Finite Group Databasebeta

← #10883521

Subgroups — #10883521

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3037C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic
3#83276C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic, complemented
5#72107C212yesmaximal, complemented
6#72103C212yesmaximal, complemented
7#71637C211yesmaximal, characteristic, complemented
8#70527C211yesmaximal, characteristic, complemented
9#70517C211yesmaximal, characteristic, complemented
10#70303C212yesmaximal, complemented
11#70301C212yesmaximal, complemented
12#28178—31nomaximal, Sylow
13C2×C3×#137C2212yes—
14C3×#532C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#949C2212yes—
17C2×#948C2212yes—
18C22×#215C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C3×#464—22no—
21C3×#462—21no—
22C3×#460—21no—
23C3×#423C2211yescharacteristic
24C3×#417C2212yes—
25#5234C2212yes—
26#5232C2212yes—
27C2×#836C2212yes—
28C2×#835C2211yescharacteristic
29C2×#823C2212yes—
30#5083—21no—
31#5072—21no—
32#5057—22no—
33#5026—22no—
34#5026—22no—
35#5011—22no—
36#5011—22no—
37C2×#809C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4924C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×#803C2211yescharacteristic
42#4558C2212yes—
43#4558C2212yes—
44#4554C2212yes—
45#4530C2212yes—
46#3037D311yescharacteristic, complemented
47#3036—31no—
48C2×#440—31no—
49#2932—31no—
50#2078—32no—
51#2019—32no—
52#1908—32no—
53#1907—32no—
54#1838—31no—
55#1801—31no—
56#1800—31no—
57C2×C3×#41C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×#137—24no—
62C3×#137—42no—
63C2×C3×Q16—22no—
64C2×C3×SD16—22no—
65C2×C3×M16C2312yes—
66C3×C4⋊Q8D411yescharacteristic
67C3×#127D411yescharacteristic
68C3×#124D412yes—
69C3×#121—21no—
70C3×#121—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×Q8—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C42—21no—
80C2×C3×C42C2311yescharacteristic
81#949—24no—
82#949—42no—
83#948—42no—
84#948—42no—
85C2×#215—22no—
86C2×#215—22no—
87C2×#215—22no—
88C2×#215—22no—
89C2×#213—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×#208—22no—
93C2×#208C2312yes—
94C2×#208C2312yes—
95C22×C3⋊C8—22no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C4×D12—21no—
99C4×D12—21no—
100C4×D12—21no—
101C4×D12—21no—
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×#103—22no—
105C3×#102—22no—
106#836—42no—
107#835—22no—
108#835—22no—
109#832—22no—
110#830—24no—
111#823—42no—
112#809—22no—
113#807—22no—
114C4⋊(C3⋊C8)—21no—
115C4⋊(C3⋊C8)—21no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×#137C2×D312yes—
121C2×AGL(1,ℤ/8)—32no—
122C22×SD16—31no—
123C2×#131—32no—
124#532C2×D311yescharacteristic
125C2×C4×Q8C2×D311yescharacteristic
126C2×C4×D4—31no—
127#464—22no—
128#463—62no—
129#462—21no—
130#461—61no—
131#460—21no—
132#459—61no—
133#440—62no—
134#440—62no—
135#435—62no—
136#435—62no—
137C2×#107—31no—
138#423—31no—
139#423—31no—
140#423C2×D311yescharacteristic
141#417—32no—
142#417C2×D312yes—
143C2×#103—31no—
144C2×#102—31no—
145C2×#101—32no—
146C2×#100—32no—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343—32no—
151#341—32no—
152#341—32no—
153C23×D3—21no—
154C3×#41—22no—
155C3×#41—22no—
156C3×#41—22no—
157C3×#41—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8C2×D412yes—
164C2×C3×Q8#4112yes—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4C2×D412yes—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4#4111yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12#4112yes—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—82no—
185C3×Q16—44no—
186C3×SD16—44no—
187C3×M16—22no—
188C3×M16—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—41no—
195C3×C4⋊C4—41no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C42—41no—
199C3×C42—41no—
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#215—28no—
205#215—44no—
206#213—44no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211#208—22no—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—44no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—44no—
223C2×C3⋊C8C2×D412yes—
224C2×C3⋊C8C2×D412yes—
225C2×#41C22×D312yes—
226C22×Q8C22×D311yescharacteristic
227C22×D4—31no—
228C23×C4—61no—
229#137—24no—
230#137—42no—
231AGL(1,ℤ/8)—64no—
232AGL(1,ℤ/8)—122no—
233C2×Q16—22no—
234C2×SD16—22no—
235C2×SD16—62no—
236C2×SD16—62no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×D8—62no—
240#131—122no—
241#131—122no—
242C2×M16—32no—
243C2×M16—32no—
244C2×M16—62no—
245C2×M16C22×D312yes—
246C22×C8—31no—
247C22×C8—31no—
248C22×C8—62no—
249C4⋊Q8#6711yescharacteristic
250#127#6711yescharacteristic
251#124#6712yes—
252#121—21no—
253#121—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263C4×D4—61no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C22⋊C4—61no—
268C2×C42—21no—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—61no—
274C4⋊C8—61no—
275C4⋊C8—61no—
276C4⋊C8—61no—
277#103—22no—
278#103—62no—
279#103—62no—
280#102—22no—
281#102—62no—
282#102—62no—
283#101—122no—
284#100—122no—
285C22⋊C8—64no—
286#97—32no—
287#97—32no—
288C23×C3—22no—
289C23×C3C2×#4111yescharacteristic
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—81no—
294C22×D3—81no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—22no—
299C3×Q8—41no—
300C3×Q8—41no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—81no—
306C3×D4—81no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—81no—
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)11yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×Dic3—41no—
328C2×Dic3—81no—
329D12—22no—
330D12—22no—
331D12—41no—
332D12—41no—
333C4×D3—82no—
334C4×D3—82no—
335C3×C8—42no—
336C3⋊C8—22no—
337C3⋊C8—22no—
338C3⋊C8—42no—
339C3⋊C8—42no—
340C24—61no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8#23712yes—
351C2×Q8C2×#6712yes—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—32no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4C2×#6712yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—61no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4—242no—
369C22×C4#23711yescharacteristic
370C22×C4#23711yescharacteristic
371C22×C4C23×D311yescharacteristic
372Q16—44no—
373SD16—44no—
374SD16—68no—
375SD16—124no—
376D8—124no—
377M16—22no—
378M16—22no—
379M16—62no—
380M16—62no—
381M16—62no—
382M16—62no—
383M16—124no—
384C2×C8—22no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8—61no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—124no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—121no—
405C22⋊C4—121no—
406C42—41no—
407C42—41no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3C2×#12111yescharacteristic
418C22×C3#55311yescharacteristic
419C22×C3#55311yescharacteristic
420C2×D3—42no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—82no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—81no—
432C3×C4#52511yescharacteristic
433C3×C4#53511yescharacteristic
434C3×C4#54611yescharacteristic
435C3×C4#54611yescharacteristic
436Dic3—81no—
437C23—22no—
438C23—62no—
439C23—121no—
440C23—121no—
441C23—241no—
442C23—241no—
443C23C2×#23711yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—41no—
449Q8—41no—
450D4—22no—
451D4—22no—
452D4—42no—
453D4—42no—
454D4—62no—
455D4—62no—
456D4—81no—
457D4—81no—
458D4—121no—
459D4—121no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—121no—
471C2×C4—242no—
472C2×C4—242no—
473C2×C4—241no—
474C2×C4#93911yescharacteristic
475C2×C4#93911yescharacteristic
476C2×C4#93911yescharacteristic
477C2×C4#93911yescharacteristic
478C2×C4C2×D3×D411yescharacteristic
479C2×C4C2×#23711yescharacteristic
480C2×C4#100411yescharacteristic
481C2×C4D3×#4111yescharacteristic
482C2×C4#100911yescharacteristic
483C2×C4C22×#6711yescharacteristic
484C8—42no—
485C8—62no—
486C8—62no—
487C8—122no—
488C8—122no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—82no—
492C2×C3#253511yescharacteristic
493C2×C3#306811yescharacteristic
494C2×C3#306911yescharacteristic
495D3—82no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—122no—
502C22—242no—
503C22—242no—
504C22—241no—
505C22—241no—
506C22#561411yescharacteristic
507C22#561811yescharacteristic
508C22C2×#93911yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—241no—
517C4#560611yescharacteristic
518C4#560711yescharacteristic
519C4#580611yescharacteristic
520C4#582911yescharacteristic
521C3#2817811yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—82no—
525C2—242no—
526C2#8266611yescharacteristic
527C2#8397811yescharacteristic
528C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3037C2
socleC22×C3C2×#121
prime radical#83297C2
hypercenter#3037D3
hypocenterC3#28178
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
1. C3×#3037C2semidirect
3. #83276C2semidirect
4. C2×#5026C2semidirect
5. #72107C2semidirect
6. #72103C2semidirect
7. #71637C2semidirect
8. #70527C2semidirect
9. #70517C2semidirect
10. #70303C2semidirect
11. #70301C2semidirect
46. #3037D3semidirect
521. C3#28178semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#832972
#832762
C2×#50262
#721072
#721032
#716372
#705272
#705172
#703032
#703012
#281783