Finite Group Databasebeta

← #72246979

Subgroups — #72246979

Order 4032.

780 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#72246811C211yesmaximal, characteristic, complemented
2#72246705C211yesmaximal, characteristic, complemented
3#72246641C212yesmaximal, complemented
4C7×#64149C211yesmaximal, characteristic, complemented
5C3×#10621415C212yesmaximal, complemented
6#13054561—32nomaximal
7C2×C32×C7×D4C2211yescharacteristic, complemented
8C2×C7×#638—22no—
9C2×C7×C4⋊(C3⋊D3)C2211yescharacteristic
10C3×#66728C2212yes—
11#11759644—22no—
12#11759634—22no—
13#11759632—22no—
14C4×C7×C3⋊Dic3C2211yescharacteristic, complemented
15C3×#66633—22no—
16C3×#66632—22no—
17C3×#66627C2212yescomplemented
18#10621565—32no—
19#10621415D312yescomplemented
20C7×#940—32no—
21C7×#940—61no—
22C3×#6495—32no—
23#10620515—32no—
24#10620512—32no—
25#10620494—32no—
26#10591522—71nomaximal, Hall
27C23×C32×C7—21no—
28C22×C7×C3⋊D3D412yes—
29C2×(C3⋊D3)⋊D7—22no—
30C32×C7×D4—22no—
31C2×C32×C4×C7C2311yescharacteristic
32C7×#638—44no—
33C2×C7×C3⋊Dic3—21no—
34C7×C4⋊(C3⋊D3)—22no—
35C3×#4268—22no—
36C3×D84D412yescomplemented
37C3×C4×D21—22no—
38C3×Dic42D412yescomplemented
39#92694—24no—
40#92693—22no—
41C3×C3⋊(C7⋊C8)—22no—
42#89849—91noHall
43C2×C3×C7×D4—21no—
44C2×C3×C7×D4C2×D312yescomplemented
45#66728C2×D312yes—
46C2×C7×#67—64no—
47C2×C7×#67—62no—
48C2×C7×#67—62no—
49C2×C7×D12—32no—
50C2×C7×D12—61no—
51C3×#1157—32no—
52D3×C22⋊D7—64no—
53#66681—62no—
54D7×D12—62no—
55#66676—62no—
56#66672—62no—
57#66633—22no—
58#66632—22no—
59#66627C2×D312yescomplemented
60C4×C7×Dic3—32no—
61C4×C7×Dic3—61no—
62C3×#1142—62no—
63C3×#1141—62no—
64C3×#1136—32no—
65#64308—71no—
66#64149D711yescharacteristic, complemented
67C3×#932—72no—
68#63568—72no—
69#63559—71no—
70C22×C32×C7—42no—
71C22×C32×C7—41no—
72C22×C32×C7C2×D411yescharacteristic
73C2×C7×C3⋊D3—22no—
74C2×C7×C3⋊D3—42no—
75C2×C3×D21—22no—
76(C3⋊D3)⋊D7—82no—
77C32×C4×C7C2×D412yes—
78C7×C3⋊Dic3—22no—
79C3×Dic21—22no—
80#6548—91no—
81C7×#127—91no—
82#6495—32no—
83#6407—91no—
84#6379—92no—
85#5198—212no—
86C23×C3×C7—21no—
87C23×C3×C7—21no—
88C23×C3×C7—22no—
89C22×D3×C7—34no—
90C22×D3×C7—62no—
91C2×D3×D7—64no—
92C3×C7×D4—22no—
93C3×C7×D4—22no—
94C3×C7×D4—42no—
95C2×C3×C4×C7—21no—
96C2×C3×C4×C7C22×D312yes—
97#4268—22no—
98D84#6712yescomplemented
99C4×D21—22no—
100Dic42#6712yescomplemented
101C7×#67—128no—
102C7×#67—124no—
103C7×#67—124no—
104C2×C7×Dic3—61no—
105C2×C7×Dic3—61no—
106C2×C7×Dic3—62no—
107C7×D12—62no—
108C7×D12—62no—
109C7×D12—122no—
110C3×C22⋊D7—62no—
111C3×D28—32no—
112C3×C4×D7—62no—
113C3×Dic14—32no—
114#4247—64no—
115#4245—64no—
116D3×Dic7—64no—
117C3⋊(C7⋊C8)—22no—
118C3×C7⋊C8—62no—
119C2×C32×D4C2×D711yescharacteristic, complemented
120C2×#638—22no—
121C2×C4⋊(C3⋊D3)C2×D711yescharacteristic
122C3×#235—72no—
123#3811—142no—
124D3×D12—142no—
125#3799—142no—
126C4×C3⋊Dic3C2×D711yescharacteristic, complemented
127C3×#214—142no—
128C3×#213—142no—
129C3×#208—72no—
130C2×C32×C7—21no—
131C2×C32×C7—81no—
132C2×C32×C7#12011yescharacteristic
133C7×C3⋊D3—42no—
134C3×D21—42no—
135C2×C7×D4—91no—
136C2×C7×D4—182no—
137C2×C7×D4D3211yescharacteristic, complemented
138C2×C22⋊D7—182no—
139#1159—182no—
140D4×D7—182no—
141#1157—32no—
142C42×C7—91no—
143#1142—62no—
144#1141—62no—
145#1136—32no—
146#1009—212no—
147C3×AGL(1,ℤ/8)—212no—
148#940—32no—
149#940—61no—
150#932—72no—
151#837—212no—
152#825—212no—
153#823—212no—
154C22×C3×C7—21no—
155C22×C3×C7—42no—
156C22×C3×C7—42no—
157C22×C3×C7—44no—
158C22×C3×C7—41no—
159C22×C3×C7—41no—
160C22×C3×C7—42no—
161C22×C3×C7D3×D412yes—
162C2×D21—22no—
163C2×D3×C7—64no—
164C2×D3×C7—122no—
165C2×D3×C7—124no—
166C2×D3×C7—122no—
167C2×D3×C7—122no—
168C2×C3×D7—62no—
169D3×D7—244no—
170C3×C4×C7—22no—
171C3×C4×C7D3×D412yes—
172C3×C4×C7C2×#6712yes—
173Dic21—22no—
174C3×Dic7—62no—
175C7×Dic3—62no—
176C7×Dic3—62no—
177C7×Dic3—64no—
178C23×C32—21no—
179C22×C3⋊D3C22⋊D712yes—
180C2×D32—142no—
181C32×D4—22no—
182C2×C32×C4C22×D711yescharacteristic
183#638—44no—
184C2×C3⋊Dic3—21no—
185C4⋊(C3⋊D3)—22no—
186C3×#67—142no—
187C3×D12—72no—
188C3×C4×D3—142no—
189C3×Dic6—72no—
190#626—144no—
191#624—142no—
192C3×C3⋊C8—142no—
193#453—631noSylow
194C32×C7#45311yescharacteristic, complemented, Hall
195C23×C7—21no—
196C23×C7—92no—
197C22×D7—182no—
198C7×D4—22no—
199C7×D4—182no—
200C7×D4—364no—
201C2×C4×C7—181no—
202C2×C4×C7C2×D3211yescharacteristic
203C22⋊D7—62no—
204C22⋊D7—184no—
205C2×Dic7—182no—
206D28—32no—
207C4×D7—62no—
208Dic14—32no—
209C7⋊C8—62no—
210C3×#41—212no—
211C2×C3×D4—21no—
212C2×C3×D4D3×D712yescomplemented
213C2×#67—64no—
214C2×#67—62no—
215C2×#67—62no—
216#239—422no—
217D3×D4—422no—
218D3×D4—424no—
219#235—72no—
220#235—422no—
221C2×D12—32no—
222C2×D12—61no—
223C2×D12—422no—
224C3×SD16—422no—
225C3×D8—422no—
226C3×M16—212no—
227#214—142no—
228#213—142no—
229C4×Dic3—32no—
230C4×Dic3—61no—
231#208—72no—
232C2×C3×C7—21no—
233C2×C3×C7—21no—
234C2×C3×C7—22no—
235C2×C3×C7—21no—
236C2×C3×C7—81no—
237C2×C3×C7—81no—
238C2×C3×C7—82no—
239C2×C3×C7#93712yes—
240D21—42no—
241C3×D7—122no—
242D3×C7—124no—
243D3×C7—242no—
244C22×C32—42no—
245C22×C32—41no—
246C22×C32C2×C22⋊D711yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—42no—
249C2×C3×D3—142no—
250D32—562no—
251C32×C4D4×D712yes—
252C3⋊Dic3—22no—
253C3×Dic3—142no—
254H(5,𝔽2)—631no—
255AGL(1,ℤ/8)—212no—
256#127—91no—
257#104—632no—
258#100—631no—
259C22×C7—42no—
260C22×C7—41no—
261C22×C7—182no—
262C22×C7—362no—
263C22×C7#381111yescharacteristic
264C2×D7—62no—
265C2×D7—722no—
266C4×C7—182no—
267C4×C7D3×#6712yes—
268Dic7—62no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—22no—
272C22×D3—34no—
273C22×D3—62no—
274C22×D3—424no—
275C3×Q8—212no—
276C3×D4—22no—
277C3×D4—22no—
278C3×D4—42no—
279C3×D4—212no—
280C3×D4—422no—
281C2×C3×C4—21no—
282C2×C3×C4—422no—
283C2×C3×C4C2×D3×D712yes—
284#67—128no—
285#67—124no—
286#67—124no—
287#67—142no—
288#67—424no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—62no—
292D12—62no—
293D12—62no—
294D12—72no—
295D12—122no—
296D12—424no—
297C4×D3—142no—
298C4×D3—424no—
299Dic6—72no—
300C3×C8—422no—
301C3⋊C8—142no—
302C3×C7—21no—
303C3×C7#519812yescomplemented
304C2×C32—21no—
305C2×C32—81no—
306C2×C32#650011yescharacteristic
307C3⋊D3—42no—
308C3×D3—282no—
309#41—212no—
310#41—1262no—
311C2×D4—91no—
312C2×D4—182no—
313C2×D4—1262no—
314C2×D4—1262no—
315C2×D4(C3⋊D3)⋊D711yescharacteristic, complemented
316SD16—422no—
317D8—422no—
318M16—212no—
319C42—91no—
320C2×C7—21no—
321C2×C7—81no—
322C2×C7—362no—
323C2×C7#6398311yescharacteristic
324D7—122no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—44no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—422no—
333C22×C3D3×C22⋊D712yes—
334C2×D3—64no—
335C2×D3—122no—
336C2×D3—124no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—142no—
340C2×D3—1684no—
341C3×C4—22no—
342C3×C4—422no—
343C3×C4D7×#6712yes—
344C3×C4#6669412yes—
345Dic3—62no—
346Dic3—62no—
347Dic3—64no—
348Dic3—142no—
349C32#8984911yescharacteristic, complemented, Sylow
350C23—21no—
351C23—92no—
352C23—1262no—
353Q8—212no—
354D4—22no—
355D4—182no—
356D4—212no—
357D4—364no—
358D4—422no—
359D4—1264no—
360C2×C4—181no—
361C2×C4—422no—
362C2×C4—1262no—
363C2×C4C2×(C3⋊D3)⋊D711yescharacteristic
364C8—422no—
365C7#1059152211yescharacteristic, complemented, Sylow
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3—82no—
373C2×C3—842no—
374C2×C3#1062113912yes—
375D3—124no—
376D3—242no—
377D3—282no—
378C22—42no—
379C22—41no—
380C22—182no—
381C22—362no—
382C22—422no—
383C22—5042no—
384C22#1175964411yescharacteristic
385C4—182no—
386C4—422no—
387C4#1175964312yes—
388C3—21no—
389C3#1305456112yescomplemented
390C2—21no—
391C2—81no—
392C2—362no—
393C2—842no—
394C2#7224697511yescharacteristic
subgroupG/H
centerC2#72246975
commutator, cosocleC2×C32×C4×C7C23
FrattiniC2×C4C2×(C3⋊D3)⋊D7
FittingC2×C32×C7×D4C22
socleC2×C32×C7#120
hypercenterC2×D4(C3⋊D3)⋊D7
hypocenterC32×C7#453
LCSG ▷ C2×C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #72246811C2semidirect
2. #72246705C2semidirect
3. #72246641C2semidirect
4. C7×#64149C2semidirect
5. C3×#10621415C2semidirect
7. C2×C32×C7×D4C22semidirect
14. C4×C7×C3⋊Dic3C22semidirect
17. C3×#66627C22semidirect
19. #10621415D3semidirect
36. C3×D84D4semidirect
38. C3×Dic42D4semidirect
44. C2×C3×C7×D4C2×D3semidirect
59. #66627C2×D3semidirect
66. #64149D7semidirect
98. D84#67semidirect
100. Dic42#67semidirect
119. C2×C32×D4C2×D7semidirect
126. C4×C3⋊Dic3C2×D7semidirect
137. C2×C7×D4D32semidirect
194. C32×C7#453semidirect
212. C2×C3×D4D3×D7semidirect
303. C3×C7#5198semidirect
315. C2×D4(C3⋊D3)⋊D7semidirect
349. C32#89849semidirect
365. C7#10591522semidirect
389. C3#13054561semidirect

Maximal subgroups

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

subgroupindex
#722468112
#722467052
#722466412
C7×#641492
C3×#106214152
#130545613
#105915227