Finite Group Databasebeta

← #10897664

Subgroups — #10897664

Order 768.

654 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
1C3×#2922C211yesmaximal, characteristic, complemented
2#78541C211yesmaximal, characteristic, complemented
3#71645C212yesmaximal, complemented
4#70547C211yesmaximal, characteristic, complemented
5#70546C212yesmaximal, complemented
6#70539C211yesmaximal, characteristic, complemented
7#70013C211yesmaximal, characteristic
8#70012C212yesmaximal, complemented
9#70005C211yesmaximal, characteristic, complemented
10#69871C211yesmaximal, characteristic, complemented
11#69793C212yesmaximal, complemented
12#29386—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5822C2212yes—
15#5748C2211yescharacteristic
16#5745C2212yes—
17C3×#433—22no—
18C3×#431—22no—
19C3×#424C2211yescharacteristic
20C3×#424C2211yescharacteristic
21C3×#408C2211yescharacteristic
22C3×#408C2211yescharacteristic
23C3×C4×M16C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25#5236C2212yes—
26#5125C2211yescharacteristic
27#5113C2212yes—
28#5110C2212yes—
29#5102C2212yes—
30#5006C2212yes—
31C2×#805C2212yes—
32#4717—22no—
33#4700—22no—
34C3⋊#338C2211yescharacteristic
35#4557C2211yescharacteristic
36#4556C2212yes—
37#4552C2212yes—
38#4533C2212yes—
39#4523C2212yes—
40#2956—31no—
41#2922D311yescharacteristic, complemented
42#2103—31no—
43#2098—31no—
44#2053—31no—
45#2017—31no—
46#1937—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841—31no—
51C2×C3×#41C2311yescharacteristic
52C2×#235C2312yes—
53C2×C3×M16C2311yescharacteristic
54C2×C3×M16C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C3×C4×Q8—21no—
58C3×C4×D4—21no—
59C3×C4×D4—22no—
60C3×#117—21no—
61C3×#117C2311yescharacteristic
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64#951—22no—
65#950—42no—
66#949—22no—
67#942—22no—
68#932—22no—
69#926C2312yes—
70C2×#208C2312yes—
71C2×#208C2312yes—
72#909—22no—
73#908—22no—
74#907—42no—
75D3×M16—42no—
76#904—22no—
77#901—24no—
78C3×C4⋊C8—22no—
79C3×C4⋊C8—22no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×#97D412yes—
83C3×C4×C8—21no—
84C3×C4×C8—21no—
85C3×C4×C8D412yes—
86#837—42no—
87#821—22no—
88(C2×D3)⋊C8—42no—
89#819—22no—
90#816—22no—
91Dic3⋊C8—42no—
92#806—24no—
93#805—22no—
94#805—22no—
95#575—32no—
96#575—31no—
97#568—32no—
98C4×#41C2×D311yescharacteristic
99#444—62no—
100#443—62no—
101#433—22no—
102#431—22no—
103#430—31no—
104#424—32no—
105#424C2×D311yescharacteristic
106#424C2×D311yescharacteristic
107#421—32no—
108C2×#104—32no—
109#416—32no—
110#410—32no—
111#408—32no—
112#408C2×D311yescharacteristic
113#408C2×D311yescharacteristic
114C4×M16C2×D311yescharacteristic
115C2×C4×C8C2×D311yescharacteristic
116#344—32no—
117#344—32no—
118#341—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)—31no—
122C3×#41—22no—
123C3×#41—22no—
124C2×C3×Q8#4111yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4#4111yescharacteristic
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4C2411yescharacteristic
131C2×#67—22no—
132#235—22no—
133#235—22no—
134#235—22no—
135#235—22no—
136C2×D12#4112yes—
137C2×C4×D3—22no—
138C2×Dic6#4112yes—
139C3×M16—22no—
140C3×M16—22no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8C2×D412yes—
146C2×C3×C8C2×D412yes—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—21no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4#4112yes—
151C3×C42—22no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156#217—22no—
157#216—42no—
158#215—42no—
159#214—42no—
160#213—42no—
161C4⋊Dic3—22no—
162C4×Dic3—22no—
163#208—22no—
164#208—22no—
165#208—22no—
166#208—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169Dic12—22no—
170D24—22no—
171#204—24no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C2×#41—32no—
175C2×#41C22×D311yescharacteristic
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—32no—
188C2×M16—122no—
189C2×M16C22×D311yescharacteristic
190C2×M16C22×D311yescharacteristic
191C22×C8C22×D311yescharacteristic
192C22×C8C22×D311yescharacteristic
193C4×Q8—21no—
194C4×D4—21no—
195C4×D4—22no—
196#117—21no—
197#117—32no—
198#117C22×D311yescharacteristic
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#108—62no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—122no—
205#104—62no—
206#104—62no—
207#104—122no—
208#103—62no—
209#102—62no—
210#99—64no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—122no—
214#97#6712yes—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8#6712yes—
218C23×C3—21no—
219C23×C3C2×#4111yescharacteristic
220C22×D3—22no—
221C3×Q8—22no—
222C3×D4—22no—
223C3×D4—42no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—42no—
229C2×C3×C4—44no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4C22×D411yescharacteristic
233C2×C3×C4C2×#4111yescharacteristic
234C2×C3×C4C2×#4112yes—
235C2×C3×C4H(5,𝔽2)12yes—
236#67—42no—
237#67—42no—
238C2×Dic3—22no—
239C2×Dic3—42no—
240D12—22no—
241D12—22no—
242C4×D3—42no—
243C4×D3—42no—
244Dic6—22no—
245Dic6—22no—
246C3×C8—22no—
247C3×C8—22no—
248C3×C8—41no—
249C3×C8—41no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252#41—22no—
253#41—22no—
254#41—62no—
255#41—62no—
256#41—62no—
257#41—62no—
258C2×Q8—32no—
259C2×Q8#23511yescharacteristic
260C2×D4—21no—
261C2×D4—32no—
262C2×D4—62no—
263C2×D4#23511yescharacteristic
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
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—62no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—124no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—124no—
290C2×C8C2×#6712yes—
291C2×C8C2×#6712yes—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—62no—
295C22⋊C4—22no—
296C22⋊C4—62no—
297C22⋊C4#23512yes—
298C42—22no—
299C42—21no—
300C42—21no—
301C42—62no—
302C42C2×#6711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—41no—
307C22×C3#52511yescharacteristic
308C22×C3#55312yes—
309C2×D3—42no—
310C2×D3—82no—
311C3×C4—22no—
312C3×C4—42no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4#52511yescharacteristic
316C3×C4#53511yescharacteristic
317C3×C4#55112yes—
318Dic3—42no—
319Dic3—82no—
320C23—21no—
321C23—62no—
322C23C2×#23511yescharacteristic
323Q8—22no—
324Q8—62no—
325Q8—62no—
326D4—22no—
327D4—42no—
328D4—62no—
329D4—62no—
330D4—122no—
331D4—122no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4C2×#23511yescharacteristic
345C2×C4D3×#4112yes—
346C2×C4#100912yes—
347C2×C4C22×#6711yescharacteristic
348C8—22no—
349C8—22no—
350C8—41no—
351C8—41no—
352C8—122no—
353C8—122no—
354C2×C3—22no—
355C2×C3—21no—
356C2×C3—81no—
357C2×C3#259811yescharacteristic
358D3—82no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—122no—
363C22—242no—
364C22#553012yes—
365C22#579111yescharacteristic
366C4—22no—
367C4—42no—
368C4—41no—
369C4—41no—
370C4—122no—
371C4—242no—
372C4#552712yes—
373C4#582811yescharacteristic
374C4#583011yescharacteristic
375C3#2938611yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—81no—
379C2—242no—
380C2#8228411yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2922C2
socleC2×C3#2598
prime radical#70013C2
hypercenter#2922D3
hypocenterC3#29386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
2. #78541C2semidirect
3. #71645C2semidirect
4. #70547C2semidirect
5. #70546C2semidirect
6. #70539C2semidirect
8. #70012C2semidirect
9. #70005C2semidirect
10. #69871C2semidirect
11. #69793C2semidirect
41. #2922D3semidirect
375. C3#29386semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
#785412
#716452
#705472
#705462
#705392
#700132
#700122
#700052
#698712
#697932
#293863