Finite Group Databasebeta

← #10876303

Subgroups — #10876303

Order 768.

720 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82536C211yesmaximal, characteristic
2#82514C211yesmaximal, characteristic, complemented
3C3×#1827C211yesmaximal, characteristic, complemented
4#70143C211yesmaximal, characteristic
5#70135C211yesmaximal, characteristic, complemented
6#70054C211yesmaximal, characteristic
7#70044C211yesmaximal, characteristic, complemented
8#68750C212yesmaximal, complemented
9#68746C212yesmaximal, complemented
10#68715C212yesmaximal, complemented
11#68700C212yesmaximal, complemented
12#27599—31nomaximal, Sylow
13#5964C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15#5819C2211yescharacteristic
16C3×#413C2211yescharacteristic
17C3×#410C2211yescharacteristic
18#5234C2211yescharacteristic
19#5234C2211yescharacteristic
20#5132C2212yes—
21#5129C2212yes—
22#5125C2212yescomplemented
23#5005—22no—
24#5005C2212yes—
25#5005C2212yes—
26#5002C2212yes—
27#5000—22no—
28#4999C2211yescharacteristic, complemented
29#4999C2211yescharacteristic, complemented
30#4999C2212yescomplemented
31C2×#805—22no—
32C2×#805C2212yes—
33#4926—22no—
34C4×#208C2211yescharacteristic, complemented
35C2×#803C2211yescharacteristic, complemented
36C3×#339C2212yescomplemented
37C3×H(3,ℤ/4)C2212yescomplemented
38#4558C2212yescomplemented
39#4533C2212yescomplemented
40#2923—31no—
41#2901—31no—
42#2098—32no—
43#2080—32no—
44#1938—32no—
45#1932—32no—
46#1868—31no—
47#1858—31no—
48#1827D311yescharacteristic, complemented
49#1794—31no—
50#1791—31no—
51C2×C3×#41C2311yescharacteristic
52#1010—22no—
53#1009—22no—
54D3×#41—42no—
55#1007—21no—
56#1004—21no—
57#1004—22no—
58C2×#235—22no—
59C2×#235C2312yes—
60C2×C3×M16C2312yes—
61C3×C4×Q8—21no—
62C3×C4×D4—22no—
63C3×C4×D4—21no—
64C3×#117D412yes—
65C3×#117C2312yes—
66C3×#117C2311yescharacteristic
67C2×C3×C42D412yes—
68C2×C3×C42C2311yescharacteristic
69#948—42no—
70C2×#208—22no—
71C2×#208C2311yescharacteristic
72C2×#208C2312yes—
73C2×#208C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C3×#101—21no—
77C3×#100—21no—
78C3×#99—22no—
79#830—22no—
80#830—22no—
81#824—22no—
82#823—22no—
83(C2×D3)⋊C8—42no—
84#819—42no—
85#810—22no—
86#809—22no—
87#805—22no—
88#805—24no—
89#805—22no—
90#805—42no—
91C4⋊(C3⋊C8)—22no—
92C4⋊(C3⋊C8)—22no—
93C4⋊(C3⋊C8)—42no—
94#803—21no—
95#803—21no—
96#803—21no—
97C4×C3⋊C8—21no—
98#585—31no—
99#568—31no—
100C4×#41C2×D311yescharacteristic
101#433—62no—
102#432—62no—
103#430—32no—
104#424—31no—
105#424—31no—
106#424—32no—
107#421—32no—
108#421—32no—
109#421—62no—
110C2×#104—32no—
111C2×#104—62no—
112#416—32no—
113#413—32no—
114#413C2×D311yescharacteristic
115#410C2×D311yescharacteristic
116#408—31no—
117#408—31no—
118#408—32no—
119C4×M16—31no—
120C2×#97—31no—
121#344—32no—
122#341—32no—
123#339C2×D312yescomplemented
124H(3,ℤ/4)C2×D312yescomplemented
125C3×#41—22no—
126C3×#41—22no—
127C2×C3×Q8#4111yescharacteristic
128C2×C3×D4#4112yes—
129C2×C3×D4#4111yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C2×#67—22no—
135C2×#67—42no—
136#239—42no—
137D3×Q8—42no—
138#237—42no—
139#237—44no—
140D3×D4—42no—
141D3×D4—44no—
142#235—24no—
143#235—22no—
144#235—22no—
145#235—22no—
146#235—22no—
147#235—42no—
148#235—42no—
149C2×D12—22no—
150C2×D12#4112yes—
151C2×C4×D3—22no—
152C2×C4×D3—42no—
153C2×Dic6—22no—
154C2×Dic6#4112yes—
155C3×M16—22no—
156C3×M16—22no—
157C2×C3×C8—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—22no—
164C3×C42—22no—
165C3×C42—21no—
166C3×C42—22no—
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42C2×D412yes—
170#208—22no—
171#208—22no—
172#208—22no—
173#208—22no—
174#208—42no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—42no—
1832−1+4—62no—
1842−1+4—61no—
185H(5,𝔽2)—62no—
186H(5,𝔽2)—62no—
187H(5,𝔽2)—61no—
188C2×#41—32no—
189C2×#41—62no—
190C2×#41—122no—
191C2×#41C22×D311yescharacteristic
192#131—122no—
193C2×M16—32no—
194C2×M16—31no—
195C2×M16—31no—
196C2×M16—62no—
197C2×M16C22×D312yes—
198C22×C8—31no—
199C22×C8—31no—
200C4×Q8—21no—
201C4×D4—22no—
202C4×D4—21no—
203#117D1212yes—
204#117C22×D312yes—
205#117C22×D311yescharacteristic
206C2×C42D1212yes—
207C2×C42C22×D311yescharacteristic
208#108—122no—
209C4⋊C8—62no—
210C4⋊C8—62no—
211C4⋊C8—122no—
212#104—62no—
213#104—62no—
214#104—64no—
215#104—122no—
216#103—62no—
217#102—62no—
218#101—21no—
219#101—62no—
220#100—21no—
221#100—62no—
222#99—22no—
223C22⋊C8—62no—
224C22⋊C8—62no—
225C22⋊C8—122no—
226#97—61no—
227#97—61no—
228#97—61no—
229C4×C8—61no—
230C23×C3#12312yes—
231C23×C3C2×#4111yescharacteristic
232C22×D3—22no—
233C22×D3—42no—
234C3×Q8—22no—
235C3×D4—24no—
236C3×D4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—42no—
243C2×C3×C4—81no—
244C2×C3×C4—82no—
245C2×C3×C4#12312yes—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4H(5,𝔽2)12yes—
250#67—44no—
251#67—42no—
252#67—42no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255D12—22no—
256D12—22no—
257D12—42no—
258C4×D3—44no—
259C4×D3—42no—
260C4×D3—42no—
261Dic6—22no—
262Dic6—22no—
263Dic6—42no—
264C3×C8—42no—
265C3⋊C8—41no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—42no—
269C3⋊C8—41no—
270#41—22no—
271#41—22no—
272#41—64no—
273#41—62no—
274#41—62no—
275#41—62no—
276#41—62no—
277#41—122no—
278#41—124no—
279#41—122no—
280#41—122no—
281#41—122no—
282C2×Q8—32no—
283C2×Q8—62no—
284C2×Q8—122no—
285C2×Q8#23711yescharacteristic
286C2×D4—32no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—122no—
290C2×D4—124no—
291C2×D4—122no—
292C2×D4#23712yes—
293C2×D4#23711yescharacteristic
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—62no—
297C22×C4—122no—
298C22×C4C2×D1212yes—
299C22×C4C23×D311yescharacteristic
300M16—22no—
301M16—22no—
302M16—62no—
303M16—62no—
304M16—62no—
305M16—62no—
306M16—122no—
307C2×C8—22no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—122no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4—22no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42C2×D1211yescharacteristic
326C42C2×D1211yescharacteristic
327C42C2×D1212yes—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3C2×#12311yescharacteristic
332C22×C3#55112yes—
333C2×D3—42no—
334C2×D3—82no—
335C2×D3—162no—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—82no—
341C3×C4#52511yescharacteristic
342C3×C4#53511yescharacteristic
343C3×C4#55112yes—
344Dic3—42no—
345C23—62no—
346C23—122no—
347C23#88612yes—
348C23C2×#23711yescharacteristic
349Q8—22no—
350Q8—62no—
351Q8—62no—
352Q8—122no—
353D4—24no—
354D4—22no—
355D4—62no—
356D4—62no—
357D4—122no—
358D4—124no—
359D4—122no—
360D4—122no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—62no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—124no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4#88612yes—
375C2×C4C22×D1211yescharacteristic
376C2×C4C2×#23711yescharacteristic
377C2×C4D3×#4112yes—
378C2×C4#100912yes—
379C8—42no—
380C8—121no—
381C8—121no—
382C8—121no—
383C8—122no—
384C8—121no—
385C2×C3—21no—
386C2×C3—22no—
387C2×C3—42no—
388C2×C3#248211yescharacteristic
389D3—82no—
390C22—22no—
391C22—41no—
392C22—42no—
393C22—122no—
394C22—242no—
395C22—482no—
396C22C2×#88611yescharacteristic
397C22#565812yes—
398C4—22no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—82no—
403C4—122no—
404C4#552411yescharacteristic
405C4#552511yescharacteristic
406C4#565812yes—
407C3#2759911yescharacteristic, complemented, Sylow
408C2—21no—
409C2—22no—
410C2—42no—
411C2—242no—
412C2#7412911yescharacteristic
subgroupG/H
centerC4#5525
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1827C2
socleC2×C3#2482
prime radical#5964C22
hypercenter#1827D3
hypocenterC3#27599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1827

Complements

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

NH ≅ G/Ntype
2. #82514C2semidirect
3. C3×#1827C2semidirect
5. #70135C2semidirect
7. #70044C2semidirect
8. #68750C2semidirect
9. #68746C2semidirect
10. #68715C2semidirect
11. #68700C2semidirect
22. #5125C22semidirect
28. #4999C22semidirect
29. #4999C22semidirect
30. #4999C22semidirect
34. C4×#208C22semidirect
35. C2×#803C22semidirect
36. C3×#339C22semidirect
37. C3×H(3,ℤ/4)C22semidirect
38. #4558C22semidirect
39. #4533C22semidirect
48. #1827D3semidirect
123. #339C2×D3semidirect
124. H(3,ℤ/4)C2×D3semidirect
407. C3#27599semidirect

Maximal subgroups

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

subgroupindex
#825362
#825142
C3×#18272
#701432
#701352
#700542
#700442
#687502
#687462
#687152
#687002
#275993