Finite Group Databasebeta

← #10802140

Subgroups — #10802140

Order 768.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83928C211yesmaximal, characteristic
2C2×#5197C211yesmaximal, characteristic, complemented
3#78989C212yesmaximal, complemented
4#78550C211yesmaximal, characteristic, complemented
5C3×#1976C211yesmaximal, characteristic, complemented
6#70665C211yesmaximal, characteristic, complemented
7#70660C211yesmaximal, characteristic, complemented
8#70401C212yesmaximal, complemented
9#70055C212yesmaximal, complemented
10#70040C212yesmaximal, complemented
11#69901C211yesmaximal, characteristic, complemented
12#21559—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5822C2212yes—
15C2×#948C2212yes—
16C2×#907C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5522C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#421C2212yes—
21C3×#410C2212yes—
22C3×#405C2211yescharacteristic
23#5237C2212yes—
24C2×#837C2212yes—
25#5197—21no—
26#5197—21no—
27#5197—21no—
28#5197—21no—
29#5192C412yescomplemented
30#5177C414yescomplemented
31#5160C412yescomplemented
32#5124C2212yes—
33Dic3×M16C2211yescharacteristic
34C2×#819C2211yescharacteristic
35C2×#819C2211yescharacteristic
36C2×#815C2211yescharacteristic
37#5005C2212yes—
38#4997C2212yes—
39#4997C2211yescharacteristic
40#4824—24no—
41#4823—24no—
42#4708—21no—
43D3×#97—21no—
44#4689—21no—
45C4×ΓL(1,𝔽25)—21no—
46#4533C2212yes—
47#4533C2212yes—
48#4532C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#444—31no—
52#2875—31no—
53#1976D311yescharacteristic, complemented
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1773—31no—
59#1757—32no—
60C2×C3×#41C2312yes—
61C2×#235C2311yescharacteristic
62C3×#135—21no—
63C3×#135—21no—
64C3×#135—21no—
65C3×#135—21no—
66C2×C3×Q16C2×C411yescharacteristic, complemented
67C2×C3×SD16C2×C412yescomplemented
68C2×C3×D8C2×C411yescharacteristic, complemented
69C2×C3×M16C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#117C2312yes—
72C3×#117C2311yescharacteristic
73#951—22no—
74#950—24no—
75#949—22no—
76#948—22no—
77#948—22no—
78#948—22no—
79#948—22no—
80#942C2×C414yes—
81#932C2×C414yes—
82C2×C4×Dic3C2311yescharacteristic
83C2×#208—22no—
84C2×#208C2312yes—
85C2×#208C2312yes—
86C2×#208C2311yescharacteristic
87C22×C3⋊C8—22no—
88C22×C3⋊C8C2311yescharacteristic
89#907—22no—
90#907D414yes—
91#901—21no—
92#901—21no—
93#901C2×C412yes—
94#901C2×C412yes—
95C2×ΓL(1,𝔽25)—21no—
96C2×ΓL(1,𝔽25)—21no—
97C2×D3×C8—21no—
98C2×D3×C8—21no—
99#893—21no—
100#887—21no—
101#67⋊C4—22no—
102#872—21no—
103C42×D3—21no—
104C3×#107—21no—
105C3×#106—21no—
106C3×#104—24no—
107C3×#99—24no—
108C3×#97—21no—
109C3×C4×C8—21no—
110#837—22no—
111#837—22no—
112#836C2×C414yescomplemented
113#833C2×C414yescomplemented
114#822—24no—
115#819—21no—
116#819—21no—
117#819C2×C412yescomplemented
118#819C2×C412yescomplemented
119#815—21no—
120#815D411yescharacteristic
121#815D411yescharacteristic
122#815D411yescharacteristic
123#815D411yescharacteristic
124C8×Dic3—21no—
125#805—24no—
126#803—21no—
127#803—22no—
128C4×C3⋊C8—21no—
129C4×C3⋊C8—22no—
130#575—32no—
131C2×#135C2×D311yescharacteristic
132C2×#131—32no—
133C2×#131—31no—
134C2×#131—31no—
135C4×#41—31no—
136#473—32no—
137#472—32no—
138#471—34no—
139#444—61no—
140#444—61no—
141#444—61no—
142#444—61no—
143#444—64no—
144#443—64no—
145#430—32no—
146C2×#108—31no—
147C2×#108—31no—
148#427C2×D311yescharacteristic
149#421—32no—
150#421C2×D312yes—
151C2×#104—32no—
152#413—32no—
153#410C2×D312yes—
154#405—31no—
155#405—32no—
156#405—61no—
157#405—61no—
158#405C2×D311yescharacteristic
159C4×M16—31no—
160C4×M16—61no—
161C2×#97—31no—
162C2×#97—61no—
163#344—32no—
164#344—32no—
165#339—32no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C2×C3×Q8C22×C412yes—
171C2×C3×D4—22no—
172C2×C3×D4C22×C412yes—
173C22×C3×C4—22no—
174C22×C3×C4C2411yescharacteristic
175C2×#67—21no—
176C22×Dic3—21no—
177#235—21no—
178#235—21no—
179#235C22×C412yes—
180#235C2×D412yes—
181C2×D12#4111yescharacteristic
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×Dic6#4111yescharacteristic
185C3×Q16—21no—
186C3×Q16—21no—
187C3×SD16—22no—
188C3×SD16—22no—
189C3×D8—21no—
190C3×D8—21no—
191C3×M16—21no—
192C3×M16C2×D412yes—
193C2×C3×C8—21no—
194C2×C3×C8C22×C411yescharacteristic
195C2×C3×C8C22×C411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8#4111yescharacteristic
199C2×C3×C8#4111yescharacteristic
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4#4112yes—
204C3×C42—21no—
205C3×C42—22no—
206#216—24no—
207#215—24no—
208#214—24no—
209#213—24no—
210(C2×D3)⋊C4—22no—
211Dic3⋊C4—22no—
212C4×Dic3—22no—
213C4×Dic3—21no—
214C4×Dic3C2×D411yescharacteristic
215C4×Dic3C2×D411yescharacteristic
216#208—21no—
217#208—22no—
218#208—22no—
219#208—22no—
220#208—42no—
221#208—42no—
222#208C22×C414yes—
223#208C22×C412yes—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—42no—
228C2×C3⋊C8—42no—
229C2×C3⋊C8C2×D412yes—
230C2×C3⋊C8C2×D411yescharacteristic
231C2×C3⋊C8C2×D411yescharacteristic
232C2×C3⋊C8#4112yes—
233ΓL(1,𝔽25)—22no—
234ΓL(1,𝔽25)—22no—
235ΓL(1,𝔽25)—24no—
236D3×C8—22no—
237D3×C8—22no—
238D3×C8—24no—
239C2×#41—31no—
240C2×#41C22×D312yes—
241#137—34no—
242#137—62no—
243AGL(1,ℤ/8)—34no—
244AGL(1,ℤ/8)—62no—
245#135—21no—
246#135—21no—
247#135—21no—
248#135—21no—
249#135—64no—
250C2×Q16C4×D311yescharacteristic, complemented
251C2×SD16C4×D312yescomplemented
252C2×D8C4×D311yescharacteristic, complemented
253#131—34no—
254#131—32no—
255#131—32no—
256#131—62no—
257#131—62no—
258#131—62no—
259#131—62no—
260#131—62no—
261#131—61no—
262#131—61no—
263C2×M16—31no—
264C2×M16—32no—
265C2×M16—32no—
266C2×M16—61no—
267C2×M16—62no—
268C2×M16—61no—
269C2×M16C22×D311yescharacteristic
270C22×C8—31no—
271C22×C8—61no—
272C22×C8—62no—
273C22×C8—61no—
274C22×C8C22×D311yescharacteristic
275C4×Q8—61no—
276C4×D4—61no—
277C4×D4—62no—
278#117—61no—
279#117C22×D312yes—
280#117C22×D311yescharacteristic
281C2×C42—31no—
282C2×C42—61no—
283#108—32no—
284#108—32no—
285#108—64no—
286#108—61no—
287#108—61no—
288#107—21no—
289#106—21no—
290#104—24no—
291#104—64no—
292#104—62no—
293#104—62no—
294#101—34no—
295#100—34no—
296#99—24no—
297#97—21no—
298#97—31no—
299#97—31no—
300#97—31no—
301#97—31no—
302#97—61no—
303#97—62no—
304#97—61no—
305C4×C8—21no—
306C4×C8—61no—
307C4×C8—62no—
308C4×C8—61no—
309C23×C3—22no—
310C23×C3C2×#4111yescharacteristic
311C22×D3—21no—
312C3×Q8—22no—
313C3×Q8—22no—
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—42no—
317C3×D4—42no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4C23×C411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329#67—22no—
330#67—22no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—81no—
335D12—21no—
336D12C4×D412yes—
337C4×D3—22no—
338C4×D3—22no—
339C4×D3—82no—
340Dic6—21no—
341Dic6C4×D412yes—
342C3×C8—22no—
343C3×C8C4×D411yescharacteristic
344C3×C8C4×D411yescharacteristic
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3⋊C8—22no—
348C3⋊C8—24no—
349C3⋊C8—42no—
350C3⋊C8#12112yes—
351C3⋊C8#12112yes—
352#41—22no—
353#41—22no—
354#41—22no—
355#41—22no—
356#41—32no—
357#41—32no—
358#41—61no—
359#41—61no—
360C2×Q8—31no—
361C2×Q8C2×C4×D312yes—
362C2×D4—22no—
363C2×D4—31no—
364C2×D4—61no—
365C2×D4C2×C4×D312yes—
366C22×C4—22no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4C23×D311yescharacteristic
371Q16—21no—
372Q16—21no—
373Q16—64no—
374SD16—22no—
375SD16—22no—
376SD16—64no—
377SD16—64no—
378D8—21no—
379D8—21no—
380D8—64no—
381M16—21no—
382M16—32no—
383M16—34no—
384M16—64no—
385M16—61no—
386M16—62no—
387M16—62no—
388M16—62no—
389M16—62no—
390M16—62no—
391M16—122no—
392M16—122no—
393M16D3×D412yes—
394C2×C8—21no—
395C2×C8—31no—
396C2×C8—32no—
397C2×C8—31no—
398C2×C8—32no—
399C2×C8—64no—
400C2×C8—61no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—122no—
406C2×C8—122no—
407C2×C8C2×C4×D311yescharacteristic
408C2×C8C2×C4×D311yescharacteristic
409C2×C8D3×D411yescharacteristic
410C2×C8D3×D411yescharacteristic
411C2×C8#23711yescharacteristic
412C2×C8#23711yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—62no—
416C22⋊C4—22no—
417C22⋊C4—62no—
418C22⋊C4#23712yes—
419C42—21no—
420C42—22no—
421C42—31no—
422C42—31no—
423C42—62no—
424C42—61no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—82no—
428C22×C3C4×#4111yescharacteristic
429C22×C3#52511yescharacteristic
430C22×C3#53211yescharacteristic
431C2×D3—22no—
432C2×D3—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—82no—
436C3×C4C2×C4×D411yescharacteristic
437C3×C4C2×C4×D411yescharacteristic
438C3×C4C2×#12111yescharacteristic
439C3×C4#52911yescharacteristic
440Dic3—22no—
441Dic3—41no—
442Dic3—41no—
443C23—22no—
444C23—61no—
445C23C2×#23711yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—32no—
449Q8—61no—
450D4—22no—
451D4—22no—
452D4—32no—
453D4—42no—
454D4—42no—
455D4—62no—
456D4—62no—
457D4—61no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—61no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—242no—
469C2×C4—241no—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—22no—
477C8—32no—
478C8—32no—
479C8—62no—
480C8—64no—
481C8—122no—
482C8#67⋊C411yescharacteristic
483C8#67⋊C411yescharacteristic
484C8#67⋊C411yescharacteristic
485C8#67⋊C411yescharacteristic
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—82no—
490C2×C3C4×#12111yescharacteristic
491D3—42no—
492C22—41no—
493C22—42no—
494C22—62no—
495C22—82no—
496C22—122no—
497C22C4×#23711yescharacteristic
498C22#560611yescharacteristic
499C22#571611yescharacteristic
500C4—42no—
501C4—42no—
502C4—62no—
503C4—82no—
504C4—121no—
505C4—121no—
506C4C2×#67⋊C411yescharacteristic
507C4C4×D3×D411yescharacteristic
508C4D3×#12111yescharacteristic
509C4#568011yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—82no—
515C2—122no—
516C2#7629111yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3C4×#121
prime radical#83928C2
cosocleC22×C3×C4C24
hypercenter#1976D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. C2×#5197C2semidirect
3. #78989C2semidirect
4. #78550C2semidirect
5. C3×#1976C2semidirect
6. #70665C2semidirect
7. #70660C2semidirect
8. #70401C2semidirect
9. #70055C2semidirect
10. #70040C2semidirect
11. #69901C2semidirect
29. #5192C4semidirect
30. #5177C4semidirect
31. #5160C4semidirect
53. #1976D3semidirect
66. C2×C3×Q16C2×C4semidirect
67. C2×C3×SD16C2×C4semidirect
68. C2×C3×D8C2×C4semidirect
112. #836C2×C4semidirect
113. #833C2×C4semidirect
117. #819C2×C4semidirect
118. #819C2×C4semidirect
250. C2×Q16C4×D3semidirect
251. C2×SD16C4×D3semidirect
252. C2×D8C4×D3semidirect
510. C3#21559semidirect

Maximal subgroups

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

subgroupindex
#839282
C2×#51972
#789892
#785502
C3×#19762
#706652
#706602
#704012
#700552
#700402
#699012
#215593