Finite Group Databasebeta

← #72246940

Subgroups — #72246940

Order 3024.

420 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊(C3⋊(C3⋊AGL(1,𝔽7)))C211yesmaximal, characteristic, complemented
2C3×Dic42⋊C3C211yesmaximal, characteristic, complemented
3C3×C4×D3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#92690⋊C3C212yesmaximal, complemented
5#92687⋊C3C212yesmaximal, complemented
6#11759605C311yesmaximal, characteristic, complemented
7#66728⋊C3—31nomaximal
8#66673⋊C3—31nomaximal
9C2×C3⋊(C3⋊AGL(1,𝔽7))C2212yes—
10C2×C3×D3×ΓL(1,𝔽8)C2211yescharacteristic
11C32×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2212yes—
13C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
14C4⋊(C3⋊D21)C2×C311yescharacteristic, complemented
15C3×Dic42C2×C311yescharacteristic, complemented
16C3×C4×D3×C7C2×C311yescharacteristic, complemented
17C2×C3×C4×ΓL(1,𝔽8)—31no—
18#92690C2×C312yescomplemented
19#92687C2×C312yescomplemented
20#4268⋊C3—32no—
21C4⋊(C3⋊AGL(1,𝔽7))—31no—
22C4⋊(C3⋊AGL(1,𝔽7))—31no—
23C4⋊(C3⋊AGL(1,𝔽7))—61no—
24C4×C3⋊AGL(1,𝔽7)—32no—
25Dic42⋊C3—21no—
26Dic42⋊C3D311yescharacteristic, complemented
27C4×D3×ΓL(1,𝔽8)—21no—
28C4×D3×ΓL(1,𝔽8)D311yescharacteristic, complemented
29C3×ΓL(1,𝔽8)⋊Q8—31no—
30#4247⋊C3—32no—
31(C3⋊AGL(1,𝔽7))⋊C4—32no—
32C3×#3799—71nomaximal, Hall
33C2×C32×ΓL(1,𝔽8)C2311yescharacteristic
34C3⋊(C3⋊AGL(1,𝔽7))—22no—
35C3×D3×ΓL(1,𝔽8)—21no—
36#66728—31no—
37#66673—31no—
38#1157⋊C3—91no—
39C2×C3⋊D21C22×C312yes—
40C2×C3×D3×C7C22×C311yescharacteristic
41C22×C3×ΓL(1,𝔽8)—31no—
42C2×C3⋊AGL(1,𝔽7)—32no—
43C2×C3⋊AGL(1,𝔽7)—32no—
44C2×C3⋊AGL(1,𝔽7)—62no—
45C2×D3×ΓL(1,𝔽8)—21no—
46C2×D3×ΓL(1,𝔽8)C2×D311yescharacteristic
47C32×C4×C7C22×C311yescharacteristic
48C3×Dic21C22×C312yes—
49C3×C7×Dic3C22×C311yescharacteristic
50C3×C4×ΓL(1,𝔽8)—21no—
51C3×C4×ΓL(1,𝔽8)—21no—
52C3×C4×ΓL(1,𝔽8)—21no—
53C3×C4×ΓL(1,𝔽8)—31no—
54C3×C4×ΓL(1,𝔽8)—41no—
55C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
56C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
57C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
58C3⋊(ΓL(1,𝔽8)⋊C4)C2×D312yes—
59Dic3×ΓL(1,𝔽8)—21no—
60Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
61C3×ΓL(1,𝔽8)⋊C4—32no—
62C3×C4⋊(C3⋊D3)—71no—
63C32×C4×D3—141no—
64C32×Dic6—71no—
65C3×#626—72no—
66C3×#624—72no—
67C32×ΓL(1,𝔽8)#4111yescharacteristic, complemented, Hall
68C2×C3×C4×C7—31no—
69#4268—32no—
70D84—31no—
71D84—31no—
72D84—61no—
73C4×D21—32no—
74Dic42C3×D311yescharacteristic, complemented
75C4×D3×C7C3×D311yescharacteristic, complemented
76C3×Dic14—31no—
77C2×C4×ΓL(1,𝔽8)—31no—
78C2×C4×ΓL(1,𝔽8)—61no—
79#4247—32no—
80D21⋊C4—32no—
81(C22⋊D7)⋊C3—92no—
82C4⋊AGL(1,𝔽7)—91no—
83C4×AGL(1,𝔽7)—92no—
84ΓL(1,𝔽8)⋊Q8—31no—
85ΓL(1,𝔽8)⋊Q8—61no—
86C3×#239—211no—
87C3×#235—211no—
88#3799—71no—
89C2×C32×C7C23×C311yescharacteristic
90C3⋊D21—22no—
91C3×D3×C7—21no—
92C2×C3×ΓL(1,𝔽8)—21no—
93C2×C3×ΓL(1,𝔽8)—21no—
94C2×C3×ΓL(1,𝔽8)—21no—
95C2×C3×ΓL(1,𝔽8)—41no—
96C2×C3×ΓL(1,𝔽8)—61no—
97C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
98C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
99C3⋊AGL(1,𝔽7)—62no—
100C3⋊AGL(1,𝔽7)—62no—
101C3⋊AGL(1,𝔽7)—122no—
102D3×ΓL(1,𝔽8)—21no—
103D3×ΓL(1,𝔽8)—21no—
104D3×ΓL(1,𝔽8)—21no—
105#1157—91noHall
106C2×C3×C3⋊D3—72no—
107C2×C32×D3—141no—
108C33×C4—141no—
109C32×Dic3—72no—
110C32×Dic3—141no—
111C22×C3×C7—31no—
112C2×D21—32no—
113C2×D21—32no—
114C2×D21—62no—
115C2×D3×C7C2×C3×D311yescharacteristic
116C22×ΓL(1,𝔽8)—31no—
117C22×ΓL(1,𝔽8)—61no—
118C2×AGL(1,𝔽7)—92no—
119C3×C4×C7—21no—
120C3×C4×C7—31no—
121C3×C4×C7C2×C3×D311yescharacteristic
122C3×C4×C7C2×C3×D311yescharacteristic
123Dic21C2×C3×D312yes—
124C3×Dic7—32no—
125C7×Dic3C2×C3×D311yescharacteristic
126C4×ΓL(1,𝔽8)—21no—
127C4×ΓL(1,𝔽8)—21no—
128C4×ΓL(1,𝔽8)—31no—
129C4×ΓL(1,𝔽8)—41no—
130C4×ΓL(1,𝔽8)—61no—
131C4×ΓL(1,𝔽8)D3211yescharacteristic
132ΓL(1,𝔽8)⋊C4—32no—
133ΓL(1,𝔽8)⋊C4—62no—
134C32×Q8—211no—
135C2×C32×C4—211no—
136C4⋊(C3⋊D3)—71no—
137C3×#67—212no—
138C3×D12—211no—
139C3×D12—211no—
140C3×D12—212no—
141C3×D12—421no—
142C3×C4×D3—71no—
143C3×C4×D3—141no—
144C3×C4×D3—212no—
145C3×C4×D3—212no—
146C3×C4×D3AGL(1,𝔽7)11yescharacteristic, complemented
147C3×Dic6—71no—
148C3×Dic6—71no—
149C3×Dic6—141no—
150#626—72no—
151#624—72no—
152C32×C7C3×#4111yescharacteristic, complemented
153C3×ΓL(1,𝔽8)—21no—
154C3×ΓL(1,𝔽8)—21no—
155C3×ΓL(1,𝔽8)—21no—
156C3×ΓL(1,𝔽8)—41no—
157C3×ΓL(1,𝔽8)#23511yescharacteristic, complemented
158C3×ΓL(1,𝔽8)#23911yescharacteristic, complemented
159C2×C4×C7—31no—
160C22⋊D7—92no—
161D28—91no—
162C4×D7—92no—
163Dic14—31no—
164C2×C33—141no—
165C3×C3⋊D3—142no—
166C32×D3—141no—
167C3×#41—631no—
168#239—211no—
169#235—211no—
170C2×C3×C7—21no—
171C2×C3×C7—61no—
172C2×C3×C7C22×C3×D311yescharacteristic
173C2×C3×C7C22×C3×D311yescharacteristic
174D21—62no—
175D21—62no—
176D21—122no—
177D3×C7—21no—
178C2×ΓL(1,𝔽8)—21no—
179C2×ΓL(1,𝔽8)—21no—
180C2×ΓL(1,𝔽8)—41no—
181C2×ΓL(1,𝔽8)—61no—
182C2×ΓL(1,𝔽8)—61no—
183C2×ΓL(1,𝔽8)—61no—
184C2×ΓL(1,𝔽8)C2×D3211yescharacteristic
185AGL(1,𝔽7)—182no—
186C22×C32—211no—
187C2×C3⋊D3—72no—
188C2×C3×D3—71no—
189C2×C3×D3—141no—
190C2×C3×D3—212no—
191C2×C3×D3—212no—
192C2×C3×D3—422no—
193C2×C3×D3C2×AGL(1,𝔽7)11yescharacteristic
194C32×C4—71no—
195C32×C4—71no—
196C32×C4—141no—
197C32×C4—211no—
198C32×C4—212no—
199C32×C4—281no—
200C32×C4—281no—
201C32×C4—281no—
202C32×C4C2×AGL(1,𝔽7)11yescharacteristic
203C3×Dic3—72no—
204C3×Dic3—72no—
205C3×Dic3—71no—
206C3×Dic3—142no—
207C3×Dic3—141no—
208C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
209C22×C7—31no—
210C2×D7—92no—
211C4×C7—31no—
212C4×C7C3×D3211yescharacteristic
213Dic7—32no—
214C33—141noSylow
215C3×Q8—211no—
216C3×Q8—211no—
217C3×Q8—421no—
218C3×D4—631no—
219C3×D4—632no—
220C2×C3×C4—31no—
221C2×C3×C4—421no—
222C2×C3×C4—421no—
223C2×C3×C4—632no—
224#67—212no—
225D12—212no—
226D12—211no—
227D12—211no—
228D12—421no—
229C4×D3—212no—
230C4×D3—212no—
231C4×D3C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
232Dic6—71no—
233C3×C7—21no—
234C3×C7C3×#23511yescharacteristic, complemented
235C3×C7C3×#23911yescharacteristic, complemented
236ΓL(1,𝔽8)—21no—
237ΓL(1,𝔽8)—21no—
238ΓL(1,𝔽8)—41no—
239ΓL(1,𝔽8)#379911yescharacteristic, complemented
240C2×C32—71no—
241C2×C32—71no—
242C2×C32—141no—
243C2×C32—281no—
244C2×C32—281no—
245C2×C32—281no—
246C2×C32—421no—
247C2×C32C22×AGL(1,𝔽7)11yescharacteristic
248C3⋊D3—142no—
249C3×D3—21no—
250C3×D3—141no—
251C3×D3—141no—
252C3×D3—141no—
253C3×D3—422no—
254C3×D3—422no—
255C3×D3—842no—
256#41—631noSylow
257C2×C7—61no—
258C2×C7C2×C3×D3211yescharacteristic
259D7—182no—
260C22×C3—31no—
261C22×C3—421no—
262C22×C3—421no—
263C22×C3—632no—
264C2×D3—212no—
265C2×D3—212no—
266C2×D3—422no—
267C2×D3C2×C3⋊AGL(1,𝔽7)11yescharacteristic
268C3×C4—21no—
269C3×C4—31no—
270C3×C4—141no—
271C3×C4—141no—
272C3×C4—141no—
273C3×C4—212no—
274C3×C4—212no—
275C3×C4—281no—
276C3×C4—421no—
277C3×C4—421no—
278C3×C4—422no—
279C3×C4D3×AGL(1,𝔽7)11yescharacteristic
280C3×C4C2×C3⋊AGL(1,𝔽7)11yescharacteristic
281Dic3—72no—
282Dic3C2×C3⋊AGL(1,𝔽7)11yescharacteristic
283C32—71no—
284C32—71no—
285C32—141no—
286C32—281no—
287C32—281no—
288C32—281no—
289C32#1157⋊C311yescharacteristic, complemented
290Q8—211no—
291D4—632no—
292D4—631no—
293C2×C4—31no—
294C2×C4—632no—
295C7C3×#379911yescharacteristic, complemented, Sylow
296C2×C3—21no—
297C2×C3—61no—
298C2×C3—141no—
299C2×C3—141no—
300C2×C3—141no—
301C2×C3—281no—
302C2×C3—421no—
303C2×C3—421no—
304C2×C3—421no—
305C2×C3—1262no—
306C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
307C2×C3C22×C3⋊AGL(1,𝔽7)11yescharacteristic
308D3—21no—
309D3—422no—
310D3—422no—
311D3—842no—
312C22—31no—
313C22—632no—
314C4—31no—
315C4—212no—
316C4D3×C3⋊AGL(1,𝔽7)11yescharacteristic
317C3—21no—
318C3—141no—
319C3—141no—
320C3—141no—
321C3—281no—
322C3#66673⋊C311yescharacteristic, complemented
323C3#66728⋊C311yescharacteristic, complemented
324C2—61no—
325C2—1262no—
326C2C2×D3×C3⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C32×C7C23×C3
FittingC32×C4×C7C22×C3
hypercenterC4D3×C3⋊AGL(1,𝔽7)
hypocenterC32×C7C3×#41
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊(C3⋊AGL(1,𝔽7)))C2semidirect
2. C3×Dic42⋊C3C2semidirect
3. C3×C4×D3×ΓL(1,𝔽8)C2semidirect
4. #92690⋊C3C2semidirect
5. #92687⋊C3C2semidirect
6. #11759605C3semidirect
14. C4⋊(C3⋊D21)C2×C3semidirect
15. C3×Dic42C2×C3semidirect
16. C3×C4×D3×C7C2×C3semidirect
18. #92690C2×C3semidirect
19. #92687C2×C3semidirect
26. Dic42⋊C3D3semidirect
28. C4×D3×ΓL(1,𝔽8)D3semidirect
67. C32×ΓL(1,𝔽8)#41semidirect
74. Dic42C3×D3semidirect
75. C4×D3×C7C3×D3semidirect
146. C3×C4×D3AGL(1,𝔽7)semidirect
152. C32×C7C3×#41semidirect
157. C3×ΓL(1,𝔽8)#235semidirect
158. C3×ΓL(1,𝔽8)#239semidirect
231. C4×D3C3⋊AGL(1,𝔽7)semidirect
234. C3×C7C3×#235semidirect
235. C3×C7C3×#239semidirect
239. ΓL(1,𝔽8)#3799semidirect
289. C32#1157⋊C3semidirect
295. C7C3×#3799semidirect
322. C3#66673⋊C3semidirect
323. C3#66728⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊(C3⋊AGL(1,𝔽7)))2
C3×Dic42⋊C32
C3×C4×D3×ΓL(1,𝔽8)2
#92690⋊C32
#92687⋊C32
#117596053
#66728⋊C33
#66673⋊C33
C3×#37997