Finite Group Databasebeta

← C2×#72214606 #72276759

Subgroups — C2×#72214606 #72276759

Order 4032.

644 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C2×ΓL(1,𝔽8)×C22⋊A4C211yesmaximal, characteristic, complemented
2#72214606C212yesmaximal, direct product factor
3C2×#10620804C312yesmaximal, complemented
4C2×#10620796C311yesmaximal, characteristic, complemented
5C2×#10620789C311yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×C22⋊A4C2211yescharacteristic, complemented
7C2×A4×AGL(1,𝔽7)—41nomaximal
8C2×C22⋊(C22⋊ΓL(1,𝔽8))C2×C312yescomplemented
9C2×C7×C22⋊A4C2×C311yescharacteristic, complemented
10C25×ΓL(1,𝔽8)C2×C311yescharacteristic, complemented
11C22×(C22⋊D7)⋊C3—31no—
12#10620804C2×C314yescomplemented
13#10620796C2×C312yescomplemented
14#10620789—32no—
15#10620789C2×C312yescomplemented
16C2×(C22⋊Dic7)⋊C3—31no—
17C2×C3×#120⋊C3—71nomaximal, Hall
18C2×A4×ΓL(1,𝔽8)—41no—
19C2×A4×ΓL(1,𝔽8)—82no—
20A4×AGL(1,𝔽7)—42no—
21C2×#6516C3211yescharacteristic, complemented, Hall
22C22⋊(C22⋊ΓL(1,𝔽8))C22×C312yescomplemented
23C7×C22⋊A4C22×C311yescharacteristic, complemented
24C24×ΓL(1,𝔽8)—31no—
25C24×ΓL(1,𝔽8)—31no—
26C24×ΓL(1,𝔽8)—62no—
27C24×ΓL(1,𝔽8)—62no—
28C24×ΓL(1,𝔽8)C22×C311yescharacteristic, complemented
29C2×D7⋊A4—42no—
30C2×A4×D7—41no—
31C23×AGL(1,𝔽7)A411yescharacteristic, complemented
32C2×(C22⋊D7)⋊C3—32no—
33C2×(C22⋊D7)⋊C3—32no—
34C2×(C22⋊D7)⋊C3—62no—
35C2×(C22⋊D7)⋊C3—62no—
36C22×ΓL(1,𝔽8)⋊C4—31no—
37(C22⋊Dic7)⋊C3—32no—
38(C22⋊Dic7)⋊C3—32no—
39C2×C3×C22⋊A4—71no—
40C3×#120⋊C3—72no—
41C2×C3×AGL(1,𝔽7)—161no—
42A4×ΓL(1,𝔽8)—41no—
43A4×ΓL(1,𝔽8)—82no—
44C25×C7C2×C3211yescharacteristic, complemented
45C22×C22⋊D7—31no—
46#6516—32no—
47#6516C2×C3212yescomplemented
48C2×C22⋊Dic7—31no—
49C2×C3×#120—71no—
50C2×#120⋊C3—72no—
51C2×#120⋊C3—71no—
52C2×C22⋊ΓL(1,𝔽8)—42no—
53C2×C22⋊ΓL(1,𝔽8)—84no—
54C2×C7×A4—41no—
55C2×C7×A4—82no—
56C23×ΓL(1,𝔽8)—22no—
57C23×ΓL(1,𝔽8)—32no—
58C23×ΓL(1,𝔽8)—32no—
59C23×ΓL(1,𝔽8)—31no—
60C23×ΓL(1,𝔽8)—31no—
61C23×ΓL(1,𝔽8)—62no—
62C23×ΓL(1,𝔽8)—62no—
63C23×ΓL(1,𝔽8)—62no—
64C23×ΓL(1,𝔽8)—62no—
65C23×ΓL(1,𝔽8)—62no—
66C23×ΓL(1,𝔽8)—64no—
67C23×ΓL(1,𝔽8)—62no—
68C23×ΓL(1,𝔽8)—62no—
69C23×ΓL(1,𝔽8)—64no—
70C23×ΓL(1,𝔽8)C2×A411yescharacteristic
71D7⋊A4—44no—
72A4×D7—42no—
73C22×AGL(1,𝔽7)—61no—
74C22×AGL(1,𝔽7)—61no—
75C22×AGL(1,𝔽7)C2×A412yescomplemented
76(C22⋊D7)⋊C3—64no—
77(C22⋊D7)⋊C3—64no—
78C2×ΓL(1,𝔽8)⋊C4—32no—
79C2×ΓL(1,𝔽8)⋊C4—61no—
80C2×ΓL(1,𝔽8)⋊C4—61no—
81C3×C22⋊A4—71no—
82C22×C3×A4—281no—
83C2×C3×ΓL(1,𝔽8)—161no—
84C3×AGL(1,𝔽7)—162no—
85C24×C7—31no—
86C24×C7—31no—
87C24×C7—62no—
88C24×C7—62no—
89C24×C7C22×C3211yescharacteristic, complemented
90C23×D7C3×A411yescharacteristic, complemented
91C2×C22⋊D7—32no—
92C2×C22⋊D7—32no—
93C2×C22⋊D7—62no—
94C2×C22⋊D7—62no—
95C22×Dic7—31no—
96C22⋊Dic7—32no—
97C22⋊Dic7—32no—
98C25×C3—71no—
99C2×C22⋊A4—72no—
100C2×C22⋊A4AGL(1,𝔽7)11yescharacteristic, complemented
101C22×C3×D4—211no—
102C3×#120—72no—
103C3×#120—212no—
104C2×C3×C22⋊C4—211no—
105#120⋊C3—74no—
106#120⋊C3—72no—
107C2×C3×D7—161no—
108C22⋊ΓL(1,𝔽8)—42no—
109C22⋊ΓL(1,𝔽8)—84no—
110C7×A4—41no—
111C7×A4—82no—
112C22×ΓL(1,𝔽8)—22no—
113C22×ΓL(1,𝔽8)—31no—
114C22×ΓL(1,𝔽8)—31no—
115C22×ΓL(1,𝔽8)—32no—
116C22×ΓL(1,𝔽8)—32no—
117C22×ΓL(1,𝔽8)—62no—
118C22×ΓL(1,𝔽8)—62no—
119C22×ΓL(1,𝔽8)—64no—
120C22×ΓL(1,𝔽8)—62no—
121C22×ΓL(1,𝔽8)—62no—
122C22×ΓL(1,𝔽8)—62no—
123C22×ΓL(1,𝔽8)—62no—
124C22×ΓL(1,𝔽8)—64no—
125C22×ΓL(1,𝔽8)—62no—
126C22×ΓL(1,𝔽8)C22×A411yescharacteristic
127C2×AGL(1,𝔽7)—41no—
128C2×AGL(1,𝔽7)—62no—
129C2×AGL(1,𝔽7)—121no—
130C2×AGL(1,𝔽7)—162no—
131ΓL(1,𝔽8)⋊C4—62no—
132C2×C3×A4—281no—
133C2×C3×A4—282no—
134C2×C3×A4—562no—
135C2×#120—71noSylow
136C3×ΓL(1,𝔽8)—161noHall
137C23×C7—22no—
138C23×C7—32no—
139C23×C7—32no—
140C23×C7—31no—
141C23×C7—31no—
142C23×C7—62no—
143C23×C7—62no—
144C23×C7—62no—
145C23×C7—62no—
146C23×C7—62no—
147C23×C7—64no—
148C23×C7—62no—
149C23×C7—62no—
150C23×C7—64no—
151C23×C7C2×C3×A411yescharacteristic
152C22×D7—61no—
153C22×D7—61no—
154C22×D7C2×C3×A412yescomplemented
155C22⋊D7—64no—
156C22⋊D7—64no—
157C2×Dic7—32no—
158C2×Dic7—61no—
159C2×Dic7—61no—
160C24×C3—71no—
161C24×C3—71no—
162C24×C3—211no—
163C24×C3—211no—
164C24×C3—422no—
165C24×C3—422no—
166C22⋊A4—72no—
167C22⋊A4C2×AGL(1,𝔽7)11yescharacteristic, complemented
168C22×A4—281no—
169C22×A4—282no—
170C2×C3×D4—212no—
171C2×C3×D4—212no—
172C2×C3×D4—422no—
173C2×C3×D4—422no—
174C22×C3×C4—211no—
175C3×C22⋊C4—212no—
176C3×C22⋊C4—212no—
177C2×C3×C7—161no—
178C3×D7—162no—
179C2×ΓL(1,𝔽8)—31no—
180C2×ΓL(1,𝔽8)—31no—
181C2×ΓL(1,𝔽8)—62no—
182C2×ΓL(1,𝔽8)—62no—
183C2×ΓL(1,𝔽8)—162no—
184C2×ΓL(1,𝔽8)#120⋊C311yescharacteristic, complemented
185AGL(1,𝔽7)—42no—
186AGL(1,𝔽7)—164no—
187C22×C32—1121no—
188C3×A4—281no—
189C3×A4—562no—
190C25C3×AGL(1,𝔽7)11yescharacteristic, complemented
191C22×D4—211no—
192#120—72no—
193#120—212no—
194C2×C22⋊C4—211no—
195C22×C7—22no—
196C22×C7—31no—
197C22×C7—31no—
198C22×C7—32no—
199C22×C7—32no—
200C22×C7—62no—
201C22×C7—62no—
202C22×C7—64no—
203C22×C7—62no—
204C22×C7—62no—
205C22×C7—62no—
206C22×C7—62no—
207C22×C7—64no—
208C22×C7—62no—
209C22×C7C22×C3×A411yescharacteristic
210C2×D7—41no—
211C2×D7—62no—
212C2×D7—121no—
213Dic7—62no—
214C23×C3—72no—
215C23×C3—71no—
216C23×C3—142no—
217C23×C3—211no—
218C23×C3—211no—
219C23×C3—212no—
220C23×C3—212no—
221C23×C3—422no—
222C23×C3—422no—
223C23×C3—422no—
224C23×C3—422no—
225C23×C3—422no—
226C23×C3—424no—
227C23×C3—422no—
228C23×C3—422no—
229C23×C3—424no—
230C23×C3—421no—
231C23×C3—421no—
232C2×A4—41no—
233C2×A4—82no—
234C2×A4—282no—
235C2×A4—284no—
236C2×A4—282no—
237C2×A4—564no—
238C3×D4—424no—
239C3×D4—424no—
240C2×C3×C4—212no—
241C2×C3×C4—421no—
242C2×C3×C4—421no—
243C3×C7—161no—
244ΓL(1,𝔽8)—162no—
245ΓL(1,𝔽8)C2×#120⋊C311yescharacteristic, complemented
246C2×C32—1121no—
247C2×C32—1122no—
248C24—31no—
249C24—31no—
250C24—62no—
251C24—62no—
252C24—71no—
253C24C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented
254C2×D4—212no—
255C2×D4—212no—
256C2×D4—422no—
257C2×D4—422no—
258C22×C4—211no—
259C22⋊C4—212no—
260C22⋊C4—212no—
261C2×C7—31no—
262C2×C7—31no—
263C2×C7—62no—
264C2×C7—62no—
265C2×C7C3×#120⋊C311yescharacteristic, complemented
266D7—42no—
267C22×C3—71no—
268C22×C3—142no—
269C22×C3—211no—
270C22×C3—211no—
271C22×C3—212no—
272C22×C3—212no—
273C22×C3—281no—
274C22×C3—422no—
275C22×C3—422no—
276C22×C3—422no—
277C22×C3—422no—
278C22×C3—424no—
279C22×C3—422no—
280C22×C3—422no—
281C22×C3—422no—
282C22×C3—424no—
283C22×C3—422no—
284C22×C3—841no—
285C22×C3—1121no—
286C22×C3—1122no—
287A4—41no—
288A4—82no—
289A4—282no—
290A4—564no—
291C3×C4—422no—
292C32—1121noSylow
293C23—22no—
294C23—32no—
295C23—32no—
296C23—31no—
297C23—31no—
298C23—62no—
299C23—62no—
300C23—62no—
301C23—62no—
302C23—62no—
303C23—64no—
304C23—62no—
305C23—62no—
306C23—64no—
307C23—72no—
308C23—421no—
309C23—421no—
310C23A4×AGL(1,𝔽7)11yescharacteristic
311D4—424no—
312D4—424no—
313C2×C4—212no—
314C2×C4—421no—
315C2×C4—421no—
316C7C2×C3×#120⋊C311yescharacteristic, complemented, Sylow
317C2×C3—71no—
318C2×C3—161no—
319C2×C3—211no—
320C2×C3—211no—
321C2×C3—282no—
322C2×C3—422no—
323C2×C3—422no—
324C2×C3—1122no—
325C2×C3—1124no—
326C2×C3—1122no—
327C22—22no—
328C22—31no—
329C22—31no—
330C22—32no—
331C22—32no—
332C22—62no—
333C22—62no—
334C22—64no—
335C22—62no—
336C22—62no—
337C22—62no—
338C22—62no—
339C22—64no—
340C22—62no—
341C22—281no—
342C22—422no—
343C22—841no—
344C22C2×A4×AGL(1,𝔽7)11yescharacteristic
345C4—422no—
346C3—71no—
347C3—161no—
348C3—1122no—
349C2—31no—
350C2—31no—
351C2—62no—
352C2—62no—
353C2—282no—
354C2#7221460611yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72214606
commutator, cosocle, hypocenterC24×C7C22×C32
FrattiniC22C2×A4×AGL(1,𝔽7)
FittingC25×C7C2×C32
socleC23×C7C2×C3×A4
LCSG ▷ C24×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×ΓL(1,𝔽8)×C22⋊A4C2semidirect
2. #72214606C2direct (N × H)
3. C2×#10620804C3semidirect
4. C2×#10620796C3semidirect
5. C2×#10620789C3semidirect
6. ΓL(1,𝔽8)×C22⋊A4C22semidirect
8. C2×C22⋊(C22⋊ΓL(1,𝔽8))C2×C3semidirect
9. C2×C7×C22⋊A4C2×C3semidirect
10. C25×ΓL(1,𝔽8)C2×C3semidirect
12. #10620804C2×C3semidirect
13. #10620796C2×C3semidirect
15. #10620789C2×C3semidirect
21. C2×#6516C32semidirect
22. C22⋊(C22⋊ΓL(1,𝔽8))C22×C3semidirect
23. C7×C22⋊A4C22×C3semidirect
28. C24×ΓL(1,𝔽8)C22×C3semidirect
31. C23×AGL(1,𝔽7)A4semidirect
44. C25×C7C2×C32semidirect
47. #6516C2×C32semidirect
75. C22×AGL(1,𝔽7)C2×A4semidirect
89. C24×C7C22×C32semidirect
90. C23×D7C3×A4semidirect
100. C2×C22⋊A4AGL(1,𝔽7)semidirect
154. C22×D7C2×C3×A4semidirect
167. C22⋊A4C2×AGL(1,𝔽7)semidirect
184. C2×ΓL(1,𝔽8)#120⋊C3semidirect
190. C25C3×AGL(1,𝔽7)semidirect
245. ΓL(1,𝔽8)C2×#120⋊C3semidirect
253. C24C2×C3×AGL(1,𝔽7)semidirect
265. C2×C7C3×#120⋊C3semidirect
316. C7C2×C3×#120⋊C3semidirect
354. C2#72214606direct (N × H)

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×C22⋊A42
#722146062
C2×#106208043
C2×#106207963
C2×#106207893
C2×A4×AGL(1,𝔽7)4
C2×C3×#120⋊C37