Finite Group Databasebeta

← #72276941

Subgroups — #72276941

Order 2688.

1176 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10621518C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#410C211yesmaximal, characteristic, complemented
3#90009⋊C3C211yesmaximal, characteristic, complemented
4#89777⋊C3C212yesmaximal, complemented
5#89773⋊C3C212yesmaximal, complemented
6#11727802C311yesmaximal, characteristic, complemented, Hall
7C2×#41×ΓL(1,𝔽8)C2211yescharacteristic
8#10621518—24no—
9#10621518—24no—
10C2×#66659—21no—
11C2×D4×AGL(1,𝔽7)—22no—
12C2×D4×AGL(1,𝔽7)—21no—
13C2×#1157⋊C3C412yescomplemented
14C2×#1157⋊C3C2211yescharacteristic
15C22×C4⋊AGL(1,𝔽7)C412yescomplemented
16C22×C4⋊AGL(1,𝔽7)C2211yescharacteristic
17ΓL(1,𝔽8)×#117C2212yescomplemented
18C2×#1136⋊C3C2212yescomplemented
19ΓL(1,𝔽8)×#99—22no—
20#6410⋊C3—21no—
21#6407⋊C3—21no—
22#6383⋊C3—24no—
23#6379⋊C3—24no—
24C2×#6553C2×C311yescharacteristic
25C7×#410C2×C311yescharacteristic, complemented
26#90009C2×C311yescharacteristic, complemented
27#89777C2×C312yescomplemented
28#89773C2×C312yescomplemented
29C3×#1795—71nomaximal, Hall
30C23×AGL(1,𝔽7)—22no—
31C23×AGL(1,𝔽7)—21no—
32#41×ΓL(1,𝔽8)—22no—
33#41×ΓL(1,𝔽8)—22no—
34C2×Q8×ΓL(1,𝔽8)D411yescharacteristic
35C2×D4×ΓL(1,𝔽8)D412yes—
36C2×D4×ΓL(1,𝔽8)D411yescharacteristic
37C22×C4×ΓL(1,𝔽8)C2×C412yes—
38C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
39C2×(C22⋊D7)⋊C3—22no—
40C2×(C22⋊D7)⋊C3—21no—
41#66659—28no—
42D4×AGL(1,𝔽7)—216no—
43D4×AGL(1,𝔽7)—28no—
44#1157⋊C3—22no—
45#1157⋊C3—24no—
46#1157⋊C3—24no—
47#1157⋊C3D414yes—
48C2×C4⋊AGL(1,𝔽7)—22no—
49C2×C4⋊AGL(1,𝔽7)—21no—
50C2×C4⋊AGL(1,𝔽7)—22no—
51C2×C4⋊AGL(1,𝔽7)—24no—
52C2×C4⋊AGL(1,𝔽7)—24no—
53C2×C4⋊AGL(1,𝔽7)C2×C411yescharacteristic
54C2×C4⋊AGL(1,𝔽7)C2×C412yes—
55C2×C4⋊AGL(1,𝔽7)D414yes—
56C2×C4×AGL(1,𝔽7)—22no—
57C2×C4×AGL(1,𝔽7)—21no—
58C2×ΓL(1,𝔽8)⋊Q8C2×C411yescharacteristic
59C4⋊C4×ΓL(1,𝔽8)—22no—
60C22⋊C4×ΓL(1,𝔽8)—22no—
61C42×ΓL(1,𝔽8)—22no—
62#1136⋊C3—22no—
63#1136⋊C3—22no—
64C2×ΓL(1,𝔽8)⋊C8—22no—
65C2×C7×#41C22×C311yescharacteristic
66#6553—24no—
67#6553—24no—
68C2×#1161—21no—
69C2×D4×D7—22no—
70C2×D4×D7—21no—
71C2×#1157C3×C412yescomplemented
72C2×#1157C22×C311yescharacteristic
73C22×D28C3×C412yescomplemented
74C22×D28C22×C311yescharacteristic
75C7×#117C22×C312yescomplemented
76C2×#1136C22×C312yescomplemented
77C7×#99—22no—
78#6410—21no—
79#6407—21no—
80#6383—24no—
81#6379—24no—
82C2×C3×H(5,𝔽2)—71no—
83C3×#421—72no—
84C3×#418—72no—
85C3×#413—71no—
86C3×#410—71no—
87C23×ΓL(1,𝔽8)C22⋊C412yes—
88C23×ΓL(1,𝔽8)C2×D411yescharacteristic
89C22×AGL(1,𝔽7)—22no—
90C22×AGL(1,𝔽7)—21no—
91C22×AGL(1,𝔽7)—28no—
92C22×AGL(1,𝔽7)—24no—
93C22×AGL(1,𝔽7)—82no—
94C22×AGL(1,𝔽7)—84no—
95Q8×ΓL(1,𝔽8)—22no—
96D4×ΓL(1,𝔽8)—24no—
97D4×ΓL(1,𝔽8)—22no—
98C2×C4×ΓL(1,𝔽8)—22no—
99C2×C4×ΓL(1,𝔽8)—22no—
100C2×C4×ΓL(1,𝔽8)—42no—
101C2×C4×ΓL(1,𝔽8)C22⋊C412yes—
102C2×C4×ΓL(1,𝔽8)C22×C411yescharacteristic
103C2×C4×ΓL(1,𝔽8)C2×D412yes—
104C2×C4×ΓL(1,𝔽8)C2×D412yes—
105C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
106(C22⋊D7)⋊C3—28no—
107(C22⋊D7)⋊C3—24no—
108C2×ΓL(1,𝔽8)⋊C4—21no—
109C4⋊AGL(1,𝔽7)—28no—
110C4⋊AGL(1,𝔽7)—21no—
111C4⋊AGL(1,𝔽7)—22no—
112C4⋊AGL(1,𝔽7)—24no—
113C4⋊AGL(1,𝔽7)C22⋊C412yes—
114C4⋊AGL(1,𝔽7)C22⋊C414yes—
115C4×AGL(1,𝔽7)—28no—
116C4×AGL(1,𝔽7)—24no—
117ΓL(1,𝔽8)⋊Q8—21no—
118ΓL(1,𝔽8)⋊Q8C22⋊C412yes—
119ΓL(1,𝔽8)⋊C8—42no—
120#1795—71noSylow
121C23×D7—22no—
122C23×D7—21no—
123C7×#41—22no—
124C7×#41—22no—
125C2×C7×Q8C3×D411yescharacteristic
126C2×C7×D4C3×D412yes—
127C2×C7×D4C3×D411yescharacteristic
128C22×C4×C7C2×C3×C412yes—
129C22×C4×C7C23×C311yescharacteristic
130C2×C22⋊D7—22no—
131C2×C22⋊D7—21no—
132#1161—28no—
133D4×D7—216no—
134D4×D7—28no—
135#1157—22no—
136#1157—24no—
137#1157—24no—
138#1157C3×D414yes—
139C2×D28—22no—
140C2×D28—21no—
141C2×D28—22no—
142C2×D28—24no—
143C2×D28—24no—
144C2×D28C2×C3×C411yescharacteristic
145C2×D28C2×C3×C412yes—
146C2×D28C3×D414yes—
147C2×C4×D7—22no—
148C2×C4×D7—21no—
149C2×Dic14C2×C3×C411yescharacteristic
150C7×C4⋊C4—22no—
151C7×C22⋊C4—22no—
152C42×C7—22no—
153#1136—22no—
154#1136—22no—
155C2×C7⋊C8—22no—
156C3×H(5,𝔽2)—144no—
157C3×H(5,𝔽2)—144no—
158C2×C3×#41—71no—
159C2×C3×#41—72no—
160C2×C3×#41—71no—
161C2×C3×#41—141no—
162C22×C3×D4—71no—
163C22×C3×D4—72no—
164C22×C3×D4—141no—
165C22×C3×D4—142no—
166C2×C3×M16—72no—
167C3×#117—72no—
168C3×#104—144no—
169C3×#102—144no—
170C3×#101—141no—
171C3×#100—141no—
172C3×#99—142no—
173C22×ΓL(1,𝔽8)—22no—
174C22×ΓL(1,𝔽8)—41no—
175C22×ΓL(1,𝔽8)—42no—
176C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
177C22×ΓL(1,𝔽8)#12012yes—
178C2×AGL(1,𝔽7)—24no—
179C2×AGL(1,𝔽7)—22no—
180C2×AGL(1,𝔽7)—44no—
181C2×AGL(1,𝔽7)—42no—
182C2×AGL(1,𝔽7)—84no—
183C2×AGL(1,𝔽7)—88no—
184C4×ΓL(1,𝔽8)—22no—
185C4×ΓL(1,𝔽8)—82no—
186C4×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
187C4×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
188C4×ΓL(1,𝔽8)#12012yes—
189ΓL(1,𝔽8)⋊C4—22no—
190C2×H(5,𝔽2)—71no—
191#421—72no—
192#418—72no—
193#413—71no—
194#410AGL(1,𝔽7)11yescharacteristic, complemented
195C23×C7C3×C22⋊C412yes—
196C23×C7C2×C3×D411yescharacteristic
197C22×D7—22no—
198C22×D7—21no—
199C22×D7—28no—
200C22×D7—24no—
201C22×D7—82no—
202C22×D7—84no—
203C7×Q8—22no—
204C7×D4—24no—
205C7×D4—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—42no—
209C2×C4×C7C3×C22⋊C412yes—
210C2×C4×C7C22×C3×C411yescharacteristic
211C2×C4×C7C2×C3×D411yescharacteristic
212C2×C4×C7C2×C3×D412yes—
213C2×C4×C7C2×C3×D412yes—
214C22⋊D7—28no—
215C22⋊D7—24no—
216C2×Dic7—21no—
217D28—28no—
218D28—21no—
219D28—22no—
220D28—24no—
221D28C3×C22⋊C412yes—
222D28C3×C22⋊C414yes—
223C4×D7—28no—
224C4×D7—24no—
225Dic14—21no—
226Dic14C3×C22⋊C412yes—
227C7⋊C8—42no—
228C24×C3—141no—
229C24×C3—142no—
230C3×#41—74no—
231C3×#41—142no—
232C3×#41—142no—
233C3×#41—142no—
234C3×#41—144no—
235C3×#41—144no—
236C3×#41—148no—
237C2×C3×Q8—71no—
238C2×C3×Q8—71no—
239C2×C3×D4—74no—
240C2×C3×D4—72no—
241C2×C3×D4—71no—
242C2×C3×D4—71no—
243C2×C3×D4—72no—
244C2×C3×D4—141no—
245C2×C3×D4—142no—
246C2×C3×D4—142no—
247C2×C3×D4—144no—
248C2×C3×D4—144no—
249C2×C3×D4—141no—
250C2×C3×D4—142no—
251C2×C3×D4—148no—
252C2×C3×D4—1416no—
253C22×C3×C4—72no—
254C22×C3×C4—71no—
255C22×C3×C4—141no—
256C22×C3×C4—142no—
257C3×M16—142no—
258C3×M16—142no—
259C2×C3×C8—142no—
260C3×C4⋊C4—142no—
261C3×C22⋊C4—142no—
262C3×C42—142no—
263C2×ΓL(1,𝔽8)—22no—
264C2×ΓL(1,𝔽8)—21no—
265C2×ΓL(1,𝔽8)—42no—
266C2×ΓL(1,𝔽8)#37911yescharacteristic
267AGL(1,𝔽7)—44no—
268AGL(1,𝔽7)—42no—
269H(5,𝔽2)—144no—
270H(5,𝔽2)—144no—
271C2×#41—72no—
272C2×#41—71no—
273C2×#41—141no—
274C2×#41C2×AGL(1,𝔽7)11yescharacteristic
275C22×D4—72no—
276C22×D4—71no—
277C22×D4—141no—
278C22×D4—142no—
279C2×M16—72no—
280#117C2×AGL(1,𝔽7)12yescomplemented
281#104—144no—
282#102—144no—
283#101—141no—
284#100—141no—
285#99—22no—
286C22×C7—22no—
287C22×C7—41no—
288C22×C7—42no—
289C22×C7C2×C3×C22⋊C411yescharacteristic
290C22×C7C3×#12012yes—
291C2×D7—24no—
292C2×D7—22no—
293C2×D7—44no—
294C2×D7—42no—
295C2×D7—84no—
296C2×D7—88no—
297C4×C7—22no—
298C4×C7—82no—
299C4×C7C2×C3×C22⋊C411yescharacteristic
300C4×C7C2×C3×C22⋊C411yescharacteristic
301C4×C7C3×#12012yes—
302Dic7—22no—
303C23×C3—71no—
304C23×C3—72no—
305C23×C3—141no—
306C23×C3—142no—
307C23×C3—144no—
308C23×C3—148no—
309C23×C3—562no—
310C23×C3—564no—
311C3×Q8—72no—
312C3×Q8—142no—
313C3×Q8—141no—
314C3×D4—74no—
315C3×D4—72no—
316C3×D4—142no—
317C3×D4—144no—
318C3×D4—141no—
319C3×D4—142no—
320C3×D4—144no—
321C3×D4—144no—
322C3×D4—148no—
323C3×D4—148no—
324C2×C3×C4—71no—
325C2×C3×C4—71no—
326C2×C3×C4—72no—
327C2×C3×C4—72no—
328C2×C3×C4—72no—
329C2×C3×C4—142no—
330C2×C3×C4—142no—
331C2×C3×C4—144no—
332C2×C3×C4—141no—
333C2×C3×C4—148no—
334C2×C3×C4—282no—
335C3×C8—282no—
336ΓL(1,𝔽8)#179511yescharacteristic, complemented, Hall
337C24—141no—
338C24—142no—
339#41—22no—
340#41—22no—
341#41—74no—
342#41—144no—
343#41—144no—
344#41—148no—
345#41—142no—
346C2×Q8—71no—
347C2×Q8C4⋊AGL(1,𝔽7)11yescharacteristic
348C2×D4—72no—
349C2×D4—71no—
350C2×D4—74no—
351C2×D4—144no—
352C2×D4—144no—
353C2×D4—1416no—
354C2×D4—148no—
355C2×D4—141no—
356C2×D4—142no—
357C2×D4—141no—
358C2×D4—142no—
359C2×D4—142no—
360C2×D4C4⋊AGL(1,𝔽7)11yescharacteristic
361C2×D4(C22⋊D7)⋊C312yes—
362C22×C4—141no—
363C22×C4—142no—
364C22×C4C4×AGL(1,𝔽7)12yes—
365C22×C4C22×AGL(1,𝔽7)11yescharacteristic
366M16—142no—
367M16—142no—
368C2×C8—142no—
369C4⋊C4—22no—
370C22⋊C4—22no—
371C42—22no—
372C2×C7—22no—
373C2×C7—21no—
374C2×C7—42no—
375C2×C7C3×#37911yescharacteristic
376D7—44no—
377D7—42no—
378C22×C3—71no—
379C22×C3—72no—
380C22×C3—142no—
381C22×C3—142no—
382C22×C3—144no—
383C22×C3—281no—
384C22×C3—282no—
385C22×C3—284no—
386C22×C3—282no—
387C22×C3—564no—
388C22×C3—568no—
389C3×C4—71no—
390C3×C4—71no—
391C3×C4—72no—
392C3×C4—142no—
393C3×C4—142no—
394C3×C4—562no—
395C23—148no—
396C23—141no—
397C23—142no—
398C23—144no—
399C23—562no—
400C23—564no—
401C23(C2×AGL(1,𝔽7))⋊C412yes—
402C23C2×(C22⋊D7)⋊C311yescharacteristic
403Q8—22no—
404Q8—72no—
405Q8—141no—
406D4—24no—
407D4—22no—
408D4—74no—
409D4—72no—
410D4—148no—
411D4—148no—
412D4—144no—
413D4—141no—
414D4—142no—
415D4—144no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—148no—
420C2×C4—141no—
421C2×C4—144no—
422C2×C4(C2×AGL(1,𝔽7))⋊C412yes—
423C2×C4C2×C4×AGL(1,𝔽7)11yescharacteristic
424C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
425C2×C4D4×AGL(1,𝔽7)12yes—
426C2×C4D4×AGL(1,𝔽7)12yes—
427C8—282no—
428C7C3×#179511yescharacteristic, complemented, Sylow
429C2×C3—71no—
430C2×C3—142no—
431C2×C3—141no—
432C2×C3—282no—
433C2×C3—284no—
434C2×C3—282no—
435C22—22no—
436C22—41no—
437C22—42no—
438C22—144no—
439C22—142no—
440C22—284no—
441C22—282no—
442C22—564no—
443C22—568no—
444C22C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
445C22#6500⋊C312yes—
446C4—22no—
447C4—82no—
448C4—142no—
449C4C22⋊C4×AGL(1,𝔽7)11yescharacteristic
450C4C22⋊C4×AGL(1,𝔽7)11yescharacteristic
451C4#6445⋊C312yes—
452C3—71noSylow
453C2—22no—
454C2—21no—
455C2—42no—
456C2—284no—
457C2—282no—
458C2#89727⋊C311yescharacteristic
subgroupG/H
centerC2#89727⋊C3
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C22×AGL(1,𝔽7)
FittingC7×#410C2×C3
socleC2×C7C3×#379
prime radicalC2×#10621518C2
cosocleC22×C4×C7C23×C3
hypercenter#410AGL(1,𝔽7)
hypocenterC7C3×#1795
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#410C2semidirect
3. #90009⋊C3C2semidirect
4. #89777⋊C3C2semidirect
5. #89773⋊C3C2semidirect
6. #11727802C3semidirect
13. C2×#1157⋊C3C4semidirect
15. C22×C4⋊AGL(1,𝔽7)C4semidirect
17. ΓL(1,𝔽8)×#117C22semidirect
18. C2×#1136⋊C3C22semidirect
25. C7×#410C2×C3semidirect
26. #90009C2×C3semidirect
27. #89777C2×C3semidirect
28. #89773C2×C3semidirect
71. C2×#1157C3×C4semidirect
73. C22×D28C3×C4semidirect
75. C7×#117C22×C3semidirect
76. C2×#1136C22×C3semidirect
194. #410AGL(1,𝔽7)semidirect
280. #117C2×AGL(1,𝔽7)semidirect
336. ΓL(1,𝔽8)#1795semidirect
428. C7C3×#1795semidirect

Maximal subgroups

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

subgroupindex
C2×#106215182
ΓL(1,𝔽8)×#4102
#90009⋊C32
#89777⋊C32
#89773⋊C32
#117278023
C3×#17957