Finite Group Databasebeta

← #72246942

Subgroups — #72246942

Order 3024.

436 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 78 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#92690⋊C3C211yesmaximal, characteristic, complemented
3#92687⋊C3C211yesmaximal, characteristic, complemented
4D3×Dic3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
5#13094715C211yesmaximal, characteristic, complemented
6C3×#4245⋊C3C211yesmaximal, characteristic, complemented
7C3×Dic3×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
8#11759438C311yesmaximal, characteristic, complemented
9#66683⋊C3—31nomaximal
10#66675⋊C3—31nomaximal
11C2×C3⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic
12C2×C3×D3×ΓL(1,𝔽8)C2211yescharacteristic
13C2×C32×AGL(1,𝔽7)C2211yescharacteristic
14ΓL(1,𝔽8)×C3⋊Dic3C2211yescharacteristic
15C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
16C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
17C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
18#92696C2×C311yescharacteristic, complemented
19#92690C2×C311yescharacteristic, complemented
20#92687C2×C311yescharacteristic, complemented
21D3×C7×Dic3C2×C311yescharacteristic, complemented
22#92677C2×C311yescharacteristic, complemented
23C3×#4245C2×C311yescharacteristic, complemented
24C3×Dic3×D7C2×C311yescharacteristic, complemented
25#4268⋊C3—31no—
26C4×C3⋊AGL(1,𝔽7)—31no—
27C2×Dic3×ΓL(1,𝔽8)—31no—
28C4×D3×ΓL(1,𝔽8)—31no—
29C3×(C22⋊D7)⋊C3—31no—
30C3×C4×AGL(1,𝔽7)—31no—
31#4248⋊C3—31no—
32#4248⋊C3—31no—
33#4247⋊C3—31no—
34#4246⋊C3—31no—
35#4246⋊C3—31no—
36#4246⋊C3—61no—
37#4245⋊C3—21no—
38#4245⋊C3D311yescharacteristic, complemented
39(C3⋊AGL(1,𝔽7))⋊C4—31no—
40Dic3×AGL(1,𝔽7)—21no—
41Dic3×AGL(1,𝔽7)D311yescharacteristic, complemented
42C3×#3804—71nomaximal, Hall
43C2×C32×ΓL(1,𝔽8)C2311yescharacteristic
44C3⋊(C3⋊AGL(1,𝔽7))—21no—
45C3×D3×ΓL(1,𝔽8)—21no—
46C32×AGL(1,𝔽7)—21no—
47#66683—31no—
48#66675—31no—
49#1157⋊C3—91no—
50C2×C3⋊D21C22×C311yescharacteristic
51C2×C3×D3×C7C22×C311yescharacteristic
52C2×C32×D7C22×C311yescharacteristic
53C22×C3×ΓL(1,𝔽8)—31no—
54C2×C3⋊AGL(1,𝔽7)—31no—
55C2×C3⋊AGL(1,𝔽7)—31no—
56C2×C3⋊AGL(1,𝔽7)—61no—
57C2×D3×ΓL(1,𝔽8)—21no—
58C2×D3×ΓL(1,𝔽8)C2×D311yescharacteristic
59C2×C3×AGL(1,𝔽7)—21no—
60C2×C3×AGL(1,𝔽7)—21no—
61C2×C3×AGL(1,𝔽7)—21no—
62C2×C3×AGL(1,𝔽7)—41no—
63C2×C3×AGL(1,𝔽7)C2×D311yescharacteristic
64C2×C3×AGL(1,𝔽7)C2×D311yescharacteristic
65C7×C3⋊Dic3C22×C311yescharacteristic
66C3×Dic21C22×C311yescharacteristic
67C3×Dic21C22×C311yescharacteristic
68C3×C7×Dic3C22×C311yescharacteristic
69C3×C4×ΓL(1,𝔽8)—31no—
70C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
71C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
72C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
73C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
74Dic3×ΓL(1,𝔽8)—21no—
75Dic3×ΓL(1,𝔽8)—31no—
76Dic3×ΓL(1,𝔽8)—31no—
77Dic3×ΓL(1,𝔽8)—61no—
78Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
79C3×ΓL(1,𝔽8)⋊C4—31no—
80C3×ΓL(1,𝔽8)⋊C4—31no—
81C3×#638—71no—
82C32×#67—71no—
83C2×C32×Dic3—71no—
84C3×#627—71no—
85C3×#626—71no—
86C3×#624—71no—
87C3×D3×Dic3—141no—
88C32×ΓL(1,𝔽8)#4111yescharacteristic, complemented, Hall
89#4268—31no—
90C4×D21—31no—
91C2×C7×Dic3—31no—
92C4×D3×C7—31no—
93C3×C22⋊D7—31no—
94C3×C4×D7—31no—
95C2×C4×ΓL(1,𝔽8)—91no—
96#4248—31no—
97#4248—31no—
98#4247—31no—
99#4246—31no—
100#4246—31no—
101#4246—61no—
102#4245C3×D311yescharacteristic, complemented
103D21⋊C4—31no—
104Dic3×D7C3×D311yescharacteristic, complemented
105(C22⋊D7)⋊C3—31no—
106(C22⋊D7)⋊C3—61no—
107(C22⋊D7)⋊C3—91no—
108C4⋊AGL(1,𝔽7)—91no—
109C4×AGL(1,𝔽7)—31no—
110C4×AGL(1,𝔽7)—61no—
111C4×AGL(1,𝔽7)—91no—
112ΓL(1,𝔽8)⋊Q8—91no—
113C3×#237—211no—
114C3×#235—211no—
115#3804—71no—
116C2×C32×C7C23×C311yescharacteristic
117C3⋊D21—21no—
118C3×D3×C7—21no—
119C32×D7—21no—
120C2×C3×ΓL(1,𝔽8)—21no—
121C2×C3×ΓL(1,𝔽8)—21no—
122C2×C3×ΓL(1,𝔽8)—21no—
123C2×C3×ΓL(1,𝔽8)—41no—
124C2×C3×ΓL(1,𝔽8)—61no—
125C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
126C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
127C3⋊AGL(1,𝔽7)—61no—
128C3⋊AGL(1,𝔽7)—61no—
129C3⋊AGL(1,𝔽7)—121no—
130D3×ΓL(1,𝔽8)—21no—
131D3×ΓL(1,𝔽8)—41no—
132C3×AGL(1,𝔽7)—21no—
133C3×AGL(1,𝔽7)—21no—
134C3×AGL(1,𝔽7)—21no—
135C3×AGL(1,𝔽7)—21no—
136C3×AGL(1,𝔽7)—41no—
137C3×AGL(1,𝔽7)—41no—
138C3×AGL(1,𝔽7)—41no—
139C3×AGL(1,𝔽7)—41no—
140#1157—91noHall
141C22×C33—71no—
142C2×C3×C3⋊D3—71no—
143C2×C32×D3—141no—
144C3×C3⋊Dic3—141no—
145C32×Dic3—71no—
146C32×Dic3—71no—
147C32×Dic3—141no—
148C22×C3×C7—31no—
149C2×D21—31no—
150C2×D21—31no—
151C2×D21—61no—
152C2×D3×C7C2×C3×D311yescharacteristic
153C2×C3×D7—21no—
154C2×C3×D7C2×C3×D311yescharacteristic
155C2×C3×D7C2×C3×D311yescharacteristic
156C22×ΓL(1,𝔽8)—31no—
157C22×ΓL(1,𝔽8)—61no—
158C2×AGL(1,𝔽7)—21no—
159C2×AGL(1,𝔽7)—21no—
160C2×AGL(1,𝔽7)—41no—
161C2×AGL(1,𝔽7)—91no—
162C2×AGL(1,𝔽7)D3211yescharacteristic
163C3×C4×C7—31no—
164Dic21C2×C3×D311yescharacteristic
165Dic21C2×C3×D311yescharacteristic
166C3×Dic7—31no—
167C3×Dic7—31no—
168C7×Dic3—31no—
169C7×Dic3—31no—
170C7×Dic3—61no—
171C7×Dic3C2×C3×D311yescharacteristic
172C4×ΓL(1,𝔽8)—31no—
173C4×ΓL(1,𝔽8)—61no—
174C4×ΓL(1,𝔽8)—91no—
175ΓL(1,𝔽8)⋊C4—31no—
176ΓL(1,𝔽8)⋊C4—31no—
177ΓL(1,𝔽8)⋊C4—61no—
178ΓL(1,𝔽8)⋊C4—61no—
179C32×D4—211no—
180C2×C32×C4—211no—
181#638—71no—
182C3×#67—71no—
183C3×#67—71no—
184C3×#67—141no—
185C3×#67—211no—
186C3×#67—211no—
187C3×#67—211no—
188C3×#67—421no—
189C2×C3×Dic3—71no—
190C2×C3×Dic3—71no—
191C2×C3×Dic3—141no—
192C2×C3×Dic3—421no—
193C3×D12—211no—
194C3×C4×D3—211no—
195C3×C4×D3—211no—
196C3×C4×D3—421no—
197C3×Dic6—211no—
198C3×Dic6—211no—
199#627—71no—
200#626—71no—
201#624—71no—
202D3×Dic3AGL(1,𝔽7)11yescharacteristic, complemented
203C32×C7C3×#4111yescharacteristic, complemented
204C3×ΓL(1,𝔽8)—21no—
205C3×ΓL(1,𝔽8)—21no—
206C3×ΓL(1,𝔽8)—21no—
207C3×ΓL(1,𝔽8)—41no—
208C3×ΓL(1,𝔽8)#23511yescharacteristic, complemented
209C3×ΓL(1,𝔽8)#23711yescharacteristic, complemented
210C2×C4×C7—91no—
211C22⋊D7—31no—
212C22⋊D7—91no—
213D28—91no—
214C4×D7—31no—
215C4×D7—91no—
216Dic14—91no—
217C2×C33—141no—
218C2×C33—141no—
219C3×C3⋊D3—141no—
220C32×D3—141no—
221C3×#41—631no—
222#237—211no—
223#235—211no—
224C2×C3×C7—21no—
225C2×C3×C7—61no—
226C2×C3×C7C22×C3×D311yescharacteristic
227C2×C3×C7C22×C3×D311yescharacteristic
228D21—61no—
229D21—61no—
230D21—121no—
231C3×D7—21no—
232C3×D7—21no—
233C3×D7—41no—
234D3×C7—21no—
235C2×ΓL(1,𝔽8)—21no—
236C2×ΓL(1,𝔽8)—21no—
237C2×ΓL(1,𝔽8)—41no—
238C2×ΓL(1,𝔽8)—61no—
239C2×ΓL(1,𝔽8)—121no—
240C2×ΓL(1,𝔽8)C2×D3211yescharacteristic
241AGL(1,𝔽7)—21no—
242AGL(1,𝔽7)—21no—
243AGL(1,𝔽7)—21no—
244AGL(1,𝔽7)—41no—
245AGL(1,𝔽7)—41no—
246AGL(1,𝔽7)—41no—
247AGL(1,𝔽7)—181no—
248C22×C32—71no—
249C22×C32—71no—
250C22×C32—71no—
251C22×C32—141no—
252C22×C32—141no—
253C22×C32—141no—
254C22×C32—281no—
255C22×C32—421no—
256C2×C3⋊D3—71no—
257C2×C3×D3—141no—
258C2×C3×D3—211no—
259C2×C3×D3—211no—
260C2×C3×D3—281no—
261C2×C3×D3—421no—
262C2×C3×D3C2×AGL(1,𝔽7)11yescharacteristic
263C32×C4—211no—
264C32×C4—211no—
265C32×C4—421no—
266C3⋊Dic3C2×AGL(1,𝔽7)11yescharacteristic
267C3×Dic3—71no—
268C3×Dic3—71no—
269C3×Dic3—71no—
270C3×Dic3—71no—
271C3×Dic3—141no—
272C3×Dic3—141no—
273C3×Dic3—141no—
274C3×Dic3—281no—
275C3×Dic3—421no—
276C3×Dic3—421no—
277C3×Dic3—841no—
278C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
279C22×C7—31no—
280C2×D7—91no—
281C2×D7C3×D3211yescharacteristic
282C4×C7—31no—
283C4×C7—91no—
284Dic7—31no—
285Dic7—31no—
286C33—141noSylow
287C3×Q8—631no—
288C3×D4—211no—
289C3×D4—211no—
290C3×D4—421no—
291C3×D4—631no—
292C3×D4—631no—
293C2×C3×C4—211no—
294C2×C3×C4—211no—
295C2×C3×C4—421no—
296C2×C3×C4—631no—
297C2×C3×C4—631no—
298#67—71no—
299#67—211no—
300#67—211no—
301#67—211no—
302#67—421no—
303C2×Dic3—31no—
304C2×Dic3—71no—
305D12—211no—
306C4×D3—31no—
307C4×D3—211no—
308C4×D3—211no—
309Dic6—211no—
310Dic6—211no—
311C3×C7—21no—
312C3×C7C3×#23511yescharacteristic, complemented
313C3×C7C3×#23711yescharacteristic, complemented
314ΓL(1,𝔽8)—21no—
315ΓL(1,𝔽8)—21no—
316ΓL(1,𝔽8)—41no—
317ΓL(1,𝔽8)#380411yescharacteristic, complemented
318C2×C32—141no—
319C2×C32—141no—
320C2×C32—141no—
321C2×C32—141no—
322C2×C32—141no—
323C2×C32—141no—
324C2×C32—141no—
325C2×C32—281no—
326C2×C32—281no—
327C2×C32—281no—
328C2×C32—281no—
329C2×C32—281no—
330C2×C32—281no—
331C2×C32—281no—
332C2×C32—421no—
333C2×C32—561no—
334C2×C32C22×AGL(1,𝔽7)11yescharacteristic
335C3⋊D3—141no—
336C3×D3—21no—
337C3×D3—141no—
338C3×D3—281no—
339C3×D3—421no—
340C3×D3—421no—
341C3×D3—841no—
342#41—631noSylow
343C2×C7—61no—
344C2×C7C2×C3×D3211yescharacteristic
345D7—21no—
346D7—181no—
347C22×C3—31no—
348C22×C3—71no—
349C22×C3—71no—
350C22×C3—71no—
351C22×C3—141no—
352C22×C3—141no—
353C22×C3—141no—
354C22×C3—211no—
355C22×C3—281no—
356C22×C3—421no—
357C22×C3—631no—
358C2×D3—211no—
359C2×D3—211no—
360C2×D3—421no—
361C2×D3D3×AGL(1,𝔽7)11yescharacteristic
362C3×C4—31no—
363C3×C4—211no—
364C3×C4—211no—
365C3×C4—211no—
366C3×C4—211no—
367C3×C4—211no—
368C3×C4—421no—
369C3×C4—421no—
370C3×C4—631no—
371C3×C4—841no—
372Dic3—31no—
373Dic3—31no—
374Dic3—61no—
375Dic3—71no—
376Dic3—71no—
377Dic3D3×AGL(1,𝔽7)11yescharacteristic
378C32—141no—
379C32—141no—
380C32—281no—
381C32—281no—
382C32—281no—
383C32—561no—
384C32#1157⋊C311yescharacteristic, complemented
385Q8—631no—
386D4—211no—
387D4—631no—
388D4—631no—
389C2×C4—91no—
390C2×C4—211no—
391C2×C4—631no—
392C7C3×#380411yescharacteristic, complemented, Sylow
393C2×C3—21no—
394C2×C3—61no—
395C2×C3—71no—
396C2×C3—141no—
397C2×C3—141no—
398C2×C3—141no—
399C2×C3—141no—
400C2×C3—141no—
401C2×C3—141no—
402C2×C3—281no—
403C2×C3—281no—
404C2×C3—281no—
405C2×C3—281no—
406C2×C3—281no—
407C2×C3—421no—
408C2×C3—561no—
409C2×C3—841no—
410C2×C3—1261no—
411C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
412C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
413D3—21no—
414D3—421no—
415D3—421no—
416D3—841no—
417C22—31no—
418C22—71no—
419C22—631no—
420C4—31no—
421C4—91no—
422C4—211no—
423C4—211no—
424C3—21no—
425C3—71no—
426C3—141no—
427C3—281no—
428C3—561no—
429C3#66675⋊C311yescharacteristic, complemented
430C3#66683⋊C311yescharacteristic, complemented
431C2—61no—
432C2—141no—
433C2—1261no—
434C2D32×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×AGL(1,𝔽7)
commutator, Fitting, socle, cosocleC2×C32×C7C23×C3
hypocenterC32×C7C3×#41
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
2. #92690⋊C3C2semidirect
3. #92687⋊C3C2semidirect
4. D3×Dic3×ΓL(1,𝔽8)C2semidirect
5. #13094715C2semidirect
6. C3×#4245⋊C3C2semidirect
7. C3×Dic3×AGL(1,𝔽7)C2semidirect
8. #11759438C3semidirect
18. #92696C2×C3semidirect
19. #92690C2×C3semidirect
20. #92687C2×C3semidirect
21. D3×C7×Dic3C2×C3semidirect
22. #92677C2×C3semidirect
23. C3×#4245C2×C3semidirect
24. C3×Dic3×D7C2×C3semidirect
38. #4245⋊C3D3semidirect
41. Dic3×AGL(1,𝔽7)D3semidirect
88. C32×ΓL(1,𝔽8)#41semidirect
102. #4245C3×D3semidirect
104. Dic3×D7C3×D3semidirect
202. D3×Dic3AGL(1,𝔽7)semidirect
203. C32×C7C3×#41semidirect
208. C3×ΓL(1,𝔽8)#235semidirect
209. C3×ΓL(1,𝔽8)#237semidirect
312. C3×C7C3×#235semidirect
313. C3×C7C3×#237semidirect
317. ΓL(1,𝔽8)#3804semidirect
384. C32#1157⋊C3semidirect
392. C7C3×#3804semidirect
429. C3#66675⋊C3semidirect
430. C3#66683⋊C3semidirect

Maximal subgroups

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

subgroupindex
#92696⋊C32
#92690⋊C32
#92687⋊C32
D3×Dic3×ΓL(1,𝔽8)2
#130947152
C3×#4245⋊C32
C3×Dic3×AGL(1,𝔽7)2
#117594383
#66683⋊C33
#66675⋊C33
C3×#38047