Finite Group Databasebeta

← #10802139

Subgroups — #10802139

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#79256C211yesmaximal, characteristic
2#78579C212yesmaximal, complemented
3C2×#4717C211yesmaximal, characteristic, complemented
4#78486C211yesmaximal, characteristic, complemented
5C3×#1773C211yesmaximal, characteristic, complemented
6#70707C211yesmaximal, characteristic, complemented
7#70665C211yesmaximal, characteristic, complemented
8#70658C211yesmaximal, characteristic, complemented
9#69898C212yesmaximal, complemented
10#68778C212yesmaximal, complemented
11#68750C212yesmaximal, complemented
12#21559—31nomaximal, Sylow
13#5748C2212yes—
14C2×#904C2211yescharacteristic
15C2×#901C2212yes—
16C2×#901C2211yescharacteristic
17C2×#901C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#427C2211yescharacteristic
21C3×#405C2212yes—
22C2×C3×#97C2211yescharacteristic
23#5132C2212yes—
24#5125C2212yes—
25C2×#819C2211yescharacteristic
26#5101C2211yescharacteristic
27#5101C2211yescharacteristic
28#5006C2212yes—
29#5005C2212yes—
30#5005C2212yes—
31C2×#805C2212yes—
32C4×#208C2211yescharacteristic
33#4898C414yescomplemented
34#4897C412yescomplemented
35#4896C412yescomplemented
36#4717—24no—
37#4717—21no—
38#4717—21no—
39#4717—21no—
40#4717—21no—
41#4708—21no—
42#4708—21no—
43#4707—21no—
44D3×#97—21no—
45#4700—24no—
46C3×#339C2212yes—
47#4559C2212yes—
48#4533C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#444—31no—
52#2875—31no—
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1773D311yescharacteristic, complemented
59#1757—32no—
60C2×#235C2312yes—
61C2×#235C2311yescharacteristic
62C2×C3×M16C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#117C2312yes—
66C2×C3×C42C2311yescharacteristic
67C4×#67—22no—
68#926C2311yescharacteristic
69C2×#208C2312yes—
70C2×#208C2311yescharacteristic
71C2×#208C2311yescharacteristic
72#909—22no—
73#909C2×C414yes—
74#908—22no—
75#908C2×C414yes—
76C2×Dic12C2×C411yescharacteristic, complemented
77#904—24no—
78#904—21no—
79#904—21no—
80#904—21no—
81#904—21no—
82C2×D24C2×C411yescharacteristic, complemented
83C2×#204C2×C412yescomplemented
84#901—21no—
85#901—21no—
86#901—22no—
87#901—22no—
88#901—22no—
89#901—22no—
90#901—22no—
91#901C2×C412yes—
92#901C2×C412yes—
93#901D414yes—
94C2×ΓL(1,𝔽25)—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—21no—
98C2×D3×C8—21no—
99C2×D3×C8—22no—
100C4×D12—21no—
101#872—21no—
102C42×D3—21no—
103C4×Dic6—21no—
104C3×#108C2×C412yescomplemented
105C3×#108C2×C412yescomplemented
106C3×#107—21no—
107C3×#106—21no—
108C3×#97—22no—
109C3×#97D411yescharacteristic
110C3×#97D411yescharacteristic
111C3×#97D411yescharacteristic
112C3×#97D411yescharacteristic
113C3×C4×C8—22no—
114#824C2×C414yescomplemented
115#823C2×C414yescomplemented
116#819—21no—
117#819—21no—
118#819—24no—
119#815—21no—
120#815—21no—
121C8×Dic3—21no—
122C8×Dic3—21no—
123#806—24no—
124#805—24no—
125#805—22no—
126#805—22no—
127#805—24no—
128#803—21no—
129C4×C3⋊C8—21no—
130#575—32no—
131C2×#135—31no—
132C2×#131—31no—
133C2×#131—31no—
134C2×#131—32no—
135C4×#41—31no—
136#473—32no—
137#472—32no—
138#471—34no—
139#444—64no—
140#444—61no—
141#444—61no—
142#444—61no—
143#444—61no—
144#443—64no—
145#430—32no—
146C2×#108—31no—
147C2×#108C2×D311yescharacteristic
148#427C2×D311yescharacteristic
149#421—32no—
150#421—32no—
151C2×#104—32no—
152#413—32no—
153#410—32no—
154#405—31no—
155#405—31no—
156#405—61no—
157#405—61no—
158#405C2×D312yes—
159C4×M16—31no—
160C4×M16—61no—
161C2×#97—61no—
162C2×#97C2×D311yescharacteristic
163#344—32no—
164#344—32no—
165#339C2×D312yes—
166C22×C3×C4—21no—
167C22×C3×C4C2411yescharacteristic
168C2×#67—21no—
169C2×#67—22no—
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—22no—
186C3×M16C22×C414yes—
187C2×C3×C8—22no—
188C2×C3×C8C22×C411yescharacteristic
189C2×C3×C8C22×C411yescharacteristic
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D412yes—
195C2×C3×C8#4111yescharacteristic
196C2×C3×C8#4111yescharacteristic
197C2×C3×C8#4112yes—
198C3×C4⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—21no—
201C3×C42—22no—
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217#4112yes—
205(C2×D3)⋊C4—22no—
206C4⋊Dic3—21no—
207Dic3⋊C4—22no—
208C4×Dic3—22no—
209C4×Dic3—21no—
210#208—21no—
211#208—21no—
212#208—22no—
213#208—22no—
214#208C22×C412yes—
215#208C2×D412yes—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
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—32no—
254#131—32no—
255#131—34no—
256#131—61no—
257#131—61no—
258#131—62no—
259#131—62no—
260#131—62no—
261#131—62no—
262#131—62no—
263C2×M16—31no—
264C2×M16—31no—
265C2×M16—32no—
266C2×M16—61no—
267C2×M16—61no—
268C2×M16—62no—
269C2×M16C22×D312yes—
270C22×C8—61no—
271C22×C8—61no—
272C22×C8—62no—
273C22×C8C22×D311yescharacteristic
274C22×C8C22×D311yescharacteristic
275C4×Q8—61no—
276C4×D4—61no—
277C4×D4—62no—
278#117—31no—
279#117—61no—
280#117C22×D312yes—
281C2×C42—61no—
282C2×C42C22×D311yescharacteristic
283#108—64no—
284#108—61no—
285#108—61no—
286#108C4×D312yescomplemented
287#108C4×D312yescomplemented
288#107—21no—
289#106—21no—
290#104—64no—
291#104—64no—
292#104—62no—
293#104—62no—
294#101—34no—
295#100—34no—
296#99—64no—
297#97—22no—
298#97—61no—
299#97—61no—
300#97—61no—
301#97D1211yescharacteristic
302#97D1211yescharacteristic
303#97D1211yescharacteristic
304#97D1211yescharacteristic
305C4×C8—22no—
306C4×C8—61no—
307C4×C8—61no—
308C4×C8—61no—
309C23×C3C2×#4111yescharacteristic
310C22×D3—21no—
311C22×D3—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—42no—
315C2×C3×C4—81no—
316C2×C3×C4C23×C411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322#67—22no—
323#67—22no—
324#67—42no—
325#67—42no—
326C2×Dic3—21no—
327C2×Dic3—22no—
328C2×Dic3—22no—
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×C8C4×D411yescharacteristic
344C3×C8C4×D411yescharacteristic
345C3×C8C4×D411yescharacteristic
346C3×C8C4×D411yescharacteristic
347C3×C8#12112yes—
348C3×C8#12112yes—
349C3⋊C8—22no—
350C3⋊C8—22no—
351C3⋊C8—42no—
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—21no—
367C22×C4—62no—
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—22no—
382M16—32no—
383M16—32no—
384M16—61no—
385M16—61no—
386M16—62no—
387M16—62no—
388M16—64no—
389M16—62no—
390M16—62no—
391M16—122no—
392M16—122no—
393M16C2×C4×D314yes—
394C2×C8—22no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—64no—
400C2×C8—62no—
401C2×C8—122no—
402C2×C8—122no—
403C2×C8C2×C4×D311yescharacteristic
404C2×C8C2×C4×D311yescharacteristic
405C2×C8C2×D1211yescharacteristic
406C2×C8C2×D1211yescharacteristic
407C2×C8C2×D1211yescharacteristic
408C2×C8C2×D1211yescharacteristic
409C2×C8#23511yescharacteristic
410C2×C8#23511yescharacteristic
411C2×C8D3×D412yes—
412C2×C8#23912yes—
413C4⋊C4—22no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C22⋊C4—22no—
417C22⋊C4—32no—
418C22⋊C4—62no—
419C42—21no—
420C42—22no—
421C42—62no—
422C42—61no—
423C42C2×D1211yescharacteristic
424C42C2×D1211yescharacteristic
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—41no—
434C3×C4—41no—
435C3×C4—82no—
436C3×C4C2×C4×D411yescharacteristic
437C3×C4C2×C4×D411yescharacteristic
438C3×C4C2×#12111yescharacteristic
439C3×C4#52911yescharacteristic
440Dic3—22no—
441Dic3—42no—
442Dic3—42no—
443C23—61no—
444C23—62no—
445C23D3×#4111yescharacteristic
446Q8—32no—
447Q8—61no—
448Q8—62no—
449Q8—62no—
450D4—32no—
451D4—62no—
452D4—62no—
453D4—62no—
454D4—62no—
455D4—61no—
456D4—122no—
457D4—122no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—61no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—81no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—242no—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×#23511yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C8—24no—
477C8—62no—
478C8—62no—
479C8—122no—
480C8C4×D1211yescharacteristic
481C8C4×D1211yescharacteristic
482C8C4×D1211yescharacteristic
483C8C4×D1211yescharacteristic
484C8#89512yes—
485C8#89512yes—
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—
497C22#552411yescharacteristic
498C22#554811yescharacteristic
499C22C4×#23911yescharacteristic
500C4—41no—
501C4—41no—
502C4—62no—
503C4—82no—
504C4—122no—
505C4—122no—
506C4C2×C4×D1211yescharacteristic
507C4C2×#89511yescharacteristic
508C4C4×D3×D411yescharacteristic
509C4#558011yescharacteristic
510C3#2155911yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—21no—
514C2—122no—
515C2—242no—
516C2C4×#89511yescharacteristic
subgroupG/H
centerC4C2×#895
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC2×C3C4×#121
prime radical#79256C2
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#21559
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #78579C2semidirect
3. C2×#4717C2semidirect
4. #78486C2semidirect
5. C3×#1773C2semidirect
6. #70707C2semidirect
7. #70665C2semidirect
8. #70658C2semidirect
9. #69898C2semidirect
10. #68778C2semidirect
11. #68750C2semidirect
33. #4898C4semidirect
34. #4897C4semidirect
35. #4896C4semidirect
58. #1773D3semidirect
76. C2×Dic12C2×C4semidirect
82. C2×D24C2×C4semidirect
83. C2×#204C2×C4semidirect
104. C3×#108C2×C4semidirect
105. C3×#108C2×C4semidirect
114. #824C2×C4semidirect
115. #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
#792562
#785792
C2×#47172
#784862
C3×#17732
#707072
#706652
#706582
#698982
#687782
#687502
#215593