Finite Group Databasebeta

← #90255⋊C3 #13057298

Subgroups — #90255⋊C3 #13057298

Order 1344.

436 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#6502⋊C3C211yesmaximal, characteristic
3#6498⋊C3C211yesmaximal, characteristic, complemented
4#6492⋊C3C211yesmaximal, characteristic, complemented
5C4×(C22⋊D7)⋊C3C211yesmaximal, characteristic, complemented
6#6487⋊C3C211yesmaximal, characteristic, complemented
7C2×(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#6459⋊C3C211yesmaximal, characteristic, complemented
9#6452⋊C3C211yesmaximal, characteristic, complemented
10#6449⋊C3C211yesmaximal, characteristic, complemented
11#6446⋊C3C211yesmaximal, characteristic, complemented
12#6442⋊C3C211yesmaximal, characteristic, complemented
13#6441⋊C3C211yesmaximal, characteristic, complemented
14#10620712C211yesmaximal, characteristic, complemented
15#6433⋊C3C211yesmaximal, characteristic, complemented
16#90255C311yesmaximal, characteristic, complemented, Hall
17C2×D4×ΓL(1,𝔽8)C2211yescharacteristic
18C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
19C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
20C2×(C22⋊D7)⋊C3C2211yescharacteristic
21C2×(C22⋊D7)⋊C3C2211yescharacteristic
22C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
23C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
24C2×C4×AGL(1,𝔽7)C2211yescharacteristic
25C2×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic
26C4⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
27C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
28C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
29C42×ΓL(1,𝔽8)C2211yescharacteristic
30(C22⋊Dic7)⋊C3C2211yescharacteristic
31(C22⋊Dic7)⋊C3C2211yescharacteristic
32(C22⋊Dic7)⋊C3C2211yescharacteristic
33(C22⋊Dic7)⋊C3C2211yescharacteristic
34(C2×AGL(1,𝔽7))⋊C4C2211yescharacteristic
35(C2×AGL(1,𝔽7))⋊C4C2211yescharacteristic
36(C2×AGL(1,𝔽7))⋊C4C2211yescharacteristic
37(C2×AGL(1,𝔽7))⋊C4C2211yescharacteristic
38C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
39C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
40C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
41(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
42(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
43(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
44(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
45(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
46(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
47(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
48(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
49(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
50(ΓL(1,𝔽8)⋊C4)⋊C4C2211yescharacteristic
51C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
52C4×C7×D4C2×C311yescharacteristic, complemented
53#6502C2×C311yescharacteristic
54#6498C2×C311yescharacteristic, complemented
55#6492C2×C311yescharacteristic, complemented
56C4×C22⋊D7C2×C311yescharacteristic, complemented
57#6487C2×C311yescharacteristic, complemented
58C2×Dic7⋊C4C2×C311yescharacteristic, complemented
59#6459C2×C311yescharacteristic, complemented
60#6452C2×C311yescharacteristic, complemented
61#6449C2×C311yescharacteristic, complemented
62#6446C2×C311yescharacteristic, complemented
63#6442C2×C311yescharacteristic, complemented
64#6441C2×C311yescharacteristic, complemented
65#6439C2×C311yescharacteristic, complemented
66#6433C2×C311yescharacteristic, complemented
67C3×#539—71nomaximal, Hall
68C23×ΓL(1,𝔽8)C2311yescharacteristic
69C23×ΓL(1,𝔽8)C2311yescharacteristic
70C22×AGL(1,𝔽7)C2311yescharacteristic
71D4×ΓL(1,𝔽8)—22no—
72C2×C4×ΓL(1,𝔽8)—41no—
73C2×C4×ΓL(1,𝔽8)—41no—
74C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
75C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
76C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
77C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
78C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
79(C22⋊D7)⋊C3—22no—
80(C22⋊D7)⋊C3—41no—
81C2×ΓL(1,𝔽8)⋊C4—22no—
82C2×ΓL(1,𝔽8)⋊C4—41no—
83C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
84C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
85C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
86C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
87C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
88C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
89C2×ΓL(1,𝔽8)⋊C4C2311yescharacteristic
90C4×AGL(1,𝔽7)—41no—
91ΓL(1,𝔽8)⋊Q8—41no—
92C2×C7×D4C22×C311yescharacteristic
93C22×C4×C7C22×C311yescharacteristic
94C22×C4×C7C22×C311yescharacteristic
95C2×C22⋊D7C22×C311yescharacteristic
96C2×C22⋊D7C22×C311yescharacteristic
97C22×Dic7C22×C311yescharacteristic
98C22×Dic7C22×C311yescharacteristic
99C2×C4×D7C22×C311yescharacteristic
100C2×Dic14C22×C311yescharacteristic
101C7×C4⋊C4C22×C311yescharacteristic
102C7×C22⋊C4C22×C311yescharacteristic
103C7×C22⋊C4C22×C311yescharacteristic
104C42×C7C22×C311yescharacteristic
105C22⋊Dic7C22×C311yescharacteristic
106C22⋊Dic7C22×C311yescharacteristic
107C22⋊Dic7C22×C311yescharacteristic
108C22⋊Dic7C22×C311yescharacteristic
109(C2×D7)⋊C4C22×C311yescharacteristic
110(C2×D7)⋊C4C22×C311yescharacteristic
111(C2×D7)⋊C4C22×C311yescharacteristic
112(C2×D7)⋊C4C22×C311yescharacteristic
113C4⋊Dic7C22×C311yescharacteristic
114C4⋊Dic7C22×C311yescharacteristic
115C4⋊Dic7C22×C311yescharacteristic
116Dic7⋊C4C22×C311yescharacteristic
117Dic7⋊C4C22×C311yescharacteristic
118Dic7⋊C4C22×C311yescharacteristic
119Dic7⋊C4C22×C311yescharacteristic
120Dic7⋊C4C22×C311yescharacteristic
121Dic7⋊C4C22×C311yescharacteristic
122Dic7⋊C4C22×C311yescharacteristic
123Dic7⋊C4C22×C311yescharacteristic
124Dic7⋊C4C22×C311yescharacteristic
125Dic7⋊C4C22×C311yescharacteristic
126C4×Dic7C22×C311yescharacteristic
127C3×#126—71no—
128C3×#126—71no—
129C3×#125—71no—
130C3×#125—71no—
131C3×#123—71no—
132C3×#123—71no—
133C3×#123—71no—
134C3×#123—71no—
135C3×C22⋊Q8—71no—
136C3×C22⋊Q8—71no—
137C3×C22⋊Q8—71no—
138C3×#121—71no—
139C3×C4×D4—71no—
140C3×C4×D4—71no—
141C2×C3×C4⋊C4—71no—
142C22×ΓL(1,𝔽8)—21no—
143C22×ΓL(1,𝔽8)—21no—
144C22×ΓL(1,𝔽8)—21no—
145C22×ΓL(1,𝔽8)—21no—
146C22×ΓL(1,𝔽8)—21no—
147C22×ΓL(1,𝔽8)#4112yes—
148C22×ΓL(1,𝔽8)C2411yescharacteristic
149C2×AGL(1,𝔽7)—21no—
150C2×AGL(1,𝔽7)—21no—
151C2×AGL(1,𝔽7)—21no—
152C4×ΓL(1,𝔽8)—21no—
153C4×ΓL(1,𝔽8)—21no—
154C4×ΓL(1,𝔽8)—21no—
155C4×ΓL(1,𝔽8)—21no—
156C4×ΓL(1,𝔽8)—21no—
157ΓL(1,𝔽8)⋊C4—21no—
158ΓL(1,𝔽8)⋊C4—21no—
159ΓL(1,𝔽8)⋊C4—21no—
160ΓL(1,𝔽8)⋊C4—21no—
161ΓL(1,𝔽8)⋊C4—21no—
162ΓL(1,𝔽8)⋊C4—21no—
163ΓL(1,𝔽8)⋊C4—21no—
164#539—71noSylow
165C23×C7C23×C311yescharacteristic
166C23×C7C23×C311yescharacteristic
167C22×D7C23×C311yescharacteristic
168C7×D4—22no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7C23×C311yescharacteristic
172C2×C4×C7C23×C311yescharacteristic
173C2×C4×C7C23×C311yescharacteristic
174C2×C4×C7C23×C311yescharacteristic
175C2×C4×C7C23×C311yescharacteristic
176C22⋊D7—22no—
177C22⋊D7—41no—
178C2×Dic7—22no—
179C2×Dic7—41no—
180C2×Dic7C23×C311yescharacteristic
181C2×Dic7C23×C311yescharacteristic
182C2×Dic7C23×C311yescharacteristic
183C2×Dic7C23×C311yescharacteristic
184C2×Dic7C23×C311yescharacteristic
185C2×Dic7C23×C311yescharacteristic
186C2×Dic7C23×C311yescharacteristic
187C4×D7—41no—
188Dic14—41no—
189C2×C3×Q8—71no—
190C2×C3×D4—71no—
191C2×C3×D4—71no—
192C2×C3×D4—71no—
193C22×C3×C4—71no—
194C22×C3×C4—71no—
195C22×C3×C4—71no—
196C22×C3×C4—71no—
197C22×C3×C4—71no—
198C3×C4⋊C4—71no—
199C3×C4⋊C4—71no—
200C3×C4⋊C4—71no—
201C3×C4⋊C4—71no—
202C3×C4⋊C4—71no—
203C3×C4⋊C4—71no—
204C3×C4⋊C4—71no—
205C3×C4⋊C4—71no—
206C3×C4⋊C4—71no—
207C3×C4⋊C4—71no—
208C3×C4⋊C4—71no—
209C3×C4⋊C4—71no—
210C3×C4⋊C4—71no—
211C3×C4⋊C4—71no—
212C3×C22⋊C4—71no—
213C3×C22⋊C4—71no—
214C3×C22⋊C4—71no—
215C3×C22⋊C4—71no—
216C3×C22⋊C4—71no—
217C3×C22⋊C4—71no—
218C3×C22⋊C4—71no—
219C3×C22⋊C4—71no—
220C3×C22⋊C4—71no—
221C3×C22⋊C4—71no—
222C3×C42—71no—
223C3×C42—71no—
224C2×ΓL(1,𝔽8)—22no—
225C2×ΓL(1,𝔽8)—41no—
226C2×ΓL(1,𝔽8)C2×#4111yescharacteristic
227C2×ΓL(1,𝔽8)H(5,𝔽2)11yescharacteristic
228C2×ΓL(1,𝔽8)2−1+411yescharacteristic
229AGL(1,𝔽7)—41no—
230#126—71no—
231#126—71no—
232#125—71no—
233#125—71no—
234#123—71no—
235#123—71no—
236#123—71no—
237#123—71no—
238C22⋊Q8—71no—
239C22⋊Q8—71no—
240C22⋊Q8—71no—
241#121—71no—
242C4×D4—71no—
243C4×D4AGL(1,𝔽7)11yescharacteristic, complemented
244C2×C4⋊C4—71no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7C3×#4112yes—
251C22×C7C24×C311yescharacteristic
252C2×D7—21no—
253C2×D7—21no—
254C2×D7—21no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260Dic7—21no—
261Dic7—21no—
262Dic7—21no—
263Dic7—21no—
264Dic7—21no—
265Dic7—21no—
266Dic7—21no—
267C23×C3—71no—
268C23×C3—71no—
269C23×C3—71no—
270C3×Q8—281no—
271C3×D4—142no—
272C3×D4—142no—
273C3×D4—281no—
274C2×C3×C4—71no—
275C2×C3×C4—71no—
276C2×C3×C4—71no—
277C2×C3×C4—71no—
278C2×C3×C4—71no—
279C2×C3×C4—71no—
280C2×C3×C4—71no—
281C2×C3×C4—71no—
282C2×C3×C4—71no—
283C2×C3×C4—71no—
284C2×C3×C4—71no—
285C2×C3×C4—71no—
286C2×C3×C4—142no—
287C2×C3×C4—281no—
288C2×C3×C4—281no—
289C2×C3×C4—281no—
290C2×C3×C4—281no—
291ΓL(1,𝔽8)#53911yescharacteristic, complemented, Hall
292C2×Q8—71no—
293C2×D4—71no—
294C2×D4—71no—
295C2×D4C2×AGL(1,𝔽7)11yescharacteristic
296C22×C4—71no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4C2×AGL(1,𝔽7)11yescharacteristic
300C22×C4C2×AGL(1,𝔽7)11yescharacteristic
301C4⋊C4—71no—
302C4⋊C4—71no—
303C4⋊C4—71no—
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—71no—
307C4⋊C4—71no—
308C4⋊C4—71no—
309C4⋊C4—71no—
310C4⋊C4—71no—
311C4⋊C4—71no—
312C4⋊C4—71no—
313C4⋊C4—71no—
314C4⋊C4C2×AGL(1,𝔽7)11yescharacteristic
315C22⋊C4—71no—
316C22⋊C4—71no—
317C22⋊C4—71no—
318C22⋊C4—71no—
319C22⋊C4—71no—
320C22⋊C4—71no—
321C22⋊C4—71no—
322C22⋊C4—71no—
323C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic
324C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic
325C42—71no—
326C42C2×AGL(1,𝔽7)11yescharacteristic
327C2×C7—22no—
328C2×C7—41no—
329C2×C7C2×C3×#4111yescharacteristic
330C2×C7C3×H(5,𝔽2)11yescharacteristic
331C2×C7C3×2−1+411yescharacteristic
332D7—41no—
333C22×C3—72no—
334C22×C3—71no—
335C22×C3—141no—
336C22×C3—141no—
337C22×C3—141no—
338C22×C3—141no—
339C22×C3—141no—
340C22×C3—141no—
341C22×C3—141no—
342C22×C3—141no—
343C3×C4—141no—
344C3×C4—141no—
345C3×C4—141no—
346C3×C4—141no—
347C3×C4—141no—
348C3×C4—141no—
349C3×C4—141no—
350C3×C4—141no—
351C3×C4—141no—
352C3×C4—141no—
353C3×C4—141no—
354C3×C4—141no—
355C23—71no—
356C23C22×AGL(1,𝔽7)11yescharacteristic
357C23C22×AGL(1,𝔽7)11yescharacteristic
358Q8—281no—
359D4—22no—
360D4—142no—
361D4—281no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—71no—
365C2×C4—71no—
366C2×C4—71no—
367C2×C4—71no—
368C2×C4—71no—
369C2×C4—71no—
370C2×C4—71no—
371C2×C4—142no—
372C2×C4—281no—
373C2×C4—281no—
374C2×C4C22×AGL(1,𝔽7)11yescharacteristic
375C2×C4C22×AGL(1,𝔽7)11yescharacteristic
376C2×C4C22×AGL(1,𝔽7)11yescharacteristic
377C2×C4C22×AGL(1,𝔽7)11yescharacteristic
378C2×C4C22×AGL(1,𝔽7)11yescharacteristic
379C7C3×#53911yescharacteristic, complemented, Sylow
380C2×C3—71no—
381C2×C3—71no—
382C2×C3—71no—
383C2×C3—142no—
384C2×C3—281no—
385C2×C3—281no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—141no—
392C22—141no—
393C22—141no—
394C22#1157⋊C312yes—
395C22C23×AGL(1,𝔽7)11yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—141no—
402C4—141no—
403C4—141no—
404C4—141no—
405C4—141no—
406C4—141no—
407C4—141no—
408C3—71noSylow
409C2—22no—
410C2—41no—
411C2—281no—
412C2C2×#1157⋊C311yescharacteristic
413C2#6548⋊C311yescharacteristic
414C2#1062151911yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC4×C7×D4C2×C3
prime radical#6502⋊C3C2
hypercenterC4×D4AGL(1,𝔽7)
hypocenterC7C3×#539
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×ΓL(1,𝔽8)C2semidirect
3. #6498⋊C3C2semidirect
4. #6492⋊C3C2semidirect
5. C4×(C22⋊D7)⋊C3C2semidirect
6. #6487⋊C3C2semidirect
7. C2×(ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
8. #6459⋊C3C2semidirect
9. #6452⋊C3C2semidirect
10. #6449⋊C3C2semidirect
11. #6446⋊C3C2semidirect
12. #6442⋊C3C2semidirect
13. #6441⋊C3C2semidirect
14. #10620712C2semidirect
15. #6433⋊C3C2semidirect
16. #90255C3semidirect
52. C4×C7×D4C2×C3semidirect
54. #6498C2×C3semidirect
55. #6492C2×C3semidirect
56. C4×C22⋊D7C2×C3semidirect
57. #6487C2×C3semidirect
58. C2×Dic7⋊C4C2×C3semidirect
59. #6459C2×C3semidirect
60. #6452C2×C3semidirect
61. #6449C2×C3semidirect
62. #6446C2×C3semidirect
63. #6442C2×C3semidirect
64. #6441C2×C3semidirect
65. #6439C2×C3semidirect
66. #6433C2×C3semidirect
243. C4×D4AGL(1,𝔽7)semidirect
291. ΓL(1,𝔽8)#539semidirect
379. C7C3×#539semidirect

Maximal subgroups

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

subgroupindex
C4×D4×ΓL(1,𝔽8)2
#6502⋊C32
#6498⋊C32
#6492⋊C32
C4×(C22⋊D7)⋊C32
#6487⋊C32
C2×(ΓL(1,𝔽8)⋊C4)⋊C42
#6459⋊C32
#6452⋊C32
#6449⋊C32
#6446⋊C32
#6442⋊C32
#6441⋊C32
#106207122
#6433⋊C32
#902553
C3×#5397