Finite Group Databasebeta

← #10951991

Subgroups — #10951991

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3415C211yesmaximal, characteristic, complemented
2#84011C212yesmaximal, complemented
3#82033C211yesmaximal, characteristic, complemented
4#72123C212yesmaximal, complemented
5#71518C212yesmaximal, complemented
6#71513C212yesmaximal, complemented
7#71511C211yesmaximal, characteristic, complemented
8#71472C211yesmaximal, characteristic, complemented
9#71465C211yesmaximal, characteristic, complemented
10#71456C211yesmaximal, characteristic, complemented
11#71452C211yesmaximal, characteristic, complemented
12#33859—31nomaximal, Sylow
13C3×#576C2212yes—
14#5836—22no—
15#5771C2212yescomplemented
16C3×#509—22no—
17C3×#509C2212yes—
18C3×#508—21no—
19C3×#508C2211yescharacteristic
20C2×C3×D16—21no—
21C2×C3×D16C2211yescharacteristic
22C3×#504C2211yescharacteristic
23#5242—22no—
24#5201—22no—
25#5200—22no—
26#5196C2212yescomplemented
27#5195C2212yescomplemented
28#5160C2212yescomplemented
29#5155C2212yescomplemented
30C2×#827C2212yescomplemented
31#5144C2211yescharacteristic
32#5143C2212yes—
33D3×Q32—21no—
34#4915—22no—
35#4912—21no—
36C2×Dic24C2211yescharacteristic, complemented
37#4901C2211yescharacteristic, complemented
38#4567C2212yescomplemented
39#4563C2212yescomplemented
40#4561C2211yescharacteristic
41#4517C2212yescomplemented
42#4515C2211yescharacteristic, complemented
43#4511C2211yescharacteristic, complemented
44#4506C2211yescharacteristic, complemented
45Dic3×C16C2211yescharacteristic, complemented
46#3417—31no—
47#3415D311yescharacteristic, complemented
48#3290—32no—
49#2249—31no—
50#2222—31no—
51#2221—31no—
52#2212—32no—
53#2194—32no—
54#2193—32no—
55#2178—31no—
56#2170—31no—
57C3×H(5,𝔽2)—22no—
58#1010—22no—
59C3×AGL(1,ℤ/8)—22no—
60C3×AGL(1,ℤ/8)—42no—
61C3×#135—22no—
62C3×#135C2312yes—
63C2×C3×D8—22no—
64C2×C3×D8C2312yes—
65C3×#131C2311yescharacteristic
66#951—22no—
67#951—22no—
68#951—42no—
69#950—42no—
70#950D411yescharacteristic
71#948—22no—
72#936—22no—
73C2×#214—22no—
74C2×#214—42no—
75#932—22no—
76#932—42no—
77D3×Q16—22no—
78#915—42no—
79#912—22no—
80C2×Dic12C2312yes—
81#901C2311yescharacteristic
82C3×Q32D411yescharacteristic
83C3×SD32—22no—
84C3×SD32D412yes—
85C3×D16—22no—
86C3×D16—21no—
87C3×D16—21no—
88C3×D16D411yescharacteristic
89C3×M32—21no—
90C3×M32C2311yescharacteristic
91C2×C3×C16—21no—
92C2×C3×C16C2311yescharacteristic
93#837—22no—
94#837D411yescharacteristic
95#833—22no—
96#829—22no—
97#827—22no—
98#827—42no—
99#825D411yescharacteristic
100#824—22no—
101#822D411yescharacteristic
102#819C2312yes—
103C8×Dic3C2311yescharacteristic
104#812C2311yescharacteristic
105C2×C3⋊C16C2311yescharacteristic
106Dic24—21no—
107Dic24—21no—
108#798—21no—
109D3×C16—21no—
110#578—32no—
111#577—62no—
112#576C2×D312yes—
113#510—32no—
114#510—62no—
115#509—22no—
116#509C2×D312yes—
117#508—21no—
118#508—61no—
119#508C2×D311yescharacteristic
120C2×Q32—31no—
121C2×Q32—61no—
122C2×SD32—32no—
123C2×D16—21no—
124C2×D16C2×D311yescharacteristic
125#504—31no—
126#504C2×D311yescharacteristic
127#495—32no—
128#473—32no—
129#472—32no—
130#471—62no—
131#455—62no—
132#454—62no—
133#443—31no—
134#443—32no—
135#368—31no—
136#364—31no—
137#362—32no—
138#361—32no—
139#359—32no—
140#350—31no—
141#350—31no—
142C4×C16—31no—
143C3×#41—22no—
144C3×#41—42no—
145C3×#41—42no—
146C3×#41C2×D411yescharacteristic
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—82no—
151D3×Q8—42no—
152#237—42no—
153#237—82no—
154#235—42no—
155#235C2×D411yescharacteristic
156C2×Dic6—22no—
157C2×Dic6—42no—
158C3×Q16C2×D412yes—
159C3×SD16—22no—
160C3×SD16—42no—
161C3×D8—22no—
162C3×D8—22no—
163C3×D8—42no—
164C3×D8C2×D412yes—
165C3×M16—21no—
166C3×M16C2×D411yescharacteristic
167C2×C3×C8—21no—
168C2×C3×C8C2411yescharacteristic
169#217—82no—
170#216—42no—
171#216—42no—
172#216D811yescharacteristic
173#215D811yescharacteristic
174#214—42no—
175#214—42no—
176#214—42no—
177#214—42no—
178#214—42no—
179#214D811yescharacteristic
180#213—42no—
181#213D811yescharacteristic
182C4×Dic3C2×D411yescharacteristic
183#208—22no—
184#208—42no—
185#208C2×D411yescharacteristic
186C2×C3⋊C8—42no—
187C2×C3⋊C8C2×D411yescharacteristic
188Dic12—22no—
189Dic12—22no—
190ΓL(1,𝔽25)—21no—
191D3×C8—21no—
192C3×C16—21no—
193C3×C16C2×D411yescharacteristic
194C3×C16C2×D411yescharacteristic
195C3⋊C16—21no—
196C3⋊C16—21no—
1972−1+4—62no—
198H(5,𝔽2)—22no—
199#137—62no—
200#137—62no—
201#137—122no—
202#137—122no—
203AGL(1,ℤ/8)—22no—
204AGL(1,ℤ/8)—42no—
205AGL(1,ℤ/8)—62no—
206AGL(1,ℤ/8)—122no—
207#135—22no—
208#135—31no—
209#135—62no—
210#135—122no—
211#135C22×D312yes—
212C2×Q16—32no—
213C2×Q16—62no—
214C2×SD16—62no—
215C2×SD16—122no—
216C2×D8—22no—
217C2×D8C22×D312yes—
218#131—31no—
219#131—62no—
220#131C22×D311yescharacteristic
221#124—62no—
222Q32—62no—
223Q32—61no—
224Q32—61no—
225Q32#6711yescharacteristic
226SD32—22no—
227SD32—62no—
228SD32—122no—
229SD32#6712yes—
230D16—22no—
231D16—21no—
232D16—21no—
233D16#6711yescharacteristic
234M32—21no—
235M32—31no—
236M32—61no—
237M32C22×D311yescharacteristic
238C2×C16—21no—
239C2×C16—31no—
240C2×C16—61no—
241C2×C16C22×D311yescharacteristic
242#108—31no—
243#108—32no—
244#104—31no—
245#104—31no—
246#104—62no—
247#101—62no—
248#100—62no—
249C4×C8—31no—
250C23×C3—42no—
251C23×C3—82no—
252C3×Q8—22no—
253C3×Q8C2×D811yescharacteristic
254C3×D4—21no—
255C3×D4—22no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—82no—
260C3×D4—82no—
261C3×D4C2×D811yescharacteristic
262C2×C3×C4—21no—
263C2×C3×C4—42no—
264C2×C3×C4—82no—
265C2×C3×C4C22×D411yescharacteristic
266#67—21no—
267#67—82no—
268C2×Dic3—21no—
269C2×Dic3—42no—
270C2×Dic3—82no—
271D12C2×D811yescharacteristic
272C4×D3—21no—
273C4×D3—82no—
274Dic6—42no—
275Dic6—42no—
276Dic6—82no—
277Dic6C2×D811yescharacteristic
278C3×C8—21no—
279C3×C8C22×D411yescharacteristic
280C3×C8H(5,𝔽2)11yescharacteristic
281C3⋊C8—21no—
282C3⋊C8—42no—
283C3⋊C8C2×D811yescharacteristic
284C3⋊C8C2×D811yescharacteristic
285#41—22no—
286#41—31no—
287#41—42no—
288#41—42no—
289#41—122no—
290#41—122no—
291#41—242no—
292#41D3×D411yescharacteristic
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—82no—
300Q16—31no—
301Q16—62no—
302Q16—62no—
303Q16—122no—
304Q16—122no—
305Q16C2×#6712yes—
306SD16—22no—
307SD16—31no—
308SD16—31no—
309SD16—42no—
310SD16—122no—
311SD16—122no—
312SD16—122no—
313SD16—122no—
314SD16—122no—
315D8—22no—
316D8—22no—
317D8—31no—
318D8—42no—
319D8—122no—
320D8C2×#6712yes—
321M16—21no—
322M16—31no—
323M16—61no—
324M16—62no—
325M16—122no—
326M16D3×D411yescharacteristic
327C2×C8—21no—
328C2×C8—31no—
329C2×C8—61no—
330C2×C8—122no—
331C2×C8C23×D311yescharacteristic
332C22⋊C4—242no—
333C42—31no—
334C16—21no—
335C16—61no—
336C16—61no—
337C16C2×#6711yescharacteristic
338C16C2×#6711yescharacteristic
339C22×C3—21no—
340C22×C3—42no—
341C22×C3—82no—
342C22×C3—162no—
343C22×C3—322no—
344C22×C3#57611yescharacteristic
345C2×D3—21no—
346C3×C4—21no—
347C3×C4—42no—
348C3×C4D4211yescharacteristic
349C3×C4C22×D811yescharacteristic
350Dic3—21no—
351Dic3—41no—
352Dic3—82no—
353C23—42no—
354C23—82no—
355Q8—22no—
356Q8—31no—
357Q8—122no—
358Q8—122no—
359Q8—242no—
360Q8D3×D811yescharacteristic
361D4—21no—
362D4—22no—
363D4—31no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—61no—
368D4—82no—
369D4—82no—
370D4—242no—
371D4D3×D811yescharacteristic
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—82no—
377C2×C4—122no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4C2×D3×D411yescharacteristic
381C8—21no—
382C8—31no—
383C8—31no—
384C8—61no—
385C8—122no—
386C8#100411yescharacteristic
387C8C22×#6711yescharacteristic
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—82no—
391C2×C3—162no—
392C2×C3D4×D811yescharacteristic
393D3—41no—
394C22—21no—
395C22—42no—
396C22—61no—
397C22—82no—
398C22—162no—
399C22—322no—
400C22#575711yescharacteristic
401C4—21no—
402C4—42no—
403C4—61no—
404C4—121no—
405C4—242no—
406C4C2×D3×D811yescharacteristic
407C4D4×#6711yescharacteristic
408C3#3385911yescharacteristic, complemented, Sylow
409C2—21no—
410C2—41no—
411C2—82no—
412C2—121no—
413C2—162no—
414C2D8×#6711yescharacteristic
subgroupG/H
centerC2D8×#67
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3415C2
socleC2×C3D4×D8
hypercenter#3415D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #3415

Complements

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

NH ≅ G/Ntype
1. C3×#3415C2semidirect
2. #84011C2semidirect
3. #82033C2semidirect
4. #72123C2semidirect
5. #71518C2semidirect
6. #71513C2semidirect
7. #71511C2semidirect
8. #71472C2semidirect
9. #71465C2semidirect
10. #71456C2semidirect
11. #71452C2semidirect
15. #5771C22semidirect
26. #5196C22semidirect
27. #5195C22semidirect
28. #5160C22semidirect
29. #5155C22semidirect
30. C2×#827C22semidirect
36. C2×Dic24C22semidirect
37. #4901C22semidirect
38. #4567C22semidirect
39. #4563C22semidirect
41. #4517C22semidirect
42. #4515C22semidirect
43. #4511C22semidirect
44. #4506C22semidirect
45. Dic3×C16C22semidirect
47. #3415D3semidirect
408. C3#33859semidirect

Maximal subgroups

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

subgroupindex
C3×#34152
#840112
#820332
#721232
#715182
#715132
#715112
#714722
#714652
#714562
#714522
#338593