Finite Group Databasebeta

← #10897677

Subgroups — #10897677

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78541C211yesmaximal, characteristic, complemented
2#78520C211yesmaximal, characteristic, complemented
3C3×#1841C211yesmaximal, characteristic, complemented
4#69035C212yesmaximal, complemented
5#69034C212yesmaximal, complemented
6#68778C212yesmaximal, complemented
7#68746C212yesmaximal, complemented
8#68727C211yesmaximal, characteristic
9#68718C211yesmaximal, characteristic, complemented
10#68715C211yesmaximal, characteristic, complemented
11#68698C211yesmaximal, characteristic, complemented
12#29387—31nomaximal, Sylow
13#5748C2211yescharacteristic
14#5748C2212yes—
15#5745C2212yes—
16C4×#235C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18#5129C2211yescharacteristic
19#5129C2211yescharacteristic
20#5129C2212yes—
21#5125C2211yescharacteristic
22#5119C2211yescharacteristic
23#5119C2211yescharacteristic
24#5006C2212yes—
25#5005C2212yes—
26#5002C2212yes—
27#4999C2212yes—
28C2×#805C2212yes—
29C2×#803C2211yescharacteristic
30#4717—22no—
31#4705—22no—
32#4700—22no—
33#4690—22no—
34C3×#344C2212yes—
35C3×#339C2212yes—
36C3×#338C2211yescharacteristic
37C3×H(3,ℤ/4)C2211yescharacteristic
38#4559C2212yes—
39#4558C2212yes—
40#2956—31no—
41#2923—31no—
42#2103—31no—
43#2098—31no—
44#2053—31no—
45#2017—31no—
46#1938—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841D311yescharacteristic, complemented
51C2×#235C2311yescharacteristic
52C2×#235C2312yes—
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58C4×#67—22no—
59#926C2311yescharacteristic
60C2×#208C2312yes—
61C22×C3⋊C8C2311yescharacteristic
62C22×C3⋊C8C2311yescharacteristic
63#909—22no—
64#909—22no—
65#909—22no—
66#908—22no—
67#908—22no—
68#908—22no—
69#907—42no—
70D3×M16—42no—
71#904—22no—
72#904—42no—
73#901—24no—
74C4×D12—21no—
75#872—21no—
76C42×D3—21no—
77C4×Dic6—21no—
78C3×#97D412yes—
79C3×C4×C8D412yes—
80(C2×D3)⋊C8—22no—
81(C2×D3)⋊C8—22no—
82(C2×D3)⋊C8—42no—
83#819—22no—
84Dic3⋊C8—22no—
85Dic3⋊C8—22no—
86#810—22no—
87#809—22no—
88#806—24no—
89#805—22no—
90#805—22no—
91#805—42no—
92C4⋊(C3⋊C8)—42no—
93#803—21no—
94#803—21no—
95#575—31no—
96#575—32no—
97#568—32no—
98C4×#41—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#432—62no—
103#430—31no—
104#424—31no—
105#424—31no—
106#424—32no—
107#421—32no—
108C2×#104—32no—
109#416—32no—
110#410—32no—
111#408—31no—
112#408—31no—
113#408—32no—
114C4×M16C2×D311yescharacteristic
115C2×#97—31no—
116#344—32no—
117#344C2×D312yes—
118#341—32no—
119#339C2×D312yes—
120#338C2×D311yescharacteristic
121H(3,ℤ/4)C2×D311yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4C2411yescharacteristic
124C2×#67—22no—
125C2×#67—21no—
126#235—22no—
127#235—22no—
128#235—22no—
129#235—22no—
130#235—22no—
131#235—22no—
132C2×D12#4112yes—
133C2×D12#4111yescharacteristic
134C2×C4×D3—21no—
135C2×C4×D3—22no—
136C2×C4×D3—22no—
137C2×Dic6#4112yes—
138C2×Dic6#4111yescharacteristic
139C3×M16—22no—
140C3×M16—22no—
141C3×M16—22no—
142C3×M16—22no—
143C2×C3×C8—22no—
144C2×C3×C8C2×D412yes—
145C2×C3×C8C2×D412yes—
146C3×C4⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C42—21no—
149C3×C42—22no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#217#4112yes—
153(C2×D3)⋊C4—22no—
154C4⋊Dic3—21no—
155Dic3⋊C4—22no—
156C4×Dic3—22no—
157C4×Dic3—21no—
158#208—22no—
159#208—22no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165Dic12—22no—
166Dic12—42no—
167D24—22no—
168D24—42no—
169#204—24no—
170#204—42no—
171#204—42no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C2×#41—31no—
175C2×#41—32no—
176#137—62no—
177#137—62no—
178#137—62no—
179AGL(1,ℤ/8)—62no—
180AGL(1,ℤ/8)—62no—
181AGL(1,ℤ/8)—62no—
182#135—62no—
183#135—122no—
184#131—64no—
185#131—122no—
186C2×M16—32no—
187C2×M16—122no—
188C2×M16C22×D312yes—
189C2×M16C22×D311yescharacteristic
190C2×M16C22×D311yescharacteristic
191C22×C8—31no—
192C22×C8—31no—
193C4×Q8—61no—
194C4×D4—61no—
195C4×D4—62no—
196#117—31no—
197#117—61no—
198#117C22×D312yes—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#108—62no—
202C4⋊C8—62no—
203C4⋊C8—62no—
204C4⋊C8—122no—
205#104—62no—
206#104—62no—
207#104—122no—
208#103—62no—
209#102—62no—
210#99—64no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—61no—
215#97—61no—
216#97D1212yes—
217C4×C8D1212yes—
218C23×C3C2×#4111yescharacteristic
219C22×D3—21no—
220C22×D3—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—42no—
224C2×C3×C4—81no—
225C2×C3×C4C22×D411yescharacteristic
226C2×C3×C4C2×#4112yes—
227C2×C3×C4C2×#4111yescharacteristic
228C2×C3×C4H(5,𝔽2)12yes—
229#67—42no—
230#67—42no—
231#67—42no—
232C2×Dic3—21no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235D12—22no—
236D12—22no—
237D12—22no—
238C4×D3—41no—
239C4×D3—42no—
240C4×D3—41no—
241C4×D3—42no—
242C4×D3—82no—
243Dic6—22no—
244Dic6—22no—
245Dic6—22no—
246C3×C8—22no—
247C3×C8—22no—
248C3×C8—42no—
249C3⋊C8—41no—
250C3⋊C8—42no—
251C3⋊C8—41no—
252#41—62no—
253#41—62no—
254#41—62no—
255#41—62no—
256#41—62no—
257#41—62no—
258C2×Q8—32no—
259C2×Q8—31no—
260C2×D4—32no—
261C2×D4—31no—
262C2×D4—62no—
263C2×D4—61no—
264C22×C4—21no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4C23×D311yescharacteristic
269Q16—62no—
270Q16—122no—
271SD16—64no—
272SD16—122no—
273SD16—122no—
274D8—62no—
275D8—122no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—62no—
281M16—62no—
282M16—124no—
283C2×C8—22no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—124no—
290C2×C8C2×D1212yes—
291C2×C8C2×D1212yes—
292C4⋊C4—22no—
293C4⋊C4—62no—
294C4⋊C4—61no—
295C22⋊C4—22no—
296C22⋊C4—32no—
297C22⋊C4—62no—
298C42—21no—
299C42—22no—
300C42—62no—
301C42—61no—
302C42C2×D1211yescharacteristic
303C42C2×D1211yescharacteristic
304C22×C3—41no—
305C22×C3#52511yescharacteristic
306C22×C3#55312yes—
307C2×D3—42no—
308C2×D3—41no—
309C2×D3—42no—
310C2×D3—82no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—82no—
314C3×C4#52511yescharacteristic
315C3×C4#53511yescharacteristic
316C3×C4#55112yes—
317Dic3—42no—
318Dic3—42no—
319Dic3—41no—
320C23—61no—
321C23—62no—
322C23D3×#4111yescharacteristic
323Q8—62no—
324Q8—62no—
325Q8—62no—
326D4—62no—
327D4—62no—
328D4—62no—
329D4—122no—
330D4—122no—
331D4—122no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—61no—
338C2×C4—81no—
339C2×C4—122no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—242no—
344C2×C4C22×D1211yescharacteristic
345C2×C4D3×#4111yescharacteristic
346C2×C4D3×#4112yes—
347C2×C4#100912yes—
348C8—22no—
349C8—22no—
350C8—42no—
351C8—121no—
352C8—121no—
353C8—122no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3#259811yescharacteristic
357D3—81no—
358D3—82no—
359C22—41no—
360C22—122no—
361C22—121no—
362C22—122no—
363C22—242no—
364C22#552411yescharacteristic
365C22#570912yes—
366C4—41no—
367C4—41no—
368C4—82no—
369C4—122no—
370C4—122no—
371C4—121no—
372C4#552411yescharacteristic
373C4#552511yescharacteristic
374C4#570512yes—
375C3#2938711yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—241no—
379C2—242no—
380C2#7430711yescharacteristic
subgroupG/H
centerC4#5525
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1841C2
socleC2×C3#2598
prime radical#68727C2
hypercenter#1841D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1841

Complements

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

NH ≅ G/Ntype
1. #78541C2semidirect
2. #78520C2semidirect
3. C3×#1841C2semidirect
4. #69035C2semidirect
5. #69034C2semidirect
6. #68778C2semidirect
7. #68746C2semidirect
9. #68718C2semidirect
10. #68715C2semidirect
11. #68698C2semidirect
50. #1841D3semidirect
375. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#785412
#785202
C3×#18412
#690352
#690342
#687782
#687462
#687272
#687182
#687152
#686982
#293873