Finite Group Databasebeta

← #72246915

Subgroups — #72246915

Order 3024.

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

normal subgroups only

This group has 396 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
1#92696⋊C3C211yesmaximal, characteristic, complemented
2#92687⋊C3C211yesmaximal, characteristic
3#92687⋊C3C211yesmaximal, characteristic
4ΓL(1,𝔽8)×#627C211yesmaximal, characteristic, complemented
5#13094713C211yesmaximal, characteristic
6C3×#4248⋊C3C211yesmaximal, characteristic, complemented
7C3×#4248⋊C3C211yesmaximal, characteristic, complemented
8#11759437C311yesmaximal, characteristic, complemented
9(C3⋊AGL(1,𝔽7))⋊Q8—31nomaximal
10(C3⋊AGL(1,𝔽7))⋊Q8—31nomaximal
11C2×C3⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic
12ΓL(1,𝔽8)×C3⋊Dic3C2211yescharacteristic
13C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
14C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
15C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
16C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
17C32×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
18#92696C2×C311yescharacteristic, complemented
19#92687C2×C311yescharacteristic
20#92687C2×C311yescharacteristic
21C7×#627C2×C311yescharacteristic, complemented
22#92675C2×C311yescharacteristic
23C3×#4248C2×C311yescharacteristic, complemented
24C3×#4248C2×C311yescharacteristic, complemented
25C4×C3⋊AGL(1,𝔽7)—31no—
26C4×C3⋊AGL(1,𝔽7)—31no—
27ΓL(1,𝔽8)×Dic6—31no—
28ΓL(1,𝔽8)×Dic6—31no—
29C3×ΓL(1,𝔽8)⋊Q8—31no—
30C3×ΓL(1,𝔽8)⋊Q8—31no—
31#4248⋊C3—21no—
32#4248⋊C3—21no—
33#4248⋊C3—31no—
34#4248⋊C3—31no—
35#4248⋊C3D311yescharacteristic, complemented
36#4248⋊C3D311yescharacteristic, complemented
37(C3⋊AGL(1,𝔽7))⋊C4—31no—
38(C3⋊AGL(1,𝔽7))⋊C4—31no—
39(C3⋊AGL(1,𝔽7))⋊C4—31no—
40(C3⋊AGL(1,𝔽7))⋊C4—31no—
41(C3⋊AGL(1,𝔽7))⋊C4—61no—
42C3×#3797—71nomaximal, Hall
43C2×C32×ΓL(1,𝔽8)C2311yescharacteristic
44C3⋊(C3⋊AGL(1,𝔽7))Q812yescomplemented
45D21⋊Q8—31no—
46D21⋊Q8—31no—
47Q8×AGL(1,𝔽7)—91no—
48C2×C3⋊D21C22×C311yescharacteristic
49C2×C3⋊AGL(1,𝔽7)—31no—
50C2×C3⋊AGL(1,𝔽7)—31no—
51C2×C3⋊AGL(1,𝔽7)—61no—
52C7×C3⋊Dic3C22×C311yescharacteristic
53C3×Dic21C22×C311yescharacteristic
54C3×Dic21C22×C311yescharacteristic
55C3×C7×Dic3C22×C311yescharacteristic
56C3×C7×Dic3C22×C311yescharacteristic
57C32×Dic7C22×C311yescharacteristic
58C3×C4×ΓL(1,𝔽8)—31no—
59C3×C4×ΓL(1,𝔽8)—31no—
60C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
61C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
62C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
63C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
64Dic3×ΓL(1,𝔽8)—21no—
65Dic3×ΓL(1,𝔽8)—21no—
66Dic3×ΓL(1,𝔽8)—31no—
67Dic3×ΓL(1,𝔽8)—31no—
68Dic3×ΓL(1,𝔽8)—61no—
69Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
70Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
71C3×ΓL(1,𝔽8)⋊C4—21no—
72C3×ΓL(1,𝔽8)⋊C4—21no—
73C3×ΓL(1,𝔽8)⋊C4—21no—
74C3×ΓL(1,𝔽8)⋊C4—31no—
75C3×ΓL(1,𝔽8)⋊C4—31no—
76C3×ΓL(1,𝔽8)⋊C4—41no—
77C3×ΓL(1,𝔽8)⋊C4C2×D311yescharacteristic
78C3×ΓL(1,𝔽8)⋊C4C2×D311yescharacteristic
79C3×C4×C3⋊D3—71no—
80C32×Dic6—71no—
81C32×Dic6—71no—
82C3×#627—71no—
83C3×#627—71no—
84C3×#624—71no—
85C3×#624—71no—
86C32×ΓL(1,𝔽8)C2×Q811yescharacteristic, complemented, Hall
87C4×D21—31no—
88C4×D21—31no—
89C7×Dic6—31no—
90C7×Dic6—31no—
91C3×Dic14—31no—
92C3×Dic14—31no—
93Q8×ΓL(1,𝔽8)—91no—
94#4248—31no—
95#4248—31no—
96#4248C3×D311yescharacteristic, complemented
97#4248C3×D311yescharacteristic, complemented
98D21⋊C4—31no—
99D21⋊C4—31no—
100D21⋊C4—31no—
101D21⋊C4—31no—
102D21⋊C4—61no—
103C4×AGL(1,𝔽7)—91no—
104C4×AGL(1,𝔽7)—91no—
105C4×AGL(1,𝔽7)—91no—
106ΓL(1,𝔽8)⋊Q8—31no—
107ΓL(1,𝔽8)⋊Q8—31no—
108ΓL(1,𝔽8)⋊Q8—61no—
109ΓL(1,𝔽8)⋊Q8—61no—
110ΓL(1,𝔽8)⋊Q8—91no—
111C3×D3×Q8—211no—
112C3×D3×Q8—211no—
113#3797—71no—
114C2×C32×C7C23×C311yescharacteristic
115C3⋊D21C3×Q812yescomplemented
116C2×C3×ΓL(1,𝔽8)—21no—
117C2×C3×ΓL(1,𝔽8)—21no—
118C2×C3×ΓL(1,𝔽8)—21no—
119C2×C3×ΓL(1,𝔽8)—41no—
120C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
121C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
122C3⋊AGL(1,𝔽7)—32no—
123C3⋊AGL(1,𝔽7)—32no—
124C3⋊AGL(1,𝔽7)—62no—
125Q8×D7—91noHall
126C2×C3×C3⋊D3—71no—
127C33×C4—71no—
128C3×C3⋊Dic3—71no—
129C32×Dic3—71no—
130C32×Dic3—71no—
131C32×Dic3—71no—
132C32×Dic3—71no—
133C2×D21—31no—
134C2×D21—31no—
135C2×D21—61no—
136C2×AGL(1,𝔽7)—91no—
137C3×C4×C7—31no—
138C3×C4×C7—31no—
139Dic21C2×C3×D311yescharacteristic
140Dic21C2×C3×D311yescharacteristic
141C3×Dic7—21no—
142C3×Dic7—31no—
143C3×Dic7—31no—
144C3×Dic7C2×C3×D311yescharacteristic
145C3×Dic7C2×C3×D311yescharacteristic
146C7×Dic3—31no—
147C7×Dic3—31no—
148C7×Dic3—61no—
149C7×Dic3C2×C3×D311yescharacteristic
150C7×Dic3C2×C3×D311yescharacteristic
151C4×ΓL(1,𝔽8)—31no—
152C4×ΓL(1,𝔽8)—31no—
153C4×ΓL(1,𝔽8)—61no—
154C4×ΓL(1,𝔽8)—61no—
155C4×ΓL(1,𝔽8)—91no—
156ΓL(1,𝔽8)⋊C4—21no—
157ΓL(1,𝔽8)⋊C4—21no—
158ΓL(1,𝔽8)⋊C4—31no—
159ΓL(1,𝔽8)⋊C4—31no—
160ΓL(1,𝔽8)⋊C4—41no—
161ΓL(1,𝔽8)⋊C4—61no—
162ΓL(1,𝔽8)⋊C4—61no—
163ΓL(1,𝔽8)⋊C4D3211yescharacteristic
164C32×Q8—211no—
165C32×Q8—211no—
166C4×C3⋊D3—71no—
167C3×C4×D3—211no—
168C3×C4×D3—211no—
169C3×C4×D3—211no—
170C3×C4×D3—211no—
171C3×C4×D3—211no—
172C3×C4×D3—211no—
173C3×C4×D3—421no—
174C3×Dic6—71no—
175C3×Dic6—71no—
176C3×Dic6—71no—
177C3×Dic6—71no—
178C3×Dic6—141no—
179C3×Dic6—141no—
180C3×Dic6—211no—
181C3×Dic6—211no—
182C3×Dic6—421no—
183C3×Dic6—421no—
184#627—71no—
185#627AGL(1,𝔽7)11yescharacteristic, complemented
186#624—71no—
187#624—71no—
188C32×C7C2×C3×Q811yescharacteristic, complemented
189C3×ΓL(1,𝔽8)—21no—
190C3×ΓL(1,𝔽8)—21no—
191C3×ΓL(1,𝔽8)—21no—
192C3×ΓL(1,𝔽8)—41no—
193C3×ΓL(1,𝔽8)D3×Q811yescharacteristic, complemented
194C3×ΓL(1,𝔽8)D3×Q811yescharacteristic, complemented
195C7×Q8—91no—
196C4×D7—91no—
197C4×D7—91no—
198C4×D7—91no—
199Dic14—31no—
200Dic14—31no—
201Dic14—91no—
202C2×C33—71no—
203C3×C3⋊D3—72no—
204C2×C3×Q8—631no—
205D3×Q8—211no—
206D3×Q8—211no—
207C2×C3×C7—21no—
208C2×C3×C7C22×C3×D311yescharacteristic
209C2×C3×C7C22×C3×D311yescharacteristic
210D21—32no—
211D21—32no—
212D21—62no—
213C2×ΓL(1,𝔽8)—21no—
214C2×ΓL(1,𝔽8)—21no—
215C2×ΓL(1,𝔽8)—41no—
216C2×ΓL(1,𝔽8)C2×D3211yescharacteristic
217AGL(1,𝔽7)—92no—
218C2×C3⋊D3—71no—
219C2×C3×D3—211no—
220C2×C3×D3—211no—
221C2×C3×D3—421no—
222C32×C4—71no—
223C32×C4—71no—
224C32×C4—71no—
225C32×C4—141no—
226C32×C4—141no—
227C32×C4—141no—
228C32×C4—211no—
229C32×C4—211no—
230C32×C4—281no—
231C32×C4—421no—
232C32×C4—421no—
233C3⋊Dic3C2×AGL(1,𝔽7)11yescharacteristic
234C3×Dic3—71no—
235C3×Dic3—71no—
236C3×Dic3—71no—
237C3×Dic3—71no—
238C3×Dic3—141no—
239C3×Dic3—141no—
240C3×Dic3—141no—
241C3×Dic3—141no—
242C3×Dic3—281no—
243C3×Dic3—281no—
244C3×Dic3—421no—
245C3×Dic3—421no—
246C3×Dic3—841no—
247C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
248C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
249C2×D7—91no—
250C4×C7—31no—
251C4×C7—31no—
252C4×C7—91no—
253Dic7—31no—
254Dic7—31no—
255Dic7C3×D3211yescharacteristic
256C33—71noSylow
257C3×Q8—211no—
258C3×Q8—211no—
259C3×Q8—211no—
260C3×Q8—211no—
261C3×Q8—421no—
262C3×Q8—421no—
263C3×Q8—631no—
264C3×Q8—1261no—
265C2×C3×C4—631no—
266C2×C3×C4—631no—
267C2×C3×C4—631no—
268C4×D3—211no—
269C4×D3—211no—
270C4×D3—211no—
271C4×D3—211no—
272C4×D3—211no—
273C4×D3—211no—
274C4×D3—421no—
275Dic6—31no—
276Dic6—31no—
277Dic6—71no—
278Dic6—71no—
279Dic6—211no—
280Dic6—211no—
281C3×C7—21no—
282C3×C7C3×D3×Q811yescharacteristic, complemented
283C3×C7C3×D3×Q811yescharacteristic, complemented
284ΓL(1,𝔽8)—21no—
285ΓL(1,𝔽8)—21no—
286ΓL(1,𝔽8)—41no—
287ΓL(1,𝔽8)#379711yescharacteristic, complemented
288C2×C32—141no—
289C2×C32—141no—
290C2×C32—281no—
291C2×C32—281no—
292C2×C32—281no—
293C2×C32—561no—
294C2×C32C22×AGL(1,𝔽7)11yescharacteristic
295C3⋊D3—72no—
296C3×D3—212no—
297C3×D3—212no—
298C3×D3—422no—
299C2×Q8—631noSylow
300C2×C7C2×C3×D3211yescharacteristic
301D7—92no—
302C22×C3—631no—
303C2×D3—211no—
304C2×D3—211no—
305C2×D3—421no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—71no—
309C3×C4—71no—
310C3×C4—71no—
311C3×C4—141no—
312C3×C4—141no—
313C3×C4—141no—
314C3×C4—211no—
315C3×C4—211no—
316C3×C4—211no—
317C3×C4—211no—
318C3×C4—281no—
319C3×C4—421no—
320C3×C4—421no—
321C3×C4—421no—
322C3×C4—421no—
323C3×C4—421no—
324C3×C4—841no—
325C3×C4—1261no—
326Dic3—31no—
327Dic3—31no—
328Dic3—61no—
329Dic3—71no—
330Dic3—71no—
331Dic3D3×AGL(1,𝔽7)11yescharacteristic
332Dic3D3×AGL(1,𝔽7)11yescharacteristic
333C32—141no—
334C32—141no—
335C32—281no—
336C32—281no—
337C32—281no—
338C32—561no—
339C32Q8×AGL(1,𝔽7)11yescharacteristic, complemented
340Q8—91no—
341Q8—211no—
342Q8—211no—
343Q8—631no—
344C2×C4—631no—
345C2×C4—631no—
346C2×C4—631no—
347C7C3×#379711yescharacteristic, complemented, Sylow
348C2×C3—21no—
349C2×C3—141no—
350C2×C3—141no—
351C2×C3—281no—
352C2×C3—561no—
353C2×C3—632no—
354C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
355C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
356D3—212no—
357D3—212no—
358D3—422no—
359C22—631no—
360C4—31no—
361C4—31no—
362C4—71no—
363C4—91no—
364C4—211no—
365C4—211no—
366C3—21no—
367C3—141no—
368C3—141no—
369C3—281no—
370C3—561no—
371C3(C3⋊AGL(1,𝔽7))⋊Q811yescharacteristic, complemented
372C3(C3⋊AGL(1,𝔽7))⋊Q811yescharacteristic, complemented
373C2—632no—
374C2D32×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×AGL(1,𝔽7)
commutator, Fitting, socle, cosocleC2×C32×C7C23×C3
prime radicalC2×C3⋊(C3⋊AGL(1,𝔽7))C22
hypocenterC32×C7C2×C3×Q8
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #92696⋊C3C2semidirect
4. ΓL(1,𝔽8)×#627C2semidirect
6. C3×#4248⋊C3C2semidirect
7. C3×#4248⋊C3C2semidirect
8. #11759437C3semidirect
18. #92696C2×C3semidirect
21. C7×#627C2×C3semidirect
23. C3×#4248C2×C3semidirect
24. C3×#4248C2×C3semidirect
35. #4248⋊C3D3semidirect
36. #4248⋊C3D3semidirect
44. C3⋊(C3⋊AGL(1,𝔽7))Q8semidirect
86. C32×ΓL(1,𝔽8)C2×Q8semidirect
96. #4248C3×D3semidirect
97. #4248C3×D3semidirect
115. C3⋊D21C3×Q8semidirect
185. #627AGL(1,𝔽7)semidirect
188. C32×C7C2×C3×Q8semidirect
193. C3×ΓL(1,𝔽8)D3×Q8semidirect
194. C3×ΓL(1,𝔽8)D3×Q8semidirect
282. C3×C7C3×D3×Q8semidirect
283. C3×C7C3×D3×Q8semidirect
287. ΓL(1,𝔽8)#3797semidirect
339. C32Q8×AGL(1,𝔽7)semidirect
347. C7C3×#3797semidirect
371. C3(C3⋊AGL(1,𝔽7))⋊Q8semidirect
372. C3(C3⋊AGL(1,𝔽7))⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#92696⋊C32
#92687⋊C32
ΓL(1,𝔽8)×#6272
#130947132
C3×#4248⋊C32
#117594373
(C3⋊AGL(1,𝔽7))⋊Q83
C3×#37977