Finite Group Databasebeta

← #10802132

Subgroups — #10802132

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#79257C212yesmaximal, complemented
2#79256C211yesmaximal, characteristic
3C2×#4894C211yesmaximal, characteristic, complemented
4#78550C211yesmaximal, characteristic, complemented
5C3×#1953C211yesmaximal, characteristic, complemented
6#70727C212yesmaximal, complemented
7#70719C211yesmaximal, characteristic, complemented
8#70703C211yesmaximal, characteristic, complemented
9#69897C211yesmaximal, characteristic, complemented
10#69039C212yesmaximal, complemented
11#69014C212yesmaximal, complemented
12#21559—31nomaximal, Sylow
13#5748C2212yes—
14C2×#907C2212yes—
15C2×#907C2211yescharacteristic
16C2×#904C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5522C2211yescharacteristic
19C3×#430C2212yes—
20C2×C3×#108C2211yescharacteristic
21C2×C3×#108C2211yescharacteristic
22C3×#405C2211yescharacteristic
23#5139C2212yes—
24#5139C2212yes—
25C2×#825C2212yes—
26#5132C2212yes—
27#5122C2212yes—
28Dic3×M16C2211yescharacteristic
29#5106C2211yescharacteristic
30C2×#815C2211yescharacteristic
31#5006C2212yes—
32#4997C2211yescharacteristic
33#4898C414yescomplemented
34#4897C412yescomplemented
35#4896C412yescomplemented
36#4895—24no—
37#4894—24no—
38#4894—21no—
39#4894—21no—
40#4894—21no—
41#4894—21no—
42#4708—21no—
43D3×#97—21no—
44#4689—21no—
45C4×ΓL(1,𝔽25)—21no—
46C3×#344C2212yes—
47#4556C2212yes—
48#4555C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#444—31no—
52#2875—31no—
53#1976—31no—
54#1953D311yescharacteristic, complemented
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1773—31no—
59#1757—32no—
60C2×#235C2312yes—
61C2×#235C2311yescharacteristic
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#117C2311yescharacteristic
67#926C2312yes—
68C2×C4×Dic3C2311yescharacteristic
69C2×#208C2311yescharacteristic
70C22×C3⋊C8C2311yescharacteristic
71#909—22no—
72#909C2×C414yes—
73#908—22no—
74#908C2×C414yes—
75#907—22no—
76#907—22no—
77#907—22no—
78#907—22no—
79#907—22no—
80#907D414yes—
81C2×Dic12C2×C411yescharacteristic, complemented
82#904—24no—
83#904—21no—
84#904—21no—
85#904—21no—
86#904—21no—
87C2×D24C2×C411yescharacteristic, complemented
88C2×#204C2×C412yescomplemented
89#901—21no—
90#901—21no—
91#901C2×C412yes—
92#901C2×C412yes—
93C2×ΓL(1,𝔽25)—21no—
94C2×ΓL(1,𝔽25)—21no—
95C2×ΓL(1,𝔽25)—22no—
96C2×D3×C8—21no—
97C2×D3×C8—21no—
98C2×D3×C8—22no—
99#893—21no—
100#887—21no—
101#67⋊C4—22no—
102#872—21no—
103C42×D3—21no—
104C3×#108—24no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#108C2×C412yescomplemented
108C3×#108C2×C412yescomplemented
109C3×#97—21no—
110C3×C4×C8—21no—
111#825—24no—
112#825—24no—
113#825—22no—
114#825—22no—
115#824C2×C414yescomplemented
116#823C2×C414yescomplemented
117#818—21no—
118#817—21no—
119#815—21no—
120#815—22no—
121#815D411yescharacteristic
122#815D411yescharacteristic
123#815D411yescharacteristic
124#815D411yescharacteristic
125C8×Dic3—21no—
126C8×Dic3—22no—
127#806—24no—
128#803—21no—
129C4×C3⋊C8—21no—
130#575—32no—
131C2×#135—31no—
132C2×#131—31no—
133C2×#131—32no—
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#430C2×D312yes—
146C2×#108C2×D311yescharacteristic
147C2×#108C2×D311yescharacteristic
148#427—31no—
149#421—32no—
150#421—32no—
151C2×#104—32no—
152#413—32no—
153#410—32no—
154#405—32no—
155#405—31no—
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#344C2×D312yes—
165#339—32no—
166C22×C3×C4C2411yescharacteristic
167C2×#67—21no—
168C2×#67—22no—
169C22×Dic3—21no—
170#235—21no—
171#235—21no—
172#235—22no—
173#235—22no—
174#235—22no—
175#235—22no—
176#235C22×C412yes—
177#235C2×D412yes—
178C2×D12C22×C412yes—
179C2×D12#4111yescharacteristic
180C2×C4×D3—21no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×Dic6C22×C412yes—
184C2×Dic6#4111yescharacteristic
185C3×M16—21no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16C22×C414yes—
190C3×M16C2×D412yes—
191C2×C3×C8—21no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
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×C22⋊C4#4112yes—
202C3×C42—21no—
203#217—22no—
204(C2×D3)⋊C4—22no—
205C4⋊Dic3—22no—
206Dic3⋊C4—22no—
207C4×Dic3—22no—
208C4×Dic3—21no—
209C4×Dic3—22no—
210C4×Dic3C2×D411yescharacteristic
211C4×Dic3C2×D411yescharacteristic
212#208—21no—
213#208C22×C412yes—
214C2×C3⋊C8—21no—
215C2×C3⋊C8C2×D411yescharacteristic
216C2×C3⋊C8C2×D411yescharacteristic
217C2×C3⋊C8C2×D412yes—
218C2×C3⋊C8#4112yes—
219Dic12—24no—
220Dic12—21no—
221Dic12—21no—
222D24—24no—
223D24—21no—
224D24—21no—
225#204—24no—
226#204—24no—
227#204—22no—
228#204—22no—
229ΓL(1,𝔽25)—22no—
230ΓL(1,𝔽25)—22no—
231ΓL(1,𝔽25)—24no—
232ΓL(1,𝔽25)—42no—
233ΓL(1,𝔽25)—42no—
234D3×C8—22no—
235D3×C8—22no—
236D3×C8—24no—
237D3×C8—42no—
238D3×C8—42no—
239C2×#41—31no—
240C2×#41—32no—
241#137—34no—
242#137—62no—
243AGL(1,ℤ/8)—34no—
244AGL(1,ℤ/8)—62no—
245#135—64no—
246#135—61no—
247#135—61no—
248#135—61no—
249#135—61no—
250C2×Q16—31no—
251C2×SD16—32no—
252C2×D8—31no—
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—61no—
265C2×M16—62no—
266C2×M16—61no—
267C2×M16C22×D312yes—
268C2×M16C22×D312yes—
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—32no—
279#117—61no—
280#117C22×D311yescharacteristic
281C2×C42—31no—
282C2×C42—61no—
283#108—24no—
284#108—21no—
285#108—21no—
286#108C4×D312yescomplemented
287#108C4×D312yescomplemented
288#107—61no—
289#106—61no—
290#104—62no—
291#104—62no—
292#104—64no—
293#104—64no—
294#101—34no—
295#100—34no—
296#99—64no—
297#97—21no—
298#97—31no—
299#97—31no—
300#97—31no—
301#97—31no—
302#97—61no—
303#97—61no—
304#97—62no—
305C4×C8—21no—
306C4×C8—61no—
307C4×C8—61no—
308C4×C8—62no—
309C23×C3C2×#4111yescharacteristic
310C22×D3—21no—
311C22×D3—22no—
312C2×C3×C4—22no—
313C2×C3×C4C23×C411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319#67—22no—
320#67—22no—
321#67—42no—
322#67—42no—
323C2×Dic3—21no—
324C2×Dic3—22no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—42no—
328C2×Dic3—81no—
329D12—22no—
330D12—22no—
331D12—21no—
332D12C4×D412yes—
333C4×D3—22no—
334C4×D3—22no—
335C4×D3—42no—
336C4×D3—42no—
337C4×D3—82no—
338Dic6—21no—
339Dic6—22no—
340Dic6—22no—
341Dic6C4×D412yes—
342C3×C8—24no—
343C3×C8—22no—
344C3×C8—42no—
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3×C8C4×D411yescharacteristic
348C3×C8C4×D411yescharacteristic
349C3⋊C8—22no—
350C3⋊C8#12112yes—
351C3⋊C8#12112yes—
352#41—32no—
353#41—32no—
354#41—61no—
355#41—61no—
356#41—62no—
357#41—62no—
358#41—62no—
359#41—62no—
360C2×Q8—32no—
361C2×Q8—31no—
362C2×D4—32no—
363C2×D4—31no—
364C2×D4—61no—
365C2×D4—62no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4C23×D311yescharacteristic
371Q16—64no—
372Q16—61no—
373Q16—61no—
374SD16—64no—
375SD16—64no—
376SD16—62no—
377SD16—62no—
378D8—64no—
379D8—61no—
380D8—61no—
381M16—21no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—32no—
386M16—64no—
387M16—61no—
388M16—62no—
389M16—62no—
390M16—122no—
391M16—122no—
392M16C2×C4×D314yes—
393M16D3×D412yes—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—31no—
398C2×C8—32no—
399C2×C8—31no—
400C2×C8—32no—
401C2×C8—64no—
402C2×C8—61no—
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#23911yescharacteristic
412C2×C8#23911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C22⋊C4—62no—
417C22⋊C4—62no—
418C22⋊C4#23912yes—
419C42—21no—
420C42—31no—
421C42—31no—
422C42—62no—
423C42—61no—
424C42—62no—
425C22×C3—41no—
426C22×C3C4×#4111yescharacteristic
427C22×C3#52511yescharacteristic
428C22×C3#53211yescharacteristic
429C2×D3—22no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—82no—
433C3×C4—42no—
434C3×C4C2×C4×D411yescharacteristic
435C3×C4C2×C4×D411yescharacteristic
436C3×C4C2×#12111yescharacteristic
437C3×C4#52911yescharacteristic
438Dic3—22no—
439Dic3—41no—
440Dic3—41no—
441Dic3—42no—
442Dic3—82no—
443C23—61no—
444C23—62no—
445C23C2×#23911yescharacteristic
446Q8—32no—
447Q8—61no—
448Q8—62no—
449Q8—62no—
450D4—32no—
451D4—62no—
452D4—62no—
453D4—61no—
454D4—62no—
455D4—62no—
456D4—122no—
457D4—122no—
458C2×C4—22no—
459C2×C4—61no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—242no—
469C2×C4—241no—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23911yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—24no—
477C8—22no—
478C8—32no—
479C8—32no—
480C8—42no—
481C8—62no—
482C8#89311yescharacteristic
483C8#89311yescharacteristic
484C8#89311yescharacteristic
485C8#89311yescharacteristic
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3C4×#12111yescharacteristic
490D3—42no—
491D3—82no—
492C22—41no—
493C22—62no—
494C22—122no—
495C22—122no—
496C22—242no—
497C22C4×#23911yescharacteristic
498C22#562711yescharacteristic
499C22#572511yescharacteristic
500C4—42no—
501C4—62no—
502C4—121no—
503C4—121no—
504C4—122no—
505C4—242no—
506C4C2×#89311yescharacteristic
507C4C4×D3×D411yescharacteristic
508C4D3×#12111yescharacteristic
509C4#569111yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—122no—
515C2—242no—
516C2#7596411yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1953C2
socleC2×C3C4×#121
prime radical#79256C2
cosocleC22×C3×C4C24
hypercenter#1953D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
1. #79257C2semidirect
3. C2×#4894C2semidirect
4. #78550C2semidirect
5. C3×#1953C2semidirect
6. #70727C2semidirect
7. #70719C2semidirect
8. #70703C2semidirect
9. #69897C2semidirect
10. #69039C2semidirect
11. #69014C2semidirect
33. #4898C4semidirect
34. #4897C4semidirect
35. #4896C4semidirect
54. #1953D3semidirect
81. C2×Dic12C2×C4semidirect
87. C2×D24C2×C4semidirect
88. C2×#204C2×C4semidirect
107. C3×#108C2×C4semidirect
108. C3×#108C2×C4semidirect
115. #824C2×C4semidirect
116. #823C2×C4semidirect
286. #108C4×D3semidirect
287. #108C4×D3semidirect
510. C3#21559semidirect

Maximal subgroups

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

subgroupindex
#792572
#792562
C2×#48942
#785502
C3×#19532
#707272
#707192
#707032
#698972
#690392
#690142
#215593