Finite Group Databasebeta

← #72277276

Subgroups — #72277276

Order 2688.

692 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6471⋊C3C211yesmaximal, characteristic
2#90212⋊C3C211yesmaximal, characteristic, complemented
3#89777⋊C3C211yesmaximal, characteristic, complemented
4#89778⋊C3C211yesmaximal, characteristic, complemented
5#89773⋊C3C211yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
7#89351⋊C3C211yesmaximal, characteristic, complemented
8#11726458C311yesmaximal, characteristic, complemented, Hall
9C2×#1157⋊C3C2211yescharacteristic
10C22×C4⋊AGL(1,𝔽7)C2211yescharacteristic
11C2×M16×ΓL(1,𝔽8)C2211yescharacteristic
12ΓL(1,𝔽8)×#117C2211yescharacteristic
13ΓL(1,𝔽8)×#117C2211yescharacteristic
14#6489⋊C3C2211yescharacteristic
15C2×#1136⋊C3C2211yescharacteristic
16#6471⋊C3—22no—
17#6471⋊C3—22no—
18C2×D56⋊C3C412yescomplemented
19C2×#1132⋊C3C412yescomplemented
20#6456⋊C3—22no—
21C22⋊C4×AGL(1,𝔽7)—22no—
22C4×C4⋊AGL(1,𝔽7)—22no—
23#6383⋊C3—22no—
24#6380⋊C3—22no—
25#6379⋊C3—22no—
26C2×#6471C2×C311yescharacteristic
27#90212C2×C311yescharacteristic, complemented
28#89778C2×C311yescharacteristic, complemented
29#89777C2×C311yescharacteristic, complemented
30#89773C2×C311yescharacteristic, complemented
31C7×H(3,ℤ/4)C2×C311yescharacteristic, complemented
32#89351C2×C311yescharacteristic, complemented
33C3×#1904—71nomaximal, Hall
34C23×AGL(1,𝔽7)—21no—
35C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
36C2×(C22⋊D7)⋊C3—21no—
37#1157⋊C3—21no—
38#1157⋊C3—21no—
39#1157⋊C3—21no—
40#1157⋊C3—21no—
41C2×C4⋊AGL(1,𝔽7)—21no—
42C2×C4⋊AGL(1,𝔽7)—21no—
43C2×C4⋊AGL(1,𝔽7)—21no—
44C2×C4⋊AGL(1,𝔽7)C2×C412yes—
45C2×C4⋊AGL(1,𝔽7)C2×C411yescharacteristic, complemented
46C2×C4⋊AGL(1,𝔽7)D412yes—
47C2×C4⋊AGL(1,𝔽7)D412yes—
48C2×C4×AGL(1,𝔽7)—21no—
49C2×C4×AGL(1,𝔽7)—24no—
50C2×ΓL(1,𝔽8)⋊Q8C2×C411yescharacteristic, complemented
51M16×ΓL(1,𝔽8)—21no—
52M16×ΓL(1,𝔽8)—21no—
53C2×C8×ΓL(1,𝔽8)C2×C412yescomplemented
54C4⋊C4×ΓL(1,𝔽8)—21no—
55C4⋊C4×ΓL(1,𝔽8)—21no—
56C22⋊C4×ΓL(1,𝔽8)—21no—
57C22⋊C4×ΓL(1,𝔽8)—21no—
58C42×ΓL(1,𝔽8)—21no—
59C42×ΓL(1,𝔽8)D412yes—
60(C22⋊Dic7)⋊C3D412yes—
61(C2×AGL(1,𝔽7))⋊C4—24no—
62C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
63C4×ΓL(1,𝔽8)⋊C4—21no—
64#1136⋊C3—21no—
65#1136⋊C3—21no—
66C2×ΓL(1,𝔽8)⋊C8—21no—
67D56⋊C3—24no—
68#1132⋊C3—24no—
69C2×#1157C22×C311yescharacteristic
70C22×D28C22×C311yescharacteristic
71C2×C7×M16C22×C311yescharacteristic
72C7×#117C22×C311yescharacteristic
73C7×#117C22×C311yescharacteristic
74#6489C22×C311yescharacteristic
75C2×#1136C22×C311yescharacteristic
76#6471—22no—
77#6471—22no—
78C2×D56C3×C412yescomplemented
79C2×#1132C3×C412yescomplemented
80#6456—22no—
81D7×C22⋊C4—22no—
82C4×D28—22no—
83#6383—22no—
84#6380—22no—
85#6379—22no—
86C2×C3×AGL(1,ℤ/8)—71no—
87C3×#518—71no—
88C3×#421—71no—
89C3×#418—71no—
90C3×#410—71no—
91C3×#344—71no—
92C3×H(3,ℤ/4)—71no—
93C23×ΓL(1,𝔽8)C2×D411yescharacteristic
94C22×AGL(1,𝔽7)—22no—
95C22×AGL(1,𝔽7)—22no—
96C22×AGL(1,𝔽7)—22no—
97C22×AGL(1,𝔽7)—21no—
98C22×AGL(1,𝔽7)—82no—
99C2×C4×ΓL(1,𝔽8)—22no—
100C2×C4×ΓL(1,𝔽8)—41no—
101C2×C4×ΓL(1,𝔽8)C22×C411yescharacteristic
102C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
103C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
104C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
105C2×C4×ΓL(1,𝔽8)#4111yescharacteristic
106C2×C4×ΓL(1,𝔽8)#4111yescharacteristic
107(C22⋊D7)⋊C3—41no—
108(C22⋊D7)⋊C3—41no—
109C2×ΓL(1,𝔽8)⋊C4—21no—
110C2×ΓL(1,𝔽8)⋊C4—22no—
111C4⋊AGL(1,𝔽7)—21no—
112C4⋊AGL(1,𝔽7)—22no—
113C4⋊AGL(1,𝔽7)—21no—
114C4⋊AGL(1,𝔽7)—22no—
115C4⋊AGL(1,𝔽7)—22no—
116C4⋊AGL(1,𝔽7)C22⋊C414yes—
117C4×AGL(1,𝔽7)—41no—
118C4×AGL(1,𝔽7)—41no—
119C4×AGL(1,𝔽7)—84no—
120ΓL(1,𝔽8)⋊Q8—21no—
121ΓL(1,𝔽8)⋊Q8—21no—
122C8×ΓL(1,𝔽8)—22no—
123ΓL(1,𝔽8)⋊C8—41no—
124#1904—71noSylow
125C23×D7—21no—
126C22×C4×C7C23×C311yescharacteristic
127C2×C22⋊D7—21no—
128#1157—21no—
129#1157—21no—
130#1157—21no—
131#1157—21no—
132C2×D28—21no—
133C2×D28—21no—
134C2×D28—21no—
135C2×D28C2×C3×C412yes—
136C2×D28C2×C3×C411yescharacteristic, complemented
137C2×D28C3×D412yes—
138C2×D28C3×D412yes—
139C2×C4×D7—21no—
140C2×C4×D7—24no—
141C2×Dic14C2×C3×C411yescharacteristic, complemented
142C7×M16—21no—
143C7×M16—21no—
144C2×C7×C8C2×C3×C412yescomplemented
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C22⋊C4—21no—
148C7×C22⋊C4—21no—
149C42×C7—21no—
150C42×C7C3×D412yes—
151C22⋊Dic7C3×D412yes—
152(C2×D7)⋊C4—24no—
153C4⋊Dic7—21no—
154C4×Dic7—21no—
155#1136—21no—
156#1136—21no—
157C2×C7⋊C8—21no—
158D56—24no—
159#1132—24no—
160C2×C3×#41—71no—
161C22×C3×D4—71no—
162C3×AGL(1,ℤ/8)—142no—
163C3×AGL(1,ℤ/8)—142no—
164C2×C3×SD16—72no—
165C2×C3×D8—72no—
166C2×C3×M16—71no—
167C2×C3×M16—71no—
168C3×C4×D4—142no—
169C3×C4×D4—142no—
170C3×#117—71no—
171C3×#117—71no—
172C3×#117—71no—
173C2×C3×C22⋊C4—142no—
174C3×#104—142no—
175C3×#102—142no—
176C3×#99—142no—
177C22×ΓL(1,𝔽8)—41no—
178C22×ΓL(1,𝔽8)C4×D411yescharacteristic
179C22×ΓL(1,𝔽8)#12011yescharacteristic
180C22×ΓL(1,𝔽8)#12111yescharacteristic
181C2×AGL(1,𝔽7)—24no—
182C2×AGL(1,𝔽7)—44no—
183C2×AGL(1,𝔽7)—41no—
184C2×AGL(1,𝔽7)—82no—
185C2×AGL(1,𝔽7)—82no—
186C2×AGL(1,𝔽7)—81no—
187C4×ΓL(1,𝔽8)—42no—
188C4×ΓL(1,𝔽8)—81no—
189C4×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
190C4×ΓL(1,𝔽8)C4×D411yescharacteristic
191C4×ΓL(1,𝔽8)#12111yescharacteristic
192C4×ΓL(1,𝔽8)#12311yescharacteristic
193ΓL(1,𝔽8)⋊C4—42no—
194ΓL(1,𝔽8)⋊C4—41no—
195C2×AGL(1,ℤ/8)—71no—
196#518—71no—
197#421—71no—
198#418—71no—
199#410—71no—
200#344—71no—
201H(3,ℤ/4)AGL(1,𝔽7)11yescharacteristic, complemented
202C23×C7C2×C3×D411yescharacteristic
203C22×D7—22no—
204C22×D7—22no—
205C22×D7—22no—
206C22×D7—21no—
207C22×D7—82no—
208C2×C4×C7—22no—
209C2×C4×C7—41no—
210C2×C4×C7C22×C3×C411yescharacteristic
211C2×C4×C7C2×C3×D411yescharacteristic
212C2×C4×C7C2×C3×D411yescharacteristic
213C2×C4×C7C2×C3×D411yescharacteristic
214C2×C4×C7C3×#4111yescharacteristic
215C2×C4×C7C3×#4111yescharacteristic
216C22⋊D7—41no—
217C22⋊D7—41no—
218C2×Dic7—21no—
219C2×Dic7—22no—
220D28—21no—
221D28—22no—
222D28—21no—
223D28—22no—
224D28—22no—
225D28C3×C22⋊C414yes—
226C4×D7—41no—
227C4×D7—41no—
228C4×D7—84no—
229Dic14—21no—
230Dic14—21no—
231C7×C8—22no—
232C7⋊C8—41no—
233C24×C3—141no—
234C3×#41—141no—
235C3×#41—141no—
236C3×#41—141no—
237C3×#41—141no—
238C2×C3×Q8—71no—
239C2×C3×D4—72no—
240C2×C3×D4—72no—
241C2×C3×D4—72no—
242C2×C3×D4—71no—
243C2×C3×D4—141no—
244C2×C3×D4—141no—
245C2×C3×D4—141no—
246C2×C3×D4—141no—
247C22×C3×C4—71no—
248C22×C3×C4—141no—
249C22×C3×C4—144no—
250C3×SD16—144no—
251C3×D8—144no—
252C3×M16—141no—
253C3×M16—141no—
254C3×M16—141no—
255C3×M16—141no—
256C2×C3×C8—72no—
257C2×C3×C8—141no—
258C3×C4⋊C4—141no—
259C3×C4⋊C4—141no—
260C3×C4⋊C4—141no—
261C3×C22⋊C4—72no—
262C3×C22⋊C4—141no—
263C3×C22⋊C4—141no—
264C3×C22⋊C4—144no—
265C3×C42—72no—
266C3×C42—141no—
267C3×C42—141no—
268C2×ΓL(1,𝔽8)—21no—
269C2×ΓL(1,𝔽8)—21no—
270C2×ΓL(1,𝔽8)—21no—
271C2×ΓL(1,𝔽8)#38611yescharacteristic
272AGL(1,𝔽7)—44no—
273AGL(1,𝔽7)—81no—
274C2×#41—71no—
275C22×D4—71no—
276AGL(1,ℤ/8)—142no—
277AGL(1,ℤ/8)—142no—
278C2×SD16—72no—
279C2×D8—72no—
280C2×M16—71no—
281C2×M16C2×AGL(1,𝔽7)11yescharacteristic
282C4×D4—142no—
283C4×D4—142no—
284#117—71no—
285#117C2×AGL(1,𝔽7)11yescharacteristic
286#117C2×AGL(1,𝔽7)11yescharacteristic
287C2×C22⋊C4—142no—
288#104—142no—
289#102—142no—
290#99—142no—
291C22×C7—41no—
292C22×C7C3×C4×D411yescharacteristic
293C22×C7C3×#12011yescharacteristic
294C22×C7C3×#12111yescharacteristic
295C2×D7—24no—
296C2×D7—44no—
297C2×D7—41no—
298C2×D7—82no—
299C2×D7—82no—
300C2×D7—81no—
301C4×C7—42no—
302C4×C7—81no—
303C4×C7C2×C3×C22⋊C411yescharacteristic
304C4×C7C3×C4×D411yescharacteristic
305C4×C7C3×#12111yescharacteristic
306C4×C7C3×#12311yescharacteristic
307Dic7—42no—
308Dic7—41no—
309C23×C3—71no—
310C23×C3—142no—
311C23×C3—142no—
312C23×C3—142no—
313C23×C3—141no—
314C23×C3—562no—
315C3×Q8—141no—
316C3×Q8—141no—
317C3×D4—74no—
318C3×D4—142no—
319C3×D4—142no—
320C3×D4—142no—
321C3×D4—141no—
322C3×D4—141no—
323C3×D4—281no—
324C3×D4—281no—
325C2×C3×C4—71no—
326C2×C3×C4—71no—
327C2×C3×C4—71no—
328C2×C3×C4—71no—
329C2×C3×C4—71no—
330C2×C3×C4—71no—
331C2×C3×C4—142no—
332C2×C3×C4—142no—
333C2×C3×C4—141no—
334C2×C3×C4—281no—
335C2×C3×C4—281no—
336C2×C3×C4—281no—
337C2×C3×C4—564no—
338C3×C8—142no—
339C3×C8—281no—
340ΓL(1,𝔽8)#190411yescharacteristic, complemented, Hall
341C24—141no—
342#41—141no—
343#41—141no—
344#41—141no—
345#41—141no—
346C2×Q8—71no—
347C2×D4—72no—
348C2×D4—71no—
349C2×D4—72no—
350C2×D4—72no—
351C2×D4—141no—
352C2×D4—141no—
353C2×D4—141no—
354C2×D4—141no—
355C22×C4—144no—
356C22×C4—141no—
357C22×C4C22×AGL(1,𝔽7)11yescharacteristic
358SD16—144no—
359D8—144no—
360M16—21no—
361M16—21no—
362M16—141no—
363M16—141no—
364C2×C8—141no—
365C2×C8C4×AGL(1,𝔽7)12yescomplemented
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—141no—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—72no—
372C22⋊C4—144no—
373C42—21no—
374C42—141no—
375C42C4⋊AGL(1,𝔽7)12yes—
376C2×C7—21no—
377C2×C7—21no—
378C2×C7—21no—
379C2×C7C3×#38611yescharacteristic
380D7—44no—
381D7—81no—
382C22×C3—71no—
383C22×C3—71no—
384C22×C3—71no—
385C22×C3—144no—
386C22×C3—281no—
387C22×C3—281no—
388C22×C3—284no—
389C22×C3—562no—
390C22×C3—562no—
391C22×C3—561no—
392C3×C4—71no—
393C3×C4—71no—
394C3×C4—71no—
395C3×C4—71no—
396C3×C4—282no—
397C3×C4—281no—
398C3×C4—282no—
399C3×C4—561no—
400C23—142no—
401C23—142no—
402C23—142no—
403C23—141no—
404C23—562no—
405C23D4×AGL(1,𝔽7)11yescharacteristic
406Q8—141no—
407Q8—141no—
408D4—74no—
409D4—142no—
410D4—141no—
411D4—141no—
412D4—142no—
413D4—142no—
414D4—281no—
415D4—281no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—142no—
419C2×C4—141no—
420C2×C4—281no—
421C2×C4—281no—
422C2×C4—564no—
423C2×C4C2×C4×AGL(1,𝔽7)11yescharacteristic
424C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
425C2×C4#1157⋊C311yescharacteristic
426C2×C4D4×AGL(1,𝔽7)11yescharacteristic
427C2×C4D4×AGL(1,𝔽7)11yescharacteristic
428C2×C4#6665911yescharacteristic
429C8—22no—
430C8—281no—
431C7C3×#190411yescharacteristic, complemented, Sylow
432C2×C3—71no—
433C2×C3—141no—
434C2×C3—141no—
435C2×C3—141no—
436C2×C3—284no—
437C2×C3—561no—
438C22—41no—
439C22—144no—
440C22—284no—
441C22—281no—
442C22—562no—
443C22—562no—
444C22—561no—
445C22#6445⋊C311yescharacteristic
446C22#6447⋊C311yescharacteristic
447C22#6456⋊C311yescharacteristic
448C4—42no—
449C4—81no—
450C4—282no—
451C4—281no—
452C4C4×C4⋊AGL(1,𝔽7)11yescharacteristic
453C4C22⋊C4×AGL(1,𝔽7)11yescharacteristic
454C4#6457⋊C311yescharacteristic
455C4#6458⋊C311yescharacteristic
456C3—71noSylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—284no—
461C2—561no—
462C2#89437⋊C311yescharacteristic
subgroupG/H
centerC2#89437⋊C3
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C22×AGL(1,𝔽7)
FittingC7×H(3,ℤ/4)C2×C3
socleC2×C7C3×#386
prime radicalC2×#6471⋊C3C2
cosocleC22×C4×C7C23×C3
hypercenterH(3,ℤ/4)AGL(1,𝔽7)
hypocenterC7C3×#1904
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #90212⋊C3C2semidirect
3. #89777⋊C3C2semidirect
4. #89778⋊C3C2semidirect
5. #89773⋊C3C2semidirect
6. ΓL(1,𝔽8)×H(3,ℤ/4)C2semidirect
7. #89351⋊C3C2semidirect
8. #11726458C3semidirect
18. C2×D56⋊C3C4semidirect
19. C2×#1132⋊C3C4semidirect
27. #90212C2×C3semidirect
28. #89778C2×C3semidirect
29. #89777C2×C3semidirect
30. #89773C2×C3semidirect
31. C7×H(3,ℤ/4)C2×C3semidirect
32. #89351C2×C3semidirect
45. C2×C4⋊AGL(1,𝔽7)C2×C4semidirect
50. C2×ΓL(1,𝔽8)⋊Q8C2×C4semidirect
53. C2×C8×ΓL(1,𝔽8)C2×C4semidirect
78. C2×D56C3×C4semidirect
79. C2×#1132C3×C4semidirect
136. C2×D28C2×C3×C4semidirect
141. C2×Dic14C2×C3×C4semidirect
144. C2×C7×C8C2×C3×C4semidirect
201. H(3,ℤ/4)AGL(1,𝔽7)semidirect
340. ΓL(1,𝔽8)#1904semidirect
365. C2×C8C4×AGL(1,𝔽7)semidirect
431. C7C3×#1904semidirect

Maximal subgroups

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

subgroupindex
C2×#6471⋊C32
#90212⋊C32
#89777⋊C32
#89778⋊C32
#89773⋊C32
ΓL(1,𝔽8)×H(3,ℤ/4)2
#89351⋊C32
#117264583
C3×#19047