Finite Group Databasebeta

← #72254409

Subgroups — #72254409

Order 4032.

780 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 99 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#72254331C212yesmaximal, complemented
2#72245537C212yesmaximal, complemented
3ΓL(1,𝔽8)×#833C211yesmaximal, characteristic, complemented
4#72244404C211yesmaximal, characteristic, complemented
5C3×#6503⋊C3C211yesmaximal, characteristic, complemented
6#13054537C311yesmaximal, characteristic, complemented
7#89519⋊C3—31nomaximal
8C2×C3×D4×ΓL(1,𝔽8)C2211yescharacteristic
9C2×#4268⋊C3—22no—
10#66730⋊C3—22no—
11D4×C3⋊AGL(1,𝔽7)—22no—
12#66728⋊C3C2212yes—
13C2×C3×(C22⋊D7)⋊C3—22no—
14C2×C3×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
15ΓL(1,𝔽8)×#208C2212yescomplemented
16C3×C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic, complemented
17#66565⋊C3—22no—
18#66561⋊C3—22no—
19#10621657C2×C311yescharacteristic, complemented
20#6548⋊C3—31no—
21C3×#6503C2×C311yescharacteristic, complemented
22#10620874C2×C312yescomplemented
23#6503⋊C3—21no—
24#6503⋊C3D311yescharacteristic, complemented
25#6471⋊C3—32no—
26C7×#833C2×C311yescharacteristic, complemented
27ΓL(1,𝔽8)×#100—31no—
28#10620516C2×C312yescomplemented
29#6399⋊C3—32no—
30C3×#5077—71nomaximal, Hall
31C23×C3×ΓL(1,𝔽8)D412yes—
32C22×C3⋊AGL(1,𝔽7)—22no—
33C22×C3×AGL(1,𝔽7)—21no—
34C3×D4×ΓL(1,𝔽8)—22no—
35C2×C3×C4×ΓL(1,𝔽8)C2311yescharacteristic
36#4268⋊C3—24no—
37#4268⋊C3—22no—
38C2×C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
39C4⋊(C3⋊AGL(1,𝔽7))D412yescomplemented
40C4×C3⋊AGL(1,𝔽7)—22no—
41Dic42⋊C3D412yescomplemented
42C3×(C22⋊D7)⋊C3—44no—
43C2×C3×ΓL(1,𝔽8)⋊C4—21no—
44C3×C4⋊AGL(1,𝔽7)—22no—
45ΓL(1,𝔽8)×C3⋊C8—22no—
46#89519—31noHall
47C2×C3×C7×D4C22×C311yescharacteristic
48C2×#4268—22no—
49#66730—22no—
50D4×D21—22no—
51#66728C22×C312yes—
52C2×C3×C22⋊D7—22no—
53C2×C3×D28C22×C311yescharacteristic, complemented
54C2×D4×ΓL(1,𝔽8)—21no—
55C2×D4×ΓL(1,𝔽8)C2×D311yescharacteristic
56C2×(C22⋊D7)⋊C3—24no—
57C2×(C22⋊D7)⋊C3—22no—
58C2×(C22⋊D7)⋊C3—62no—
59#1159⋊C3—62no—
60D4×AGL(1,𝔽7)—62no—
61#1157⋊C3—32no—
62C2×C4⋊AGL(1,𝔽7)—21no—
63C2×C4⋊AGL(1,𝔽7)C2×D311yescharacteristic, complemented
64C7×#208C22×C312yescomplemented
65C3×C4×Dic7C22×C311yescharacteristic, complemented
66M16×ΓL(1,𝔽8)—32no—
67#66565—22no—
68#66561—22no—
69C4×ΓL(1,𝔽8)⋊C4—21no—
70C4×ΓL(1,𝔽8)⋊C4C2×D311yescharacteristic, complemented
71D56⋊C3—62no—
72#1132⋊C3—62no—
73C3×#1004—71no—
74C32×#127—71no—
75C3×#932—72no—
76C3×#833—141no—
77C3×#805—72no—
78C22×C3×ΓL(1,𝔽8)—22no—
79C22×C3×ΓL(1,𝔽8)—42no—
80C22×C3×ΓL(1,𝔽8)C2×D411yescharacteristic
81C2×C3⋊AGL(1,𝔽7)—22no—
82C2×C3⋊AGL(1,𝔽7)—82no—
83C2×C3×AGL(1,𝔽7)—42no—
84C2×C3×AGL(1,𝔽7)—41no—
85C3×C4×ΓL(1,𝔽8)C2×D412yes—
86C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
87C3×ΓL(1,𝔽8)⋊C4—22no—
88#6548—31no—
89#6503C3×D311yescharacteristic, complemented
90#6471—32no—
91C7×#100—31no—
92#6399—32no—
93C3×#453—211no—
94#5077—71no—
95C23×C3×C7C3×D412yes—
96C22×D21—22no—
97C22×C3×D7—21no—
98C23×ΓL(1,𝔽8)—22no—
99C23×ΓL(1,𝔽8)#6712yes—
100C22×AGL(1,𝔽7)—22no—
101C22×AGL(1,𝔽7)—21no—
102C22×AGL(1,𝔽7)—62no—
103C3×C7×D4—22no—
104C2×C3×C4×C7C23×C311yescharacteristic
105#4268—24no—
106#4268—22no—
107C2×Dic21—22no—
108D84C3×D412yescomplemented
109C4×D21—22no—
110Dic42C3×D412yescomplemented
111C3×C22⋊D7—44no—
112C2×C3×Dic7—21no—
113C3×D28—22no—
114D4×ΓL(1,𝔽8)—22no—
115D4×ΓL(1,𝔽8)—42no—
116C2×C4×ΓL(1,𝔽8)—21no—
117C2×C4×ΓL(1,𝔽8)C22×D311yescharacteristic
118(C22⋊D7)⋊C3—44no—
119(C22⋊D7)⋊C3—48no—
120(C22⋊D7)⋊C3—64no—
121(C22⋊D7)⋊C3—62no—
122C2×ΓL(1,𝔽8)⋊C4—22no—
123C2×ΓL(1,𝔽8)⋊C4—21no—
124C2×ΓL(1,𝔽8)⋊C4—62no—
125C4⋊AGL(1,𝔽7)—22no—
126C4⋊AGL(1,𝔽7)—32no—
127C4⋊AGL(1,𝔽7)—42no—
128C4×AGL(1,𝔽7)—62no—
129ΓL(1,𝔽8)⋊Q8—32no—
130C7×C3⋊C8—22no—
131C8×ΓL(1,𝔽8)—62no—
132C2×C32×D4—71no—
133C2×C32×D4—142no—
134C2×C32×D4—141no—
135C2×C3×#67—142no—
136C3×#237—142no—
137C3×D3×D4—142no—
138C3×#235—72no—
139C32×C42—71no—
140C3×#214—142no—
141C3×#213—142no—
142C3×#208—142no—
143C2×C3×ΓL(1,𝔽8)—21no—
144C2×C3×ΓL(1,𝔽8)—42no—
145C2×C3×ΓL(1,𝔽8)#12011yescharacteristic
146C3⋊AGL(1,𝔽7)—42no—
147C3×AGL(1,𝔽7)—81no—
148C2×C7×D4C2×C3×D311yescharacteristic
149C2×C22⋊D7—22no—
150C2×C22⋊D7—62no—
151#1159—62no—
152D4×D7—62no—
153#1157—32no—
154C2×D28C2×C3×D311yescharacteristic, complemented
155C7×M16—32no—
156C4×Dic7C2×C3×D311yescharacteristic, complemented
157D56—62no—
158#1132—62no—
159C3×H(5,𝔽2)—211no—
160#1004—71no—
161C3×AGL(1,ℤ/8)—212no—
162C3×#127—71no—
163C3×#127—71no—
164C3×#127—141no—
165#932—72no—
166C3×#104—212no—
167C3×#100—421no—
168#833AGL(1,𝔽7)11yescharacteristic, complemented
169#805—72no—
170C22×C3×C7—22no—
171C22×C3×C7—42no—
172C22×C3×C7C2×C3×D411yescharacteristic
173C2×D21—22no—
174C2×D21—82no—
175C2×C3×D7—42no—
176C2×C3×D7—41no—
177C22×ΓL(1,𝔽8)—22no—
178C22×ΓL(1,𝔽8)—21no—
179C22×ΓL(1,𝔽8)—42no—
180C22×ΓL(1,𝔽8)—42no—
181C22×ΓL(1,𝔽8)—44no—
182C22×ΓL(1,𝔽8)C2×#6711yescharacteristic
183C2×AGL(1,𝔽7)—42no—
184C2×AGL(1,𝔽7)—44no—
185C2×AGL(1,𝔽7)—42no—
186C2×AGL(1,𝔽7)—41no—
187C2×AGL(1,𝔽7)—62no—
188C2×AGL(1,𝔽7)—242no—
189C3×C4×C7C2×C3×D412yes—
190Dic21—22no—
191C3×Dic7—22no—
192C4×ΓL(1,𝔽8)—22no—
193C4×ΓL(1,𝔽8)D3×D412yes—
194ΓL(1,𝔽8)⋊C4—24no—
195ΓL(1,𝔽8)⋊C4—22no—
196ΓL(1,𝔽8)⋊C4—62no—
197C23×C32—141no—
198C23×C32—142no—
199C22×C3×D3—142no—
200C32×D4—142no—
201C32×D4—282no—
202C32×D4—284no—
203C2×C32×C4—141no—
204C2×C32×C4—141no—
205C3×#67—142no—
206C3×#67—144no—
207C2×C3×Dic3—142no—
208C3×D12—72no—
209C3×C4×D3—142no—
210C3×Dic6—72no—
211C3×C3⋊C8—282no—
212#453—211noSylow
213C3×ΓL(1,𝔽8)#45311yescharacteristic, complemented, Hall
214C23×C7C3×#6712yes—
215C22×D7—21no—
216C22×D7—62no—
217C7×D4—22no—
218C2×C4×C7C22×C3×D311yescharacteristic
219C22⋊D7—44no—
220C22⋊D7—64no—
221C22⋊D7—62no—
222C2×Dic7—21no—
223C2×Dic7—62no—
224D28—22no—
225D28—32no—
226C4×D7—62no—
227Dic14—32no—
228C7×C8—62no—
229C3×#41—212no—
230C3×#41—422no—
231C2×C3×D4—71no—
232C2×C3×D4—71no—
233C2×C3×D4—141no—
234C2×C3×D4—142no—
235C2×C3×D4—144no—
236C2×C3×D4—142no—
237C2×C3×D4—141no—
238C2×C3×D4—281no—
239C2×C3×D4—422no—
240C2×C3×D4—422no—
241C2×C3×D4C2×AGL(1,𝔽7)11yescharacteristic
242C2×#67—142no—
243#237—142no—
244D3×D4—142no—
245#235—72no—
246C3×SD16—422no—
247C3×D8—422no—
248C3×M16—422no—
249C3×C42—71no—
250C3×C42—71no—
251C3×C42—141no—
252#214—142no—
253#213—142no—
254#208C2×AGL(1,𝔽7)12yescomplemented
255C2×C3×C7—21no—
256C2×C3×C7—42no—
257C2×C3×C7C3×#12011yescharacteristic
258D21—42no—
259C3×D7—81no—
260C2×ΓL(1,𝔽8)—22no—
261C2×ΓL(1,𝔽8)—21no—
262C2×ΓL(1,𝔽8)—21no—
263C2×ΓL(1,𝔽8)—42no—
264C2×ΓL(1,𝔽8)—82no—
265C2×ΓL(1,𝔽8)#93711yescharacteristic
266AGL(1,𝔽7)—82no—
267AGL(1,𝔽7)—81no—
268AGL(1,𝔽7)—122no—
269C22×C32—141no—
270C22×C32—282no—
271C22×C32—281no—
272C22×C32—282no—
273C22×C32—562no—
274C2×C3×D3—142no—
275C2×C3×D3—562no—
276C32×C4—142no—
277C32×C4—142no—
278C3×Dic3—142no—
279H(5,𝔽2)—211no—
280AGL(1,ℤ/8)—212no—
281#127—71no—
282#104—212no—
283#100—31no—
284C22×C7—22no—
285C22×C7—42no—
286C22×C7C2×C3×#6711yescharacteristic
287C2×D7—42no—
288C2×D7—41no—
289C2×D7—62no—
290C2×D7—242no—
291C4×C7C3×D3×D412yes—
292Dic7—22no—
293Dic7—62no—
294C23×C3—141no—
295C23×C3—142no—
296C23×C3—141no—
297C23×C3—142no—
298C23×C3—282no—
299C23×C3—422no—
300C23×C3C4⋊AGL(1,𝔽7)12yes—
301C22×D3—142no—
302C3×Q8—212no—
303C3×D4—22no—
304C3×D4—142no—
305C3×D4—142no—
306C3×D4—212no—
307C3×D4—282no—
308C3×D4—282no—
309C3×D4—282no—
310C3×D4—284no—
311C3×D4—284no—
312C3×D4—288no—
313C3×D4—422no—
314C3×D4—424no—
315C2×C3×C4—141no—
316C2×C3×C4—142no—
317C2×C3×C4—141no—
318C2×C3×C4—141no—
319C2×C3×C4—281no—
320C2×C3×C4—422no—
321C2×C3×C4—422no—
322C2×C3×C4C22×AGL(1,𝔽7)11yescharacteristic
323#67—144no—
324#67—142no—
325C2×Dic3—142no—
326D12—72no—
327C4×D3—142no—
328Dic6—72no—
329C3×C8—842no—
330C3⋊C8—22no—
331C3×C7C3×#45311yescharacteristic, complemented
332ΓL(1,𝔽8)—21no—
333ΓL(1,𝔽8)#507711yescharacteristic, complemented
334C2×C32—141no—
335C2×C32—281no—
336C2×C32—562no—
337C2×C32—561no—
338C3×D3—282no—
339#41—212no—
340#41—422no—
341C2×D4—71no—
342C2×D4—142no—
343C2×D4—422no—
344C2×D4—422no—
345C2×D4D3×AGL(1,𝔽7)11yescharacteristic
346SD16—422no—
347D8—422no—
348M16—32no—
349C42—71no—
350C2×C7—21no—
351C2×C7—42no—
352C2×C7C3×#93711yescharacteristic
353D7—81no—
354D7—122no—
355C22×C3—22no—
356C22×C3—42no—
357C22×C3—141no—
358C22×C3—282no—
359C22×C3—282no—
360C22×C3—282no—
361C22×C3—281no—
362C22×C3—282no—
363C22×C3—284no—
364C22×C3—282no—
365C22×C3—281no—
366C22×C3—281no—
367C22×C3—422no—
368C22×C3—562no—
369C22×C3—564no—
370C22×C3—1682no—
371C22×C3C2×C4⋊AGL(1,𝔽7)11yescharacteristic
372C2×D3—142no—
373C2×D3—562no—
374C3×C4—142no—
375C3×C4—142no—
376C3×C4—144no—
377C3×C4—142no—
378C3×C4—282no—
379C3×C4—422no—
380C3×C4D4×AGL(1,𝔽7)12yes—
381Dic3—142no—
382C32—141noSylow
383C23—141no—
384C23—422no—
385C23#4246⋊C312yes—
386Q8—212no—
387D4—22no—
388D4—142no—
389D4—212no—
390D4—284no—
391D4—424no—
392D4—422no—
393C2×C4—141no—
394C2×C4—422no—
395C2×C4—422no—
396C2×C4C2×D3×AGL(1,𝔽7)11yescharacteristic
397C8—62no—
398C7C3×#507711yescharacteristic, complemented, Sylow
399C2×C3—21no—
400C2×C3—42no—
401C2×C3—141no—
402C2×C3—281no—
403C2×C3—282no—
404C2×C3—281no—
405C2×C3—562no—
406C2×C3—562no—
407C2×C3—561no—
408C2×C3—562no—
409C2×C3—561no—
410C2×C3—842no—
411C2×C3#6445⋊C311yescharacteristic
412D3—282no—
413C22—22no—
414C22—42no—
415C22—282no—
416C22—281no—
417C22—422no—
418C22—1682no—
419C22C2×#4246⋊C311yescharacteristic
420C4—142no—
421C4—422no—
422C4#66694⋊C312yes—
423C3—141no—
424C3—281no—
425C3#89519⋊C311yescharacteristic, complemented
426C2—21no—
427C2—42no—
428C2—561no—
429C2—842no—
430C2#7224415111yescharacteristic
subgroupG/H
centerC2#72244151
commutator, cosocleC2×C3×C4×C7C23×C3
FrattiniC2×C4C2×D3×AGL(1,𝔽7)
FittingC2×C3×C7×D4C22×C3
socleC2×C3×C7C3×#120
hypercenterC2×D4D3×AGL(1,𝔽7)
hypocenterC3×C7C3×#453
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #72254331C2semidirect
2. #72245537C2semidirect
3. ΓL(1,𝔽8)×#833C2semidirect
4. #72244404C2semidirect
5. C3×#6503⋊C3C2semidirect
6. #13054537C3semidirect
14. C2×C3×C4⋊AGL(1,𝔽7)C22semidirect
15. ΓL(1,𝔽8)×#208C22semidirect
16. C3×C4×ΓL(1,𝔽8)⋊C4C22semidirect
19. #10621657C2×C3semidirect
21. C3×#6503C2×C3semidirect
22. #10620874C2×C3semidirect
24. #6503⋊C3D3semidirect
26. C7×#833C2×C3semidirect
28. #10620516C2×C3semidirect
39. C4⋊(C3⋊AGL(1,𝔽7))D4semidirect
41. Dic42⋊C3D4semidirect
53. C2×C3×D28C22×C3semidirect
63. C2×C4⋊AGL(1,𝔽7)C2×D3semidirect
64. C7×#208C22×C3semidirect
65. C3×C4×Dic7C22×C3semidirect
70. C4×ΓL(1,𝔽8)⋊C4C2×D3semidirect
89. #6503C3×D3semidirect
108. D84C3×D4semidirect
110. Dic42C3×D4semidirect
154. C2×D28C2×C3×D3semidirect
156. C4×Dic7C2×C3×D3semidirect
168. #833AGL(1,𝔽7)semidirect
213. C3×ΓL(1,𝔽8)#453semidirect
254. #208C2×AGL(1,𝔽7)semidirect
331. C3×C7C3×#453semidirect
333. ΓL(1,𝔽8)#5077semidirect
398. C7C3×#5077semidirect
425. C3#89519⋊C3semidirect

Maximal subgroups

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

subgroupindex
#722543312
#722455372
ΓL(1,𝔽8)×#8332
#722444042
C3×#6503⋊C32
#130545373
#89519⋊C33
C3×#50777