Finite Group Databasebeta

← #10883584

Subgroups — #10883584

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2950C211yesmaximal, characteristic, complemented
2C2×#5192C211yesmaximal, characteristic, complemented
3#83887C211yesmaximal, characteristic, complemented
4#78579C211yesmaximal, characteristic
5#71663C212yesmaximal, complemented
6#71662C212yesmaximal, complemented
7#70665C211yesmaximal, characteristic, complemented
8#70664C211yesmaximal, characteristic, complemented
9#70351C212yesmaximal
10#70341C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#28184—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×#519C2212yes—
15#5816C2212yes—
16C2×#942C2212yes—
17#5748C2211yescharacteristic
18C2×#901C2211yescharacteristic
19C2×#901C2211yescharacteristic
20C3×#441—22no—
21C3×C4×Q16—22no—
22C3×#427C2211yescharacteristic
23C3×#419C2212yes—
24C3×#405C2211yescharacteristic
25C2×#836C2212yes—
26#5225C2212yes—
27#5192—22no—
28#5192—22no—
29#5189—22no—
30#5188—22no—
31#5125C2211yescharacteristic
32C2×#819C2211yescharacteristic
33#5106C2211yescharacteristic
34#5006C2212yes—
35#5005C2211yescharacteristic
36#5005C2211yescharacteristic
37#5005C2212yes—
38#4717—22no—
39#4700—22no—
40#4556C2212yes—
41#4533C2212yes—
42#4531C2212yes—
43#4531C2212yes—
44C2×#473—31no—
45#3059—31no—
46#2955—31no—
47#2950D311yescharacteristic, complemented
48#2082—32no—
49#2009—32no—
50#1912—32no—
51#1905—32no—
52#1833—31no—
53#1814—31no—
54#1812—31no—
55C22×C3×Q8C2312yes—
56C2×#235C2311yescharacteristic
57C2×#235C2311yescharacteristic
58C2×C3×Q16—21no—
59C2×C3×Q16—21no—
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C2×C3×Q16D412yes—
63C2×C3×Q16D412yes—
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—24no—
67C3×C4×Q8—42no—
68C3×#117—42no—
69C3×#117C2312yes—
70C3×#117C2311yescharacteristic
71#947—22no—
72#946—24no—
73#944—22no—
74C2×#216—22no—
75#942—24no—
76#942—42no—
77C2×#215—22no—
78#929—42no—
79#926C2312yes—
80C2×#208C2311yescharacteristic
81C2×#208C2311yescharacteristic
82C2×#208C2312yes—
83#909—21no—
84#909—21no—
85#908—21no—
86#908—21no—
87#904—22no—
88#901—21no—
89#901—21no—
90#901—21no—
91#901—21no—
92#901—21no—
93#901—21no—
94#901—21no—
95#901—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×D3×C8—21no—
99C2×D3×C8—21no—
100C3×#107—21no—
101C3×#106—21no—
102C3×#103—24no—
103C3×#97D412yes—
104C3×C4×C8D412yes—
105#836—24no—
106#835—24no—
107#819—22no—
108#819—21no—
109#819—21no—
110#818—21no—
111#817—21no—
112#806—42no—
113#805—22no—
114#805—22no—
115#805—42no—
116#575—31no—
117C2×#137—32no—
118C22×Q16C2×D311yescharacteristic
119C2×#131—31no—
120C2×#131—31no—
121#536—32no—
122#519C2×D312yes—
123#473—62no—
124#473—62no—
125#470—62no—
126#467—62no—
127#444—62no—
128#443—62no—
129#441—22no—
130C4×Q16—22no—
131#430—31no—
132C2×#108—31no—
133#427—31no—
134#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421—32no—
138#419—32no—
139#419C2×D312yes—
140C2×#101—32no—
141#410—32no—
142#405C2×D311yescharacteristic
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—22no—
154C22×C3×C4C2411yescharacteristic
155C2×#67—21no—
156C2×#67—21no—
157#235—21no—
158#235—21no—
159#235—21no—
160#235—21no—
161#235—21no—
162#235—21no—
163#235—21no—
164#235—21no—
165C2×D12#4111yescharacteristic
166C2×D12#4111yescharacteristic
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×Dic6#4111yescharacteristic
170C2×Dic6#4111yescharacteristic
171C3×Q16—22no—
172C3×Q16—22no—
173C3×Q16—22no—
174C3×Q16—22no—
175C3×M16—21no—
176C3×M16—21no—
177C2×C3×C8C2×D411yescharacteristic
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8C2×D412yes—
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D411yescharacteristic
182C2×C3×C8#4111yescharacteristic
183C2×C3×C8#4111yescharacteristic
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—44no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—22no—
189C3×C42—22no—
190C3×C42—44no—
191C3×C42C2×D412yes—
192#217—22no—
193#216—44no—
194#215—44no—
195(C2×D3)⋊C4—44no—
196C4⋊Dic3—22no—
197Dic3⋊C4—44no—
198C4×Dic3—22no—
199#208—21no—
200#208—21no—
201#208—22no—
202#208—21no—
203#208—21no—
204#208—22no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—22no—
208Dic12—22no—
209D24—22no—
210#204—22no—
211#204—22no—
212ΓL(1,𝔽25)—41no—
213ΓL(1,𝔽25)—41no—
214ΓL(1,𝔽25)—41no—
215ΓL(1,𝔽25)—41no—
216D3×C8—41no—
217D3×C8—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41—31no—
222C22×Q8C22×D312yes—
223#137—61no—
224#137—61no—
225#137—64no—
226#137—122no—
227AGL(1,ℤ/8)—61no—
228AGL(1,ℤ/8)—61no—
229#135—62no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—62no—
235C2×Q16D1212yes—
236C2×Q16D1212yes—
237C2×SD16—62no—
238#131—61no—
239#131—61no—
240#131—61no—
241#131—61no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—31no—
247C2×M16—31no—
248C2×M16—32no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16C22×D311yescharacteristic
252C22×C8—61no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255C4⋊Q8—62no—
256#124—62no—
257#123—122no—
258C22⋊Q8—64no—
259C4×Q8—24no—
260C4×Q8—42no—
261#117—32no—
262#117—42no—
263#117C22×D312yes—
264#117C22×D311yescharacteristic
265#108—61no—
266#108—61no—
267#108—62no—
268#107—21no—
269#107—61no—
270#106—21no—
271#106—61no—
272#104—62no—
273#104—62no—
274#104—122no—
275#103—24no—
276#103—64no—
277#101—64no—
278#99—122no—
279#97#6712yes—
280C4×C8#6712yes—
281C23×C3H(5,𝔽2)11yescharacteristic
282C22×D3—21no—
283C22×D3—21no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—42no—
287C3×Q8—42no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4H(5,𝔽2)11yescharacteristic
299#67—41no—
300#67—41no—
301#67—41no—
302#67—41no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—42no—
306D12—21no—
307D12—21no—
308D12—21no—
309D12—21no—
310C4×D3—41no—
311C4×D3—41no—
312C4×D3—41no—
313C4×D3—41no—
314Dic6—21no—
315Dic6—21no—
316Dic6—21no—
317Dic6—21no—
318C3×C8—22no—
319C3×C8#12111yescharacteristic
320C3×C8#12111yescharacteristic
321C3×C8#12111yescharacteristic
322C3×C8#12111yescharacteristic
323C3⋊C8—41no—
324C3⋊C8—41no—
325C3⋊C8—42no—
326#41—61no—
327#41—61no—
328#41—61no—
329#41—61no—
330#41—61no—
331#41—61no—
332#41—61no—
333#41—61no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—31no—
340C2×Q8—31no—
341C2×Q8C2×D1214yes—
342C2×D4—31no—
343C2×D4—31no—
344C2×D4—61no—
345C2×D4—61no—
346C22×C4—22no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4C23×D311yescharacteristic
350Q16—22no—
351Q16—22no—
352Q16—22no—
353Q16—22no—
354Q16—62no—
355Q16—124no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—62no—
360M16—21no—
361M16—21no—
362M16—61no—
363M16—62no—
364M16—61no—
365M16—62no—
366M16—61no—
367M16—61no—
368M16—121no—
369M16—121no—
370M16—121no—
371M16—121no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8—62no—
375C2×C8—121no—
376C2×C8—121no—
377C2×C8—121no—
378C2×C8—121no—
379C2×C8C2×D1211yescharacteristic
380C2×C8C2×D1211yescharacteristic
381C2×C8#23511yescharacteristic
382C2×C8#23511yescharacteristic
383C2×C8C2×#6712yes—
384C2×C8C2×#6711yescharacteristic
385C2×C8C2×#6711yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—44no—
389C4⋊C4—62no—
390C4⋊C4—124no—
391C22⋊C4—21no—
392C22⋊C4—22no—
393C22⋊C4—62no—
394C22⋊C4—124no—
395C42—22no—
396C42—44no—
397C42—62no—
398C42C2×#6712yes—
399C22×C3—41no—
400C22×C3#53511yescharacteristic
401C22×C3#54611yescharacteristic
402C22×C3#55311yescharacteristic
403C2×D3—42no—
404C2×D3—41no—
405C2×D3—41no—
406C2×D3—81no—
407C3×C4—42no—
408C3×C4—44no—
409C3×C4—82no—
410C3×C4C2×#12111yescharacteristic
411C3×C4#52511yescharacteristic
412C3×C4#54611yescharacteristic
413C3×C4#55311yescharacteristic
414Dic3—41no—
415Dic3—41no—
416Dic3—82no—
417C23—61no—
418C23—61no—
419C23#100411yescharacteristic
420Q8—24no—
421Q8—24no—
422Q8—42no—
423Q8—42no—
424Q8—61no—
425Q8—61no—
426Q8—61no—
427Q8—61no—
428D4—61no—
429D4—61no—
430D4—61no—
431D4—61no—
432D4—121no—
433D4—121no—
434D4—121no—
435D4—121no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4C22×D1211yescharacteristic
449C2×C4C2×#23511yescharacteristic
450C2×C4#100411yescharacteristic
451C2×C4D3×#4111yescharacteristic
452C2×C4D3×#4111yescharacteristic
453C2×C4C22×#6711yescharacteristic
454C8—22no—
455C8—121no—
456C8—121no—
457C8—122no—
458C8#93011yescharacteristic
459C8#93011yescharacteristic
460C8#93011yescharacteristic
461C8#93011yescharacteristic
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3#253511yescharacteristic
466D3—81no—
467D3—81no—
468C22—41no—
469C22—122no—
470C22—121no—
471C22—121no—
472C22—241no—
473C22#555411yescharacteristic
474C22#555811yescharacteristic
475C22#580511yescharacteristic
476C4—42no—
477C4—44no—
478C4—82no—
479C4—121no—
480C4—121no—
481C4—242no—
482C4#555411yescharacteristic
483C4#555811yescharacteristic
484C4C2×#93011yescharacteristic
485C4#582811yescharacteristic
486C3#2818411yescharacteristic, complemented, Sylow
487C2—21no—
488C2—21no—
489C2—21no—
490C2—241no—
491C2—241no—
492C2#8241611yescharacteristic
subgroupG/H
centerC2#82416
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2950C2
socleC2×C3#2535
prime radical#5748C22
hypercenter#2950D3
hypocenterC3#28184
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2950

Complements

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

NH ≅ G/Ntype
1. C3×#2950C2semidirect
2. C2×#5192C2semidirect
3. #83887C2semidirect
5. #71663C2semidirect
6. #71662C2semidirect
7. #70665C2semidirect
8. #70664C2semidirect
10. #70341C2semidirect
11. #69896C2semidirect
47. #2950D3semidirect
486. C3#28184semidirect

Maximal subgroups

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

subgroupindex
C3×#29502
C2×#51922
#838872
#785792
#716632
#716622
#706652
#706642
#703512
#703412
#698962
#281843