Finite Group Databasebeta

← #10948598

Subgroups — #10948598

Order 768.

652 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#443C211yesmaximal, characteristic, complemented
2#83894C211yesmaximal, characteristic
3C2×#4700C211yesmaximal, characteristic, complemented
4#72080C211yesmaximal, characteristic, complemented
5#70922C214yesmaximal, complemented
6#70921C214yesmaximal, complemented
7#70690C211yesmaximal, characteristic, complemented
8#70683C211yesmaximal, characteristic, complemented
9C4×#812C211yesmaximal, characteristic, complemented
10#33463—31nomaximal, Sylow
11C2×C3×#135C2211yescharacteristic
12C2×C3×#131—21no—
13C2×#904C2211yescharacteristic
14C2×#901—21no—
15C3×#443—21no—
16C3×#443—21no—
17C3×#443C2212yescomplemented
18C3×#443C2212yescomplemented
19C2×C3×#108C2211yescharacteristic
20C2×C3×#104—21no—
21C2×C3×C4×C8C2211yescharacteristic
22#5195C412yescomplemented
23#5194C2214yes—
24#5193C412yescomplemented
25#5179C412yescomplemented
26#5147C412yescomplemented
27C2×#819C2211yescharacteristic
28C2×#812C2211yescharacteristic
29C2×#812C2211yescharacteristic
30C2×#805—21no—
31#4700—21no—
32#4700—21no—
33#4700C2212yescomplemented
34#4700C2212yescomplemented
35#4567C2214yescomplemented
36#4497C2214yescomplemented
37#4493C2214yescomplemented
38#4461C2212yescomplemented
39C4×C3⋊C16C2212yescomplemented
40#3414—31no—
41C2×#443—31no—
42C2×#443D311yescharacteristic, complemented
43#2179—34no—
44#2178—34no—
45#2161—31no—
46#2143—31no—
47#2143—31no—
48C4×M32—31no—
49C2×C3×#41—21no—
50C2×#235—21no—
51C3×#135—21no—
52C3×#135—21no—
53C3×#135C2×C412yes—
54C3×#135C2312yes—
55C2×C3×Q16C2×C411yescharacteristic
56C2×C3×SD16—21no—
57C2×C3×D8C2×C411yescharacteristic
58C3×#131—22no—
59C3×#131—22no—
60C3×#131—41no—
61C3×#131—41no—
62C2×C3×M16—21no—
63C2×C3×M16—41no—
64C22×C3×C8—41no—
65C22×C3×C8D411yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×#208—21no—
69C2×Dic12C2×C411yescharacteristic
70#904—21no—
71#904—21no—
72#904C2×C412yes—
73#904C2312yes—
74C2×D24C2×C411yescharacteristic
75C2×#204—21no—
76#901—22no—
77#901—22no—
78#901—41no—
79#901—41no—
80C2×ΓL(1,𝔽25)—41no—
81C2×D3×C8—41no—
82C3×#108—21no—
83C3×#108C2312yes—
84C3×#104—22no—
85C3×#104—22no—
86C3×C4×C8D411yescharacteristic
87C3×C4×C8D411yescharacteristic
88C3×C4×C8C2311yescharacteristic
89C3×C4×C8C2311yescharacteristic
90#829C2×C414yescomplemented
91#828C2×C414yescomplemented
92#827C2×C414yescomplemented
93#826C2×C414yescomplemented
94#819—21no—
95#819C2312yes—
96#812—21no—
97#812—21no—
98#812C2×C412yes—
99#812C2×C412yes—
100C2×C3⋊C16C2312yes—
101C2×C3⋊C16C2312yes—
102#805—22no—
103#805—22no—
104C2×#135—31no—
105C2×#135C2×D311yescharacteristic
106C2×#131—21no—
107C2×#131—61no—
108#510—32no—
109#510—32no—
110#509—32no—
111#509—32no—
112#508—34no—
113C2×M32—31no—
114C2×M32—31no—
115#443—21no—
116#443—21no—
117#443—32no—
118#443—32no—
119#443—61no—
120#443—61no—
121#443C2×D312yescomplemented
122#443C2×D312yescomplemented
123C2×#108—31no—
124C2×#108C2×D311yescharacteristic
125C2×#104—21no—
126C2×#104—61no—
127C2×C4×C8C2×D311yescharacteristic
128#368—34no—
129#359—34no—
130#359—34no—
131#346—32no—
132C4×C16—32no—
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—41no—
136C3×#41—41no—
137C2×C3×Q8—21no—
138C2×C3×D4—21no—
139C2×C3×D4—41no—
140C22×C3×C4—21no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C2×#67—41no—
144#235—22no—
145#235—22no—
146#235—41no—
147#235—41no—
148C2×D12—21no—
149C2×C4×D3—41no—
150C2×Dic6—21no—
151C3×Q16—21no—
152C3×Q16C22×C412yes—
153C3×SD16—22no—
154C3×SD16—22no—
155C3×D8—21no—
156C3×D8C22×C412yes—
157C3×M16—22no—
158C3×M16—41no—
159C3×M16—42no—
160C3×M16—42no—
161C2×C3×C8—41no—
162C2×C3×C8—42no—
163C2×C3×C8—42no—
164C2×C3×C8D811yescharacteristic
165C2×C3×C8D811yescharacteristic
166C2×C3×C8D811yescharacteristic
167C2×C3×C8D811yescharacteristic
168C2×C3×C8C22×C411yescharacteristic
169C2×C3×C8C22×C411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8#4111yescharacteristic
174C2×C3×C8#4111yescharacteristic
175C2×C3×C8C2411yescharacteristic
176C3×C42—21no—
177C3×C42C2×D411yescharacteristic
178C3×C42C2×D411yescharacteristic
179#208—22no—
180#208—41no—
181C2×C3⋊C8—41no—
182Dic12—21no—
183Dic12C22×C412yes—
184D24—21no—
185D24C22×C412yes—
186#204—22no—
187#204—22no—
188ΓL(1,𝔽25)—42no—
189ΓL(1,𝔽25)—42no—
190D3×C8—42no—
191D3×C8—42no—
192C3⋊C16—22no—
193C3⋊C16C22×C414yes—
194C2×#41—21no—
195C2×#41—61no—
196#135—21no—
197#135—21no—
198#135—32no—
199#135—32no—
200#135—61no—
201#135—61no—
202#135C4×D312yes—
203#135C22×D312yes—
204C2×Q16—31no—
205C2×Q16C4×D311yescharacteristic
206C2×SD16—21no—
207C2×SD16—61no—
208C2×D8—31no—
209C2×D8C4×D311yescharacteristic
210#131—22no—
211#131—22no—
212#131—41no—
213#131—41no—
214#131—62no—
215#131—62no—
216#131—121no—
217#131—121no—
218C2×M16—21no—
219C2×M16—41no—
220C2×M16—61no—
221C2×M16—121no—
222C22×C8—41no—
223C22×C8—121no—
224C22×C8#6711yescharacteristic
225C22×C8C22×D311yescharacteristic
226C2×C42#6711yescharacteristic
227Q32—34no—
228SD32—34no—
229SD32—34no—
230D16—34no—
231M32—32no—
232M32—32no—
233M32—61no—
234M32—61no—
235C2×C16—32no—
236C2×C16—32no—
237#108—21no—
238#108—32no—
239#108—61no—
240#108C22×D312yes—
241#104—22no—
242#104—22no—
243#104—62no—
244#104—62no—
245C4×C8#6711yescharacteristic
246C4×C8#6711yescharacteristic
247C4×C8C22×D311yescharacteristic
248C4×C8C22×D311yescharacteristic
249C23×C3—41no—
250C23×C3#13511yescharacteristic
251C22×D3—41no—
252C3×Q8—22no—
253C3×Q8—41no—
254C3×D4—22no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—41no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4C2×D811yescharacteristic
267C2×C3×C4C2×D811yescharacteristic
268C2×C3×C4#13511yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270#67—42no—
271#67—42no—
272C2×Dic3—41no—
273D12—22no—
274D12—41no—
275C4×D3—42no—
276C4×D3—42no—
277Dic6—22no—
278Dic6—41no—
279C3×C8—42no—
280C3×C8C4×D411yescharacteristic
281C3×C8C4×D411yescharacteristic
282C3×C8C2×D811yescharacteristic
283C3×C8C2×D811yescharacteristic
284C3×C8C2×D811yescharacteristic
285C3×C8C2×D811yescharacteristic
286C3×C8C23×C411yescharacteristic
287C3×C8C2×#4111yescharacteristic
288C3⋊C8—42no—
289#41—22no—
290#41—22no—
291#41—41no—
292#41—41no—
293#41—62no—
294#41—62no—
295#41—121no—
296#41—121no—
297C2×Q8—21no—
298C2×Q8—61no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—61no—
302C2×D4—121no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—121no—
306C22×C4C2×#6711yescharacteristic
307Q16—21no—
308Q16—32no—
309Q16—61no—
310Q16C2×C4×D312yes—
311SD16—22no—
312SD16—22no—
313SD16—62no—
314SD16—62no—
315D8—21no—
316D8—32no—
317D8—61no—
318D8C2×C4×D312yes—
319M16—22no—
320M16—41no—
321M16—42no—
322M16—42no—
323M16—62no—
324M16—121no—
325M16—122no—
326M16—122no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—121no—
331C2×C8—122no—
332C2×C8—122no—
333C2×C8#21311yescharacteristic
334C2×C8#21311yescharacteristic
335C2×C8#21311yescharacteristic
336C2×C8#21311yescharacteristic
337C2×C8C2×C4×D311yescharacteristic
338C2×C8C2×C4×D311yescharacteristic
339C2×C8#23511yescharacteristic
340C2×C8#23511yescharacteristic
341C2×C8C2×#6711yescharacteristic
342C2×C8C2×#6711yescharacteristic
343C2×C8C2×#6711yescharacteristic
344C2×C8C23×D311yescharacteristic
345C42—21no—
346C42C2×#6711yescharacteristic
347C42C2×#6711yescharacteristic
348C16—34no—
349C16—62no—
350C22×C3—41no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3C4×D811yescharacteristic
354C22×C3C4×D811yescharacteristic
355C22×C3C2×#13511yescharacteristic
356C2×D3—42no—
357C2×D3—82no—
358C3×C4—22no—
359C3×C4—41no—
360C3×C4—42no—
361C3×C4C4×D811yescharacteristic
362C3×C4C4×D811yescharacteristic
363C3×C4C2×C4×D411yescharacteristic
364C3×C4C22×D811yescharacteristic
365Dic3—42no—
366C23—41no—
367C23—121no—
368C23#95011yescharacteristic
369Q8—22no—
370Q8—41no—
371Q8—62no—
372Q8—121no—
373D4—22no—
374D4—42no—
375D4—42no—
376D4—41no—
377D4—62no—
378D4—122no—
379D4—122no—
380D4—121no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4C4×#6711yescharacteristic
391C2×C4C4×#6711yescharacteristic
392C2×C4C2×#21311yescharacteristic
393C2×C4C2×#21311yescharacteristic
394C2×C4#95011yescharacteristic
395C2×C4C22×#6711yescharacteristic
396C8—42no—
397C8—122no—
398C8C4×#6711yescharacteristic
399C8C4×#6711yescharacteristic
400C8C2×#21311yescharacteristic
401C8C2×#21311yescharacteristic
402C8C2×#21311yescharacteristic
403C8C2×#21311yescharacteristic
404C8C22×C4×D311yescharacteristic
405C8C2×#23511yescharacteristic
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—82no—
410C2×C3C2×C4×D811yescharacteristic
411D3—82no—
412C22—41no—
413C22—42no—
414C22—82no—
415C22—122no—
416C22—242no—
417C22C4×#21311yescharacteristic
418C22C4×#21311yescharacteristic
419C22C2×#95011yescharacteristic
420C4—22no—
421C4—41no—
422C4—42no—
423C4—122no—
424C4C4×#21311yescharacteristic
425C4C4×#21311yescharacteristic
426C4C2×C4×#6711yescharacteristic
427C4C22×#21311yescharacteristic
428C3#3346311yescharacteristic, complemented, Sylow
429C2—21no—
430C2—21no—
431C2—21no—
432C2—82no—
433C2—242no—
434C2C2×C4×#21311yescharacteristic
subgroupG/H
centerC8C2×#213
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#443C2
socleC2×C3C2×C4×D8
prime radical#83894C2
cosocleC2×C3×C8C24
hypercenterC2×#443D3
hypocenterC3#33463
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×C3×#443C2semidirect
3. C2×#4700C2semidirect
4. #72080C2semidirect
5. #70922C2semidirect
6. #70921C2semidirect
7. #70690C2semidirect
8. #70683C2semidirect
9. C4×#812C2semidirect
17. C3×#443C22semidirect
18. C3×#443C22semidirect
22. #5195C4semidirect
24. #5193C4semidirect
25. #5179C4semidirect
26. #5147C4semidirect
33. #4700C22semidirect
34. #4700C22semidirect
35. #4567C22semidirect
36. #4497C22semidirect
37. #4493C22semidirect
38. #4461C22semidirect
39. C4×C3⋊C16C22semidirect
42. C2×#443D3semidirect
90. #829C2×C4semidirect
91. #828C2×C4semidirect
92. #827C2×C4semidirect
93. #826C2×C4semidirect
121. #443C2×D3semidirect
122. #443C2×D3semidirect
428. C3#33463semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4432
#838942
C2×#47002
#720802
#709222
#709212
#706902
#706832
C4×#8122
#334633