Finite Group Databasebeta

← #10819913

Subgroups — #10819913

Order 768.

880 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83812C212yesmaximal, complemented
2#83286C211yesmaximal, characteristic
3#78990C212yesmaximal, complemented
4C3×#1884C211yesmaximal, characteristic, complemented
5#71817C211yesmaximal, characteristic, complemented
6#70397C212yesmaximal, complemented
7#70042C212yesmaximal, complemented
8#68777C212yesmaximal, complemented
9#68742C212yesmaximal, complemented
10#23197—31nomaximal, Sylow
11C3×#568C2211yescharacteristic
12C3×#532C2211yescharacteristic
13C2×#950C2212yes—
14C2×#950C2212yes—
15#5819C2211yescharacteristic
16#5745C2212yes—
17#5477C2211yescharacteristic
18C3×#421C2212yes—
19C3×#410C2211yescharacteristic
20#5240C2211yescharacteristic
21#5236C2212yes—
22#5145—24no—
23#5144—24no—
24#5139C2212yes—
25C2×#822C2212yes—
26#5122C2212yes—
27#5086C412yescomplemented
28#5074C412yescomplemented
29#5060C411yescharacteristic, complemented
30#5060C411yescharacteristic, complemented
31#5058C411yescharacteristic, complemented
32#5058C411yescharacteristic, complemented
33#5006C2212yes—
34#5005C2212yes—
35#4998C2212yes—
36#4997C2212yes—
37C2×#803C2211yescharacteristic
38#4824—24no—
39#4823—24no—
40C3×#339C2212yes—
41#4523C2212yes—
42#3150—31no—
43#2957—32no—
44#2957—32no—
45#1976—32no—
46#1976—32no—
47#1884—32no—
48#1884—31no—
49#1884D311yescharacteristic, complemented
50#1773—32no—
51C2×C3×#41C2311yescharacteristic
52C2×C3×#41C2311yescharacteristic
53C2×#235C2312yes—
54C3×#131—24no—
55C2×C3×M16—41no—
56C2×C3×M16C2312yes—
57C3×C4⋊Q8C2×C411yescharacteristic
58C3×#127C2×C411yescharacteristic
59C3×#124C2×C411yescharacteristic
60C3×#124C2×C411yescharacteristic
61C3×#121—41no—
62C3×C4×D4—41no—
63C3×#117C2312yes—
64C2×C3×C42C2311yescharacteristic
65#950—22no—
66#950—22no—
67#950—22no—
68#950—22no—
69#950—22no—
70#950—22no—
71#950—22no—
72#950—22no—
73#948—24no—
74C2×#216C2×C412yes—
75C2×#216C2×C412yes—
76C2×#215C2×C412yes—
77C2×#215C2×C412yes—
78C2×#214C2×C412yes—
79C2×#214C2×C412yes—
80C2×#213C2×C412yes—
81C2×#213C2×C412yes—
82#930—41no—
83#926C2312yes—
84C2×#208—41no—
85C2×#208C2312yes—
86C22×C3⋊C8C2312yes—
87#907—24no—
88D3×M16—42no—
89#901—24no—
90#875C2×C412yes—
91#874C2×C411yescharacteristic
92C4×D12—41no—
93#869C2×C411yescharacteristic
94C3×#104—24no—
95C3×#99—41no—
96#837—24no—
97#834—41no—
98#825—24no—
99#822D414yescomplemented
100#822D414yescomplemented
101#815—22no—
102C8×Dic3—22no—
103#808—22no—
104#807—22no—
105#806—42no—
106#805—24no—
107#803—22no—
108#803C2×C411yescharacteristic
109#803C2×C411yescharacteristic
110#803C2×C412yes—
111C4×C3⋊C8—22no—
112C2×#135—32no—
113C2×#135—32no—
114#568—31no—
115#568—32no—
116#568C2×D311yescharacteristic
117#532—31no—
118#532C2×D311yescharacteristic
119#497—32no—
120#497—31no—
121#497—31no—
122#496—32no—
123#496—31no—
124#496—31no—
125#444—64no—
126#444—64no—
127#443—64no—
128#443—64no—
129C2×#108—32no—
130#427—32no—
131#421—32no—
132#421—32no—
133#421—32no—
134#421C2×D312yes—
135#410—31no—
136#410—32no—
137#410C2×D311yescharacteristic
138#405—32no—
139#405—32no—
140C2×#97—31no—
141#339—32no—
142#339C2×D312yes—
143C3×#41—22no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41—22no—
147C2×C3×Q8C22×C411yescharacteristic
148C2×C3×Q8C22×C411yescharacteristic
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—41no—
152C2×C3×D4C22×C411yescharacteristic
153C2×C3×D4C22×C411yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2411yescharacteristic
158C2×#67—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163C2×D12—41no—
164C2×D12C22×C412yes—
165C2×C4×D3—22no—
166C2×Dic6C22×C412yes—
167C3×M16—22no—
168C3×M16—42no—
169C3×M16C2×D414yes—
170C2×C3×C8—22no—
171C2×C3×C8—42no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—41no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C42—21no—
178C3×C42—22no—
179C3×C42C22×C411yescharacteristic
180C3×C42C22×C411yescharacteristic
181#217—22no—
182#216—22no—
183#216—22no—
184#216—22no—
185#216—22no—
186#215—22no—
187#215—22no—
188#215—22no—
189#215—22no—
190#214—22no—
191#214—22no—
192#214—22no—
193#214—22no—
194#213—22no—
195#213—22no—
196#213—22no—
197#213—22no—
198(C2×D3)⋊C4—42no—
199C4⋊Dic3—22no—
200C4⋊Dic3—41no—
201C4×Dic3—22no—
202#208—22no—
203#208—42no—
204#208C2×D414yes—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—42no—
207C2×C3⋊C8C22×C412yes—
208C2×C3⋊C8C22×C412yes—
209C2×C3⋊C8C2×D412yes—
210C2×C3⋊C8C2×D412yes—
211C2×C3⋊C8#4112yes—
212C2×C3⋊C8#4112yes—
213ΓL(1,𝔽25)—44no—
214D3×C8—44no—
215C2×#41—32no—
216C2×#41C22×D311yescharacteristic
217C2×#41C22×D311yescharacteristic
218#135—62no—
219#135—62no—
220#135—62no—
221#135—62no—
222#135—62no—
223#135—62no—
224#135—62no—
225#135—62no—
226C2×Q16—32no—
227C2×Q16—32no—
228C2×SD16—32no—
229C2×SD16—32no—
230C2×SD16—32no—
231C2×SD16—32no—
232C2×D8—32no—
233C2×D8—32no—
234#131—24no—
235#131—64no—
236#131—64no—
237#131—64no—
238C2×M16—32no—
239C2×M16—41no—
240C2×M16—122no—
241C2×M16—121no—
242C2×M16C22×D312yes—
243C22×C8—32no—
244C4⋊Q8—31no—
245C4⋊Q8C4×D311yescharacteristic
246#127—31no—
247#127C4×D311yescharacteristic
248#124—32no—
249#124C4×D311yescharacteristic
250#124C4×D311yescharacteristic
251#121—41no—
252#121—121no—
253C4×D4—41no—
254C4×D4—121no—
255#117—32no—
256#117C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#108—34no—
259#108—34no—
260#107—62no—
261#106—62no—
262#104—24no—
263#104—64no—
264#104—64no—
265#104—64no—
266#99—41no—
267#99—122no—
268#99—121no—
269#97—32no—
270#97—31no—
271#97—31no—
272#97—62no—
273#97—62no—
274C4×C8—62no—
275C4×C8—62no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3H(5,𝔽2)11yescharacteristic
279C22×D3—22no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×D4—22no—
283C3×D4—22no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—82no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—81no—
297C2×C3×C4C23×C411yescharacteristic
298C2×C3×C4C22×D412yes—
299C2×C3×C4C2×#4112yes—
300C2×C3×C4H(5,𝔽2)11yescharacteristic
301#67—42no—
302#67—42no—
303C2×Dic3—22no—
304C2×Dic3—42no—
305D12—22no—
306D12—22no—
307D12—82no—
308C4×D3—42no—
309C4×D3—42no—
310Dic6—22no—
311Dic6—22no—
312C3×C8—24no—
313C3⋊C8—24no—
314C3⋊C8C4×D412yes—
315C3⋊C8C4×D412yes—
316C3⋊C8C4×D412yes—
317C3⋊C8C4×D412yes—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—62no—
323#41—62no—
324#41—62no—
325#41—62no—
326C2×Q8—32no—
327C2×Q8C2×C4×D311yescharacteristic
328C2×Q8C2×C4×D311yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—32no—
332C2×D4—41no—
333C2×D4—62no—
334C2×D4—121no—
335C2×D4C2×C4×D311yescharacteristic
336C2×D4C2×C4×D311yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—62no—
341C22×C4C23×D311yescharacteristic
342Q16—62no—
343Q16—62no—
344Q16—62no—
345Q16—62no—
346SD16—62no—
347SD16—62no—
348SD16—62no—
349SD16—62no—
350SD16—62no—
351SD16—62no—
352SD16—62no—
353SD16—62no—
354D8—62no—
355D8—62no—
356D8—62no—
357D8—62no—
358M16—22no—
359M16—34no—
360M16—42no—
361M16—62no—
362M16—124no—
363M16—122no—
364M16D3×D414yes—
365C2×C8—22no—
366C2×C8—32no—
367C2×C8—32no—
368C2×C8—32no—
369C2×C8—32no—
370C2×C8—32no—
371C2×C8—32no—
372C2×C8—42no—
373C2×C8—62no—
374C2×C8—124no—
375C2×C8—122no—
376C4⋊C4—22no—
377C4⋊C4—41no—
378C4⋊C4—62no—
379C4⋊C4—121no—
380C22⋊C4—22no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—62no—
384C22⋊C4—122no—
385C42—21no—
386C42—22no—
387C42—62no—
388C42C2×C4×D311yescharacteristic
389C42C2×C4×D311yescharacteristic
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3#51811yescharacteristic
396C22×C3#55012yes—
397C2×D3—42no—
398C2×D3—82no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—82no—
404C3×C4C2×C4×D412yes—
405C3×C4D4211yescharacteristic
406C3×C4#55911yescharacteristic
407Dic3—42no—
408Dic3—82no—
409C23—21no—
410C23—21no—
411C23—62no—
412C23#100411yescharacteristic
413Q8—22no—
414Q8—22no—
415Q8—62no—
416Q8—62no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—42no—
421D4—62no—
422D4—62no—
423D4—82no—
424D4—122no—
425D4—122no—
426D4—242no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—62no—
437C2×C4—81no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4C22×C4×D311yescharacteristic
442C2×C4C2×D3×D412yes—
443C2×C4#100411yescharacteristic
444C2×C4D3×#4112yes—
445C8—24no—
446C8—32no—
447C8—32no—
448C8—32no—
449C8—32no—
450C8—64no—
451C2×C3—22no—
452C2×C3—21no—
453C2×C3—81no—
454C2×C3—81no—
455C2×C3#236611yescharacteristic
456D3—82no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22—81no—
462C22—122no—
463C22—242no—
464C22#548611yescharacteristic
465C22#559912yes—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—82no—
471C4—122no—
472C4—242no—
473C4C4×D3×D412yes—
474C4#559011yescharacteristic
475C4#566411yescharacteristic
476C3#2319711yescharacteristic, complemented, Sylow
477C2—21no—
478C2—22no—
479C2—81no—
480C2—81no—
481C2—242no—
482C2#7467911yescharacteristic
subgroupG/H
centerC4#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1884C2
socleC2×C3#2366
prime radical#83286C2
cosocleC22×C3×C4C24
hypercenter#1884D3
hypocenterC3#23197
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1884

Complements

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

NH ≅ G/Ntype
1. #83812C2semidirect
3. #78990C2semidirect
4. C3×#1884C2semidirect
5. #71817C2semidirect
6. #70397C2semidirect
7. #70042C2semidirect
8. #68777C2semidirect
9. #68742C2semidirect
27. #5086C4semidirect
28. #5074C4semidirect
29. #5060C4semidirect
30. #5060C4semidirect
31. #5058C4semidirect
32. #5058C4semidirect
49. #1884D3semidirect
99. #822D4semidirect
100. #822D4semidirect
476. C3#23197semidirect

Maximal subgroups

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

subgroupindex
#838122
#832862
#789902
C3×#18842
#718172
#703972
#700422
#687772
#687422
#231973