Finite Group Databasebeta

← #10951993

Subgroups — #10951993

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
1#82015C211yesmaximal, characteristic, complemented
2#82013C211yesmaximal, characteristic, complemented
3#79891C212yesmaximal, complemented
4C3×#2170C211yesmaximal, characteristic, complemented
5#71456C211yesmaximal, characteristic, complemented
6#71454C211yesmaximal, characteristic, complemented
7#69304C212yesmaximal, complemented
8#69128C212yesmaximal, complemented
9#69127C212yesmaximal, complemented
10#69067C211yesmaximal, characteristic, complemented
11#69062C211yesmaximal, characteristic, complemented
12#33859—31nomaximal, Sylow
13#5770—22no—
14#5753C2212yescomplemented
15#5752C2212yes—
16C3×#504C2211yescharacteristic
17C3×#504C2211yescharacteristic, complemented
18C3×#443C2211yescharacteristic
19#5143C2212yes—
20#5142C2212yescomplemented
21#4909—22no—
22#4909C2212yescomplemented
23#4908—22no—
24#4908C2212yes—
25C2×Dic24—21no—
26C2×Dic24C2211yescharacteristic, complemented
27#4904—21no—
28#4904—21no—
29#4904C2211yescharacteristic
30C2×#800C2212yescomplemented
31C2×D48—21no—
32C2×D48C2211yescharacteristic
33#4898—22no—
34#4827—22no—
35#4825—22no—
36#4700C2212yescomplemented
37#4696C2212yescomplemented
38C3×#364C2211yescharacteristic, complemented
39C3×#350C2211yescharacteristic
40C3×#350C2211yescharacteristic, complemented
41C3×C4×C16C2211yescharacteristic, complemented
42#4517C2212yescomplemented
43#4516C2212yescomplemented
44#4510C2212yescomplemented
45#4506C2211yescharacteristic, complemented
46#3417—31no—
47#3415—31no—
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#2170D311yescharacteristic, complemented
57#1010—22no—
58#1009—22no—
59C3×#135D411yescharacteristic
60C3×#131C2311yescharacteristic
61C3×#131C2311yescharacteristic
62#951—22no—
63#949—22no—
64#918—42no—
65#916—42no—
66D3×SD16—42no—
67#911—42no—
68#909—22no—
69#909—42no—
70#908—22no—
71#908—42no—
72C2×Dic12—22no—
73C2×Dic12C2312yes—
74#904—22no—
75#904—22no—
76#904C2312yes—
77C2×D24—22no—
78C2×D24C2312yes—
79C2×#204—22no—
80#901—22no—
81#875—22no—
82C3×M32—21no—
83C3×M32—21no—
84C3×M32C2311yescharacteristic
85C3×M32C2311yescharacteristic
86C2×C3×C16—21no—
87C2×C3×C16—21no—
88C2×C3×C16C2311yescharacteristic
89C2×C3×C16C2311yescharacteristic
90C3×#108D411yescharacteristic
91C3×#104D411yescharacteristic
92C3×#104D411yescharacteristic
93C3×C4×C8C2311yescharacteristic
94#824—22no—
95#823—22no—
96#819C2312yes—
97#805—22no—
98Dic24—22no—
99Dic24—21no—
100Dic24—21no—
101Dic24D411yescharacteristic
102#800—22no—
103#800—22no—
104#800—42no—
105#800D412yes—
106D48—22no—
107D48—21no—
108D48—21no—
109D48D411yescharacteristic
110#578—32no—
111#577—62no—
112#576—32no—
113#510—32no—
114#510—62no—
115#509—32no—
116#509—62no—
117#508—31no—
118#508—61no—
119#508—61no—
120C2×Q32—31no—
121C2×Q32—61no—
122C2×SD32—32no—
123C2×D16—31no—
124C2×D16—61no—
125#504C2×D311yescharacteristic
126#504C2×D311yescharacteristic, complemented
127#495—32no—
128#473—32no—
129#472—32no—
130#471—62no—
131#455—62no—
132#454—62no—
133#443—32no—
134#443C2×D311yescharacteristic
135#368—31no—
136#364C2×D311yescharacteristic, complemented
137#362—32no—
138#361—32no—
139#359—32no—
140#350C2×D311yescharacteristic
141#350C2×D311yescharacteristic, complemented
142C4×C16C2×D311yescharacteristic, complemented
143C3×#41C2×D411yescharacteristic
144C3×#41C2×D411yescharacteristic
145#239—42no—
146D3×Q8—42no—
147#237—42no—
148#237—82no—
149D3×D4—42no—
150D3×D4—82no—
151#235—22no—
152#235—42no—
153#235—42no—
154C2×D12—22no—
155C2×D12—42no—
156C2×Dic6—22no—
157C2×Dic6—42no—
158C3×Q16D811yescharacteristic
159C3×SD16D811yescharacteristic
160C3×SD16D811yescharacteristic
161C3×D8D811yescharacteristic
162C3×M16—21no—
163C3×M16—21no—
164C3×M16C2×D411yescharacteristic
165C3×M16C2×D411yescharacteristic
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2411yescharacteristic
170C3×C42C2×D411yescharacteristic
171#216—42no—
172#215—42no—
173#214—42no—
174#213—42no—
175(C2×D3)⋊C4—82no—
176#208—22no—
177Dic12—22no—
178Dic12—22no—
179Dic12—42no—
180Dic12C2×D412yes—
181D24—22no—
182D24—22no—
183D24—42no—
184D24C2×D412yes—
185#204—22no—
186#204—42no—
187#204—42no—
188#204—42no—
189#204—42no—
190ΓL(1,𝔽25)—42no—
191D3×C8—42no—
192C3×C16—21no—
193C3×C16—21no—
194C3×C16—21no—
195C3×C16C2×D411yescharacteristic
196C3×C16C2×D411yescharacteristic
1972−1+4—62no—
198H(5,𝔽2)—62no—
199#137—62no—
200#137—62no—
201#137—122no—
202#137—122no—
203AGL(1,ℤ/8)—62no—
204AGL(1,ℤ/8)—62no—
205AGL(1,ℤ/8)—122no—
206AGL(1,ℤ/8)—122no—
207#135—32no—
208#135—62no—
209#135—62no—
210#135—122no—
211#135D1211yescharacteristic
212C2×Q16—32no—
213C2×Q16—62no—
214C2×SD16—62no—
215C2×SD16—122no—
216C2×D8—32no—
217C2×D8—62no—
218#131—62no—
219#131C22×D311yescharacteristic
220#131C22×D311yescharacteristic
221#124—62no—
222Q32—31no—
223Q32—62no—
224Q32—61no—
225Q32—61no—
226SD32—32no—
227SD32—62no—
228SD32—62no—
229SD32—122no—
230D16—31no—
231D16—62no—
232D16—61no—
233D16—61no—
234M32—21no—
235M32—21no—
236M32C22×D311yescharacteristic
237M32C22×D311yescharacteristic
238C2×C16—21no—
239C2×C16—21no—
240C2×C16C22×D311yescharacteristic
241C2×C16C22×D311yescharacteristic
242#108—32no—
243#108D1211yescharacteristic
244#104—62no—
245#104D1211yescharacteristic
246#104D1211yescharacteristic
247#101—62no—
248#100—62no—
249C4×C8C22×D311yescharacteristic
250C22×D3—42no—
251C22×D3—82no—
252C3×Q8C2×D811yescharacteristic
253C3×Q8C2×D811yescharacteristic
254C3×D4—21no—
255C3×D4—21no—
256C3×D4C2×D811yescharacteristic
257C3×D4C2×D811yescharacteristic
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4C22×D411yescharacteristic
262#67—42no—
263#67—82no—
264#67—82no—
265C2×Dic3—42no—
266C2×Dic3—82no—
267D12—22no—
268D12—42no—
269D12—42no—
270D12—82no—
271C4×D3—42no—
272C4×D3—82no—
273C4×D3—82no—
274Dic6—22no—
275Dic6—42no—
276Dic6—42no—
277Dic6—82no—
278C3×C8—21no—
279C3×C8—21no—
280C3×C8C2×D811yescharacteristic
281C3×C8C2×D811yescharacteristic
282C3×C8C22×D411yescharacteristic
283C3×C8H(5,𝔽2)11yescharacteristic
284C3⋊C8—42no—
285#41—62no—
286#41—122no—
287#41—122no—
288#41—122no—
289#41—122no—
290#41—242no—
291#41C2×D1211yescharacteristic
292#41C2×D1211yescharacteristic
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×D4—62no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—242no—
300Q16—32no—
301Q16—62no—
302Q16—62no—
303Q16—122no—
304Q16—122no—
305Q16D2411yescharacteristic
306SD16—62no—
307SD16—122no—
308SD16—122no—
309SD16—122no—
310SD16—122no—
311SD16—122no—
312SD16—122no—
313SD16D2411yescharacteristic
314SD16D2411yescharacteristic
315D8—32no—
316D8—62no—
317D8—62no—
318D8—122no—
319D8—122no—
320D8D2411yescharacteristic
321M16—21no—
322M16—21no—
323M16—62no—
324M16—122no—
325M16C2×D1211yescharacteristic
326M16C2×D1211yescharacteristic
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—122no—
330C2×C8C2×D1211yescharacteristic
331C2×C8C23×D311yescharacteristic
332C22⋊C4—242no—
333C42C2×D1211yescharacteristic
334C16—21no—
335C16—21no—
336C16—21no—
337C16D3×D411yescharacteristic
338C16D3×D411yescharacteristic
339C22×C3—21no—
340C22×C3—21no—
341C22×C3#57611yescharacteristic
342C2×D3—42no—
343C2×D3—82no—
344C2×D3—162no—
345C2×D3—322no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4D4211yescharacteristic
350C3×C4C22×D811yescharacteristic
351Dic3—42no—
352Dic3—82no—
353C23—122no—
354C23—242no—
355Q8—62no—
356Q8—122no—
357Q8—122no—
358Q8—242no—
359Q8C2×D2411yescharacteristic
360Q8C2×D2411yescharacteristic
361D4—21no—
362D4—21no—
363D4—62no—
364D4—122no—
365D4—122no—
366D4—122no—
367D4—242no—
368D4—242no—
369D4—242no—
370D4C2×D2411yescharacteristic
371D4C2×D2411yescharacteristic
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—242no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4C22×D1211yescharacteristic
381C8—21no—
382C8—21no—
383C8—122no—
384C8C2×D2411yescharacteristic
385C8C2×D2411yescharacteristic
386C8C2×D3×D411yescharacteristic
387C8#100911yescharacteristic
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3D4×D811yescharacteristic
392D3—82no—
393D3—162no—
394C22—21no—
395C22—21no—
396C22—122no—
397C22—242no—
398C22—482no—
399C22—962no—
400C22#575211yescharacteristic
401C4—21no—
402C4—21no—
403C4—41no—
404C4—122no—
405C4—242no—
406C4D4×D1211yescharacteristic
407C4C22×D2411yescharacteristic
408C3#3385911yescharacteristic, complemented, Sylow
409C2—21no—
410C2—41no—
411C2—41no—
412C2—242no—
413C2—482no—
414C2D4×D2411yescharacteristic
subgroupG/H
centerC2D4×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2170C2
socleC2×C3D4×D8
hypercenter#2170D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170

Complements

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

NH ≅ G/Ntype
1. #82015C2semidirect
2. #82013C2semidirect
3. #79891C2semidirect
4. C3×#2170C2semidirect
5. #71456C2semidirect
6. #71454C2semidirect
7. #69304C2semidirect
8. #69128C2semidirect
9. #69127C2semidirect
10. #69067C2semidirect
11. #69062C2semidirect
14. #5753C22semidirect
17. C3×#504C22semidirect
20. #5142C22semidirect
22. #4909C22semidirect
26. C2×Dic24C22semidirect
30. C2×#800C22semidirect
36. #4700C22semidirect
37. #4696C22semidirect
38. C3×#364C22semidirect
40. C3×#350C22semidirect
41. C3×C4×C16C22semidirect
42. #4517C22semidirect
43. #4516C22semidirect
44. #4510C22semidirect
45. #4506C22semidirect
56. #2170D3semidirect
126. #504C2×D3semidirect
136. #364C2×D3semidirect
141. #350C2×D3semidirect
142. C4×C16C2×D3semidirect
408. C3#33859semidirect

Maximal subgroups

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

subgroupindex
#820152
#820132
#798912
C3×#21702
#714562
#714542
#693042
#691282
#691272
#690672
#690622
#338593