Finite Group Databasebeta

← #72247121

Subgroups — #72247121

Order 3024.

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

normal subgroups only

This group has 428 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
1C4×C3⋊(C3⋊AGL(1,𝔽7))C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C3×C4×D3×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#92691⋊C3C211yesmaximal, characteristic, complemented
5#92690⋊C3C211yesmaximal, characteristic, complemented
6#92689⋊C3C211yesmaximal, characteristic, complemented
7#92688⋊C3C211yesmaximal, characteristic, complemented
8#11759607C311yesmaximal, characteristic, complemented
9#66728⋊C3—31nomaximal
10#66675⋊C3—31nomaximal
11C2×C3⋊(C3⋊AGL(1,𝔽7))C2211yescharacteristic
12C2×C3×C3⋊AGL(1,𝔽7)C2211yescharacteristic
13C2×C3×D3×ΓL(1,𝔽8)C2211yescharacteristic
14C32×C4×ΓL(1,𝔽8)C2211yescharacteristic
15C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2211yescharacteristic
16C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
17C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
18C4×C3⋊D21C2×C311yescharacteristic, complemented
19C3×C4×D21C2×C311yescharacteristic, complemented
20C3×C4×D3×C7C2×C311yescharacteristic, complemented
21C2×C3×C4×ΓL(1,𝔽8)—31no—
22#92691C2×C311yescharacteristic, complemented
23#92690C2×C311yescharacteristic, complemented
24#92689C2×C311yescharacteristic, complemented
25#92688C2×C311yescharacteristic, complemented
26#4268⋊C3—31no—
27#4268⋊C3—31no—
28C4⋊(C3⋊AGL(1,𝔽7))—31no—
29C4×C3⋊AGL(1,𝔽7)—21no—
30C4×C3⋊AGL(1,𝔽7)—31no—
31C4×C3⋊AGL(1,𝔽7)—31no—
32C4×C3⋊AGL(1,𝔽7)—61no—
33C4×C3⋊AGL(1,𝔽7)D311yescharacteristic, complemented
34Dic42⋊C3—31no—
35C4×D3×ΓL(1,𝔽8)—21no—
36C4×D3×ΓL(1,𝔽8)D311yescharacteristic, complemented
37C3×C4×AGL(1,𝔽7)—31no—
38#4248⋊C3—31no—
39#4247⋊C3—31no—
40#4246⋊C3—31no—
41#4245⋊C3—31no—
42C3×#3798—71nomaximal, Hall
43C2×C32×ΓL(1,𝔽8)C2311yescharacteristic
44C3⋊(C3⋊AGL(1,𝔽7))—21no—
45C3×C3⋊AGL(1,𝔽7)—21no—
46C3×D3×ΓL(1,𝔽8)—21no—
47#66728—31no—
48#66675—31no—
49#1157⋊C3—91no—
50C2×C3⋊D21C22×C311yescharacteristic
51C2×C3×D21C22×C311yescharacteristic
52C2×C3×D3×C7C22×C311yescharacteristic
53C22×C3×ΓL(1,𝔽8)—31no—
54C2×C3⋊AGL(1,𝔽7)—21no—
55C2×C3⋊AGL(1,𝔽7)—31no—
56C2×C3⋊AGL(1,𝔽7)—31no—
57C2×C3⋊AGL(1,𝔽7)—61no—
58C2×C3⋊AGL(1,𝔽7)C2×D311yescharacteristic
59C2×D3×ΓL(1,𝔽8)—21no—
60C2×D3×ΓL(1,𝔽8)C2×D311yescharacteristic
61C2×C3×AGL(1,𝔽7)—31no—
62C32×C4×C7C22×C311yescharacteristic
63C3⋊Dic21C22×C311yescharacteristic
64C3×Dic21C22×C311yescharacteristic
65C3×C7×Dic3C22×C311yescharacteristic
66C3×C4×ΓL(1,𝔽8)—21no—
67C3×C4×ΓL(1,𝔽8)—21no—
68C3×C4×ΓL(1,𝔽8)—21no—
69C3×C4×ΓL(1,𝔽8)—31no—
70C3×C4×ΓL(1,𝔽8)—41no—
71C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
72C3×C4×ΓL(1,𝔽8)C2×D311yescharacteristic
73C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
74C3⋊(ΓL(1,𝔽8)⋊C4)—31no—
75C3⋊(ΓL(1,𝔽8)⋊C4)—31no—
76C3⋊(ΓL(1,𝔽8)⋊C4)—61no—
77C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
78Dic3×ΓL(1,𝔽8)—21no—
79Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
80C3×ΓL(1,𝔽8)⋊C4—31no—
81C3×C4×C3⋊D3—71no—
82C32×C4×D3—71no—
83C32×C4×D3—141no—
84C3×#627—71no—
85C3×#626—71no—
86C3×#626—71no—
87C3×#625—71no—
88C32×ΓL(1,𝔽8)#4111yescharacteristic, complemented, Hall
89C2×C3×C4×C7—31no—
90#4268—31no—
91#4268—31no—
92D84—31no—
93C4×D21—31no—
94C4×D21—31no—
95C4×D21—61no—
96C4×D21C3×D311yescharacteristic, complemented
97Dic42—31no—
98C4×D3×C7C3×D311yescharacteristic, complemented
99C3×C4×D7—31no—
100C2×C4×ΓL(1,𝔽8)—31no—
101C2×C4×ΓL(1,𝔽8)—61no—
102#4248—31no—
103#4247—31no—
104#4246—31no—
105#4245—31no—
106(C22⋊D7)⋊C3—91no—
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×#235—211no—
114C3×#235—211no—
115#3798—71no—
116C2×C32×C7C23×C311yescharacteristic
117C3⋊D21—21no—
118C3×D21—21no—
119C3×D3×C7—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)—21no—
128C3⋊AGL(1,𝔽7)—21no—
129C3⋊AGL(1,𝔽7)—21no—
130C3⋊AGL(1,𝔽7)—61no—
131C3⋊AGL(1,𝔽7)—61no—
132C3⋊AGL(1,𝔽7)—61no—
133C3⋊AGL(1,𝔽7)—61no—
134D3×ΓL(1,𝔽8)—21no—
135D3×ΓL(1,𝔽8)—21no—
136D3×ΓL(1,𝔽8)—21no—
137C3×AGL(1,𝔽7)—61no—
138#1157—91noHall
139C2×C3×C3⋊D3—71no—
140C2×C32×D3—71no—
141C2×C32×D3—141no—
142C33×C4—141no—
143C3×C3⋊Dic3—71no—
144C32×Dic3—71no—
145C32×Dic3—141no—
146C22×C3×C7—31no—
147C2×D21—31no—
148C2×D21—31no—
149C2×D21—61no—
150C2×D21C2×C3×D311yescharacteristic
151C2×D3×C7C2×C3×D311yescharacteristic
152C2×C3×D7—31no—
153C22×ΓL(1,𝔽8)—31no—
154C22×ΓL(1,𝔽8)—61no—
155C2×AGL(1,𝔽7)—31no—
156C2×AGL(1,𝔽7)—61no—
157C2×AGL(1,𝔽7)—91no—
158C3×C4×C7—21no—
159C3×C4×C7—31no—
160C3×C4×C7C2×C3×D311yescharacteristic
161C3×C4×C7C2×C3×D311yescharacteristic
162Dic21—31no—
163Dic21—31no—
164Dic21—61no—
165Dic21C2×C3×D311yescharacteristic
166C3×Dic7—31no—
167C7×Dic3C2×C3×D311yescharacteristic
168C4×ΓL(1,𝔽8)—21no—
169C4×ΓL(1,𝔽8)—21no—
170C4×ΓL(1,𝔽8)—31no—
171C4×ΓL(1,𝔽8)—41no—
172C4×ΓL(1,𝔽8)—61no—
173C4×ΓL(1,𝔽8)D3211yescharacteristic
174ΓL(1,𝔽8)⋊C4—31no—
175ΓL(1,𝔽8)⋊C4—61no—
176ΓL(1,𝔽8)⋊C4—91no—
177C2×C32×C4—211no—
178C2×C32×C4—421no—
179C4×C3⋊D3—71no—
180C3×#67—211no—
181C3×#67—211no—
182C3×#67—211no—
183C3×#67—211no—
184C3×D12—211no—
185C3×D12—211no—
186C3×C4×D3—71no—
187C3×C4×D3—71no—
188C3×C4×D3—141no—
189C3×C4×D3—141no—
190C3×C4×D3—141no—
191C3×C4×D3—211no—
192C3×C4×D3—211no—
193C3×C4×D3—421no—
194C3×C4×D3AGL(1,𝔽7)11yescharacteristic, complemented
195C3×Dic6—211no—
196C3×Dic6—211no—
197#627—71no—
198#626—71no—
199#626—71no—
200#625—71no—
201C32×C7C3×#4111yescharacteristic, complemented
202C3×ΓL(1,𝔽8)—21no—
203C3×ΓL(1,𝔽8)—21no—
204C3×ΓL(1,𝔽8)—21no—
205C3×ΓL(1,𝔽8)—41no—
206C3×ΓL(1,𝔽8)#23511yescharacteristic, complemented
207C3×ΓL(1,𝔽8)#23511yescharacteristic, complemented
208C2×C4×C7—31no—
209C22⋊D7—91no—
210C22⋊D7—91no—
211D28—91no—
212C4×D7—31no—
213C4×D7—91no—
214Dic14—91no—
215C2×C33—141no—
216C3×C3⋊D3—141no—
217C32×D3—141no—
218C32×D3—141no—
219C3×#41—631no—
220#235—211no—
221#235—211no—
222C2×C3×C7—21no—
223C2×C3×C7—61no—
224C2×C3×C7C22×C3×D311yescharacteristic
225C2×C3×C7C22×C3×D311yescharacteristic
226D21—21no—
227D21—61no—
228D21—61no—
229D21—61no—
230D21—61no—
231C3×D7—61no—
232D3×C7—21no—
233C2×ΓL(1,𝔽8)—21no—
234C2×ΓL(1,𝔽8)—21no—
235C2×ΓL(1,𝔽8)—41no—
236C2×ΓL(1,𝔽8)—61no—
237C2×ΓL(1,𝔽8)—61no—
238C2×ΓL(1,𝔽8)—61no—
239C2×ΓL(1,𝔽8)C2×D3211yescharacteristic
240AGL(1,𝔽7)—61no—
241AGL(1,𝔽7)—61no—
242AGL(1,𝔽7)—61no—
243AGL(1,𝔽7)—181no—
244C22×C32—211no—
245C22×C32—421no—
246C2×C3⋊D3—71no—
247C2×C3×D3—71no—
248C2×C3×D3—71no—
249C2×C3×D3—141no—
250C2×C3×D3—141no—
251C2×C3×D3—141no—
252C2×C3×D3—211no—
253C2×C3×D3—211no—
254C2×C3×D3—421no—
255C2×C3×D3C2×AGL(1,𝔽7)11yescharacteristic
256C32×C4—141no—
257C32×C4—141no—
258C32×C4—141no—
259C32×C4—141no—
260C32×C4—211no—
261C32×C4—281no—
262C32×C4—281no—
263C32×C4—421no—
264C32×C4C2×AGL(1,𝔽7)11yescharacteristic
265C3⋊Dic3—71no—
266C3×Dic3—71no—
267C3×Dic3—71no—
268C3×Dic3—141no—
269C3×Dic3—141no—
270C3×Dic3—141no—
271C3×Dic3—211no—
272C3×Dic3—211no—
273C3×Dic3—421no—
274C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
275C22×C7—31no—
276C2×D7—31no—
277C2×D7—91no—
278C4×C7—31no—
279C4×C7C3×D3211yescharacteristic
280Dic7—31no—
281Dic7—91no—
282C33—141noSylow
283C3×Q8—631no—
284C3×D4—631no—
285C3×D4—631no—
286C3×D4—631no—
287C2×C3×C4—31no—
288C2×C3×C4—211no—
289C2×C3×C4—211no—
290C2×C3×C4—211no—
291C2×C3×C4—421no—
292C2×C3×C4—421no—
293C2×C3×C4—631no—
294#67—211no—
295#67—211no—
296#67—211no—
297#67—211no—
298D12—211no—
299D12—211no—
300C4×D3—71no—
301C4×D3—211no—
302C4×D3—211no—
303C4×D3—421no—
304C4×D3C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
305Dic6—211no—
306Dic6—211no—
307C3×C7—21no—
308C3×C7C3×#23511yescharacteristic, complemented
309C3×C7C3×#23511yescharacteristic, complemented
310ΓL(1,𝔽8)—21no—
311ΓL(1,𝔽8)—21no—
312ΓL(1,𝔽8)—41no—
313ΓL(1,𝔽8)#379811yescharacteristic, complemented
314C2×C32—141no—
315C2×C32—141no—
316C2×C32—141no—
317C2×C32—141no—
318C2×C32—281no—
319C2×C32—281no—
320C2×C32—421no—
321C2×C32—421no—
322C2×C32C22×AGL(1,𝔽7)11yescharacteristic
323C3⋊D3—141no—
324C3×D3—21no—
325C3×D3—141no—
326C3×D3—141no—
327C3×D3—141no—
328C3×D3—141no—
329C3×D3—141no—
330C3×D3—141no—
331C3×D3—141no—
332C3×D3—421no—
333C3×D3—421no—
334C3×D3—421no—
335C3×D3—421no—
336#41—631noSylow
337C2×C7—61no—
338C2×C7C2×C3×D3211yescharacteristic
339D7—61no—
340D7—181no—
341C22×C3—31no—
342C22×C3—211no—
343C22×C3—211no—
344C22×C3—211no—
345C22×C3—421no—
346C22×C3—421no—
347C22×C3—631no—
348C2×D3—71no—
349C2×D3—211no—
350C2×D3—211no—
351C2×D3—421no—
352C2×D3C2×C3⋊AGL(1,𝔽7)11yescharacteristic
353C3×C4—21no—
354C3×C4—31no—
355C3×C4—71no—
356C3×C4—141no—
357C3×C4—211no—
358C3×C4—211no—
359C3×C4—211no—
360C3×C4—281no—
361C3×C4—281no—
362C3×C4—421no—
363C3×C4—421no—
364C3×C4—631no—
365C3×C4D3×AGL(1,𝔽7)11yescharacteristic
366C3×C4C2×C3⋊AGL(1,𝔽7)11yescharacteristic
367Dic3—71no—
368Dic3—211no—
369Dic3—211no—
370Dic3—421no—
371Dic3C2×C3⋊AGL(1,𝔽7)11yescharacteristic
372C32—141no—
373C32—141no—
374C32—141no—
375C32—141no—
376C32—281no—
377C32—281no—
378C32#1157⋊C311yescharacteristic, complemented
379Q8—631no—
380D4—631no—
381D4—631no—
382D4—631no—
383C2×C4—31no—
384C2×C4—211no—
385C2×C4—631no—
386C7C3×#379811yescharacteristic, complemented, Sylow
387C2×C3—21no—
388C2×C3—61no—
389C2×C3—71no—
390C2×C3—141no—
391C2×C3—281no—
392C2×C3—281no—
393C2×C3—421no—
394C2×C3—421no—
395C2×C3—421no—
396C2×C3—421no—
397C2×C3—421no—
398C2×C3—421no—
399C2×C3—421no—
400C2×C3—1261no—
401C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
402C2×C3C22×C3⋊AGL(1,𝔽7)11yescharacteristic
403D3—21no—
404D3—141no—
405D3—421no—
406D3—421no—
407D3—421no—
408D3—421no—
409C22—31no—
410C22—211no—
411C22—631no—
412C4—31no—
413C4—211no—
414C4—631no—
415C4D3×C3⋊AGL(1,𝔽7)11yescharacteristic
416C3—21no—
417C3—71no—
418C3—141no—
419C3—281no—
420C3—281no—
421C3#66675⋊C311yescharacteristic, complemented
422C3#66728⋊C311yescharacteristic, complemented
423C2—61no—
424C2—421no—
425C2—1261no—
426C2C2×D3×C3⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, hypercenterC4D3×C3⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C32×C7C23×C3
FrattiniC2C2×D3×C3⋊AGL(1,𝔽7)
FittingC32×C4×C7C22×C3
hypocenterC32×C7C3×#41
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊(C3⋊AGL(1,𝔽7))C2semidirect
2. C3×C4×C3⋊AGL(1,𝔽7)C2semidirect
3. C3×C4×D3×ΓL(1,𝔽8)C2semidirect
4. #92691⋊C3C2semidirect
5. #92690⋊C3C2semidirect
6. #92689⋊C3C2semidirect
7. #92688⋊C3C2semidirect
8. #11759607C3semidirect
18. C4×C3⋊D21C2×C3semidirect
19. C3×C4×D21C2×C3semidirect
20. C3×C4×D3×C7C2×C3semidirect
22. #92691C2×C3semidirect
23. #92690C2×C3semidirect
24. #92689C2×C3semidirect
25. #92688C2×C3semidirect
33. C4×C3⋊AGL(1,𝔽7)D3semidirect
36. C4×D3×ΓL(1,𝔽8)D3semidirect
88. C32×ΓL(1,𝔽8)#41semidirect
96. C4×D21C3×D3semidirect
98. C4×D3×C7C3×D3semidirect
194. C3×C4×D3AGL(1,𝔽7)semidirect
201. C32×C7C3×#41semidirect
206. C3×ΓL(1,𝔽8)#235semidirect
207. C3×ΓL(1,𝔽8)#235semidirect
304. C4×D3C3⋊AGL(1,𝔽7)semidirect
308. C3×C7C3×#235semidirect
309. C3×C7C3×#235semidirect
313. ΓL(1,𝔽8)#3798semidirect
378. C32#1157⋊C3semidirect
386. C7C3×#3798semidirect
421. C3#66675⋊C3semidirect
422. C3#66728⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊AGL(1,𝔽7))2
C3×C4×C3⋊AGL(1,𝔽7)2
C3×C4×D3×ΓL(1,𝔽8)2
#92691⋊C32
#92690⋊C32
#92689⋊C32
#92688⋊C32
#117596073
#66728⋊C33
#66675⋊C33
C3×#37987