#882—21no—
104C4×D12—22no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#104—24no—
108C3×#97D412yescomplemented
109C3×C4×C8D412yescomplemented
110#836—22no—
111#835—22no—
112#833—22no—
113#832—22no—
114#825—42no—
115(C2×D3)⋊C8—22no—
116(C2×D3)⋊C8—22no—
117(C2×D3)⋊C8—42no—
118#819—21no—
119#819—21no—
120#818—21no—
121#817—21no—
122C4⋊(C3⋊C8)—22no—
123#575—32no—
124C2×#135C2×D311yescharacteristic
125#568C2×D312yes—
126C2×#131—31no—
127C23×C8—31no—
128#525—32no—
129#525—31no—
130#473—61no—
131#472—61no—
132#471—62no—
133#467—61no—
134#466—61no—
135#465—62no—
136#444—22no—
137#443—22no—
138#435—62no—
139C8×D4—62no—
140C2×#108—31no—
141C2×#108C2×D311yescharacteristic
142#427—31no—
143#424—31no—
144#421—32no—
145#421C2×D312yes—
146#416—32no—
147#413—32no—
148#408—31no—
149#407—31no—
150#407—32no—
151#405C2×D311yescharacteristic
152#344—32no—
153#344—32no—
154#341—32no—
155H(3,ℤ/4)—32no—
156C23×D3#4111yescharacteristic
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—22no—
160C3×#41—22no—
161C2×C3×Q8C2×D412yes—
162C2×C3×D4—22no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4C2411yescharacteristic
166C2×#67—21no—
167C2×#67—42no—
168C22×Dic3#4111yescharacteristic
169#235—21no—
170#235—21no—
171#235—21no—
172#235—21no—
173C2×D12—22no—
174C2×D12#4111yescharacteristic
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×Dic6#4111yescharacteristic
183C3×Q16—21no—
184C3×Q16—21no—
185C3×SD16—22no—
186C3×SD16—22no—
187C3×D8—21no—
188C3×D8—21no—
189C3×M16—21no—
190C3×M16—21no—
191C3×M16—22no—
192C3×M16—22no—
193C3×M16—44no—
194C2×C3×C8—22no—
195C2×C3×C8—44no—
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8C2×D411yescharacteristic
201C2×C3×C8#4111yescharacteristic
202C2×C3×C8#4111yescharacteristic
203C3×C4⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C42C2×D412yes—
206#217—22no—
207#217—42no—
208#216—42no—
209#215—42no—
210#214—42no—
211#213—42no—
212(C2×D3)⋊C4—22no—
213(C2×D3)⋊C4—22no—
214(C2×D3)⋊C4—42no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217Dic3⋊C4—42no—
218C4×Dic3—22no—
219#208—22no—
220#208—21no—
221#208—21no—
222#208—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228ΓL(1,𝔽25)—41no—
229ΓL(1,𝔽25)—41no—
230D3×C8—41no—
231D3×C8—41no—
232D3×C8—41no—
233D3×C8—41no—
234D3×C8—41no—
235D3×C8—41no—
236C2×#41—31no—
237C2×#41C22×D312yes—
238C23×C4—31no—
239#137—62no—
240#137—62no—
241AGL(1,ℤ/8)—62no—
242AGL(1,ℤ/8)—62no—
243#135—21no—
244#135—21no—
245#135—21no—
246#135—21no—
247#135—122no—
248C2×Q16D1211yescharacteristic, complemented
249C2×SD16D1212yescomplemented
250C2×D8D1211yescharacteristic, complemented
251#131—24no—
252#131—42no—
253#131—61no—
254#131—61no—
255#131—61no—
256#131—61no—
257C2×M16—31no—
258C2×M16—32no—
259C2×M16—42no—
260C2×M16—61no—
261C2×M16C22×D312yes—
262C2×M16C22×D311yescharacteristic
263C22×C8—31no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8C22×D311yescharacteristic
272#123—61no—
273#123—62no—
274C22⋊Q8—61no—
275C22⋊Q8—62no—
276#121—61no—
277#121—62no—
278#120—61no—
279#120—62no—
280C4×D4—62no—
281C4×D4—122no—
282#117—32no—
283#117C22×D311yescharacteristic
284#108—21no—
285#108—21no—
286#108—61no—
287#108—61no—
288#107—61no—
289#106—61no—
290C4⋊C8—62no—
291#104—24no—
292#104—122no—
293#103—62no—
294#102—62no—
295#101—62no—
296#100—62no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299C22⋊C8—122no—
300#97#6712yescomplemented
301C4×C8#6712yescomplemented
302C23×C3—22no—
303C23×C3C2×#4111yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—42no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×D4—22no—
312C3×D4—22no—
313C3×D4—42no—
314C3×D4—42no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4H(5,𝔽2)11yescharacteristic
324C2×C3×C4H(5,𝔽2)11yescharacteristic
325#67—41no—
326#67—41no—
327#67—84no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—42no—
333D12—21no—
334D12—21no—
335D12—44no—
336C4×D3—41no—
337C4×D3—41no—
338C4×D3—41no—
339C4×D3—41no—
340C4×D3—41no—
341C4×D3—41no—
342Dic6—21no—
343Dic6—21no—
344C3×C8—22no—
345C3×C8—42no—
346C3×C8#12111yescharacteristic
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3×C8#12111yescharacteristic
350C3⋊C8—41no—
351C3⋊C8—41no—
352C3⋊C8—42no—
353C24—31no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—61no—
359#41—61no—
360#41—61no—
361#41—61no—
362C2×Q8—31no—
363C2×Q8C2×D1212yes—
364C2×D4—22no—
365C2×D4—31no—
366C2×D4—62no—
367C2×D4—61no—
368C2×D4—122no—
369C2×D4C2×D1212yes—
370C22×C4—22no—
371C22×C4—31no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4C23×D311yescharacteristic
380Q16—21no—
381Q16—21no—
382Q16—122no—
383SD16—22no—
384SD16—22no—
385SD16—122no—
386SD16—122no—
387D8—21no—
388D8—21no—
389D8—122no—
390M16—21no—
391M16—21no—
392M16—22no—
393M16—22no—
394M16—44no—
395M16—61no—
396M16—62no—
397M16—61no—
398M16—62no—
399M16—121no—
400M16—121no—
401C2×C8—22no—
402C2×C8—44no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—62no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—121no—
409C2×C8—121no—
410C2×C8—121no—
411C2×C8—121no—
412C2×C8—121no—
413C2×C8—121no—
414C2×C8C2×D1211yescharacteristic
415C2×C8C2×D1211yescharacteristic
416C2×C8#23511yescharacteristic
417C2×C8#23511yescharacteristic
418C2×C8C2×#6712yes—
419C2×C8C2×#6711yescharacteristic
420C2×C8C2×#6711yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—122no—
425C22⋊C4—21no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C42—62no—
432C42C2×#6712yes—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3#52511yescharacteristic
437C22×C3#54611yescharacteristic
438C22×C3#55311yescharacteristic
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—22no—
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—42no—
447C2×D3—41no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4C2×#12111yescharacteristic
451C3×C4#53511yescharacteristic
452C3×C4#54611yescharacteristic
453C3×C4#55311yescharacteristic
454Dic3—41no—
455Dic3—41no—
456Dic3—82no—
457C23—22no—
458C23—61no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—122no—
463C23D3×#4111yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—61no—
467Q8—61no—
468D4—22no—
469D4—22no—
470D4—42no—
471D4—42no—
472D4—61no—
473D4—61no—
474D4—124no—
475D4—121no—
476D4—121no—
477D4—244no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4C22×D1211yescharacteristic
494C2×C4C2×#23511yescharacteristic
495C2×C4#100411yescharacteristic
496C2×C4#100411yescharacteristic
497C2×C4D3×#4111yescharacteristic
498C2×C4C22×#6711yescharacteristic
499C8—22no—
500C8—42no—
501C8—121no—
502C8—121no—
503C8—122no—
504C8#93011yescharacteristic
505C8#93011yescharacteristic
506C8#93011yescharacteristic
507C8#93011yescharacteristic
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—82no—
512C2×C3#253511yescharacteristic
513D3—42no—
514D3—81no—
515C22—41no—
516C22—42no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—82no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22—122no—
525C22—121no—
526C22—121no—
527C22#555411yescharacteristic
528C22#555811yescharacteristic
529C22#582811yescharacteristic
530C4—42no—
531C4—42no—
532C4—121no—
533C4—121no—
534C4—242no—
535C4#555411yescharacteristic
536C4#555811yescharacteristic
537C4C2×#93011yescharacteristic
538C4#580511yescharacteristic
539C3#2818011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—82no—
544C2—122no—
545C2—241no—
546C2#8241611yescharacteristic
subgroupG/H
centerC4#5805
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2957C2
socleC2×C3#2535
hypercenter#2957D3
hypocenterC3#28180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2957

Complements

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

NH ≅ G/Ntype
1. C3×#2957C2semidirect
2. #83928C2semidirect
3. #83914C2semidirect
4. #78553C2semidirect
5. #72029C2semidirect
6. #72020C2semidirect
7. #70719C2semidirect
8. #70711C2semidirect
9. #70406C2semidirect
10. #70404C2semidirect
11. #69896C2semidirect
48. #2957D3semidirect
64. C2×C3×Q16D4semidirect
65. C2×C3×SD16D4semidirect
66. C2×C3×D8D4semidirect
108. C3×#97D4semidirect
109. C3×C4×C8D4semidirect
248. C2×Q16D12semidirect
249. C2×SD16D12semidirect
250. C2×D8D12semidirect
300. #97#67semidirect
301. C4×C8#67semidirect
539. C3#28180semidirect

Maximal subgroups

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

subgroupindex
C3×#29572
#839282
#839142
#785532
#720292
#720202
#707192
#707112
#704062
#704042
#698962
#281803