Finite Group Databasebeta

← #72276615

Subgroups — #72276615

Order 2688.

1276 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#10620789C212yesmaximal, complemented
2#90184⋊C3C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#379C211yesmaximal, characteristic, complemented
4C2×#6380⋊C3C212yesmaximal, complemented
5#89321⋊C3C211yesmaximal, characteristic, complemented
6#11727254C311yesmaximal, characteristic, complemented, Hall
7C25×ΓL(1,𝔽8)C2211yescharacteristic
8C22×(C22⋊D7)⋊C3—22no—
9C22×(C22⋊D7)⋊C3C2212yes—
10C2×C22⋊C4×ΓL(1,𝔽8)—22no—
11C2×C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
12C2×C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic
13#10620789—24no—
14#10620789—24no—
15C2×(C22⋊Dic7)⋊C3—22no—
16C2×(C22⋊Dic7)⋊C3C2212yes—
17#6493⋊C3—22no—
18#6449⋊C3—22no—
19#6445⋊C3—22no—
20#6380⋊C3—24no—
21C2×#6516C2×C312yescomplemented
22#90184C2×C311yescharacteristic, complemented
23C7×#379C2×C311yescharacteristic, complemented
24C2×#6380C2×C312yescomplemented
25#89321C2×C311yescharacteristic, complemented
26C3×#2021—71nomaximal, Hall
27C24×ΓL(1,𝔽8)—22no—
28C24×ΓL(1,𝔽8)—21no—
29C24×ΓL(1,𝔽8)—22no—
30C24×ΓL(1,𝔽8)—21no—
31C24×ΓL(1,𝔽8)—44no—
32C24×ΓL(1,𝔽8)D412yescomplemented
33C24×ΓL(1,𝔽8)C2311yescharacteristic
34C23×AGL(1,𝔽7)D412yescomplemented
35C22×C4×ΓL(1,𝔽8)—21no—
36C22×C4×ΓL(1,𝔽8)D412yescomplemented
37C2×(C22⋊D7)⋊C3—28no—
38C2×(C22⋊D7)⋊C3—22no—
39C2×(C22⋊D7)⋊C3—22no—
40C2×(C22⋊D7)⋊C3—22no—
41C2×(C22⋊D7)⋊C3—22no—
42C2×(C22⋊D7)⋊C3—22no—
43C2×(C22⋊D7)⋊C3—22no—
44C2×(C22⋊D7)⋊C3—48no—
45C22×ΓL(1,𝔽8)⋊C4—22no—
46C22×ΓL(1,𝔽8)⋊C4D412yescomplemented
47C2×C4⋊AGL(1,𝔽7)—22no—
48C22⋊C4×ΓL(1,𝔽8)—22no—
49C22⋊C4×ΓL(1,𝔽8)—22no—
50C22⋊C4×ΓL(1,𝔽8)—44no—
51(C22⋊Dic7)⋊C3—28no—
52(C22⋊Dic7)⋊C3—22no—
53(C22⋊Dic7)⋊C3D414yescomplemented
54(C2×AGL(1,𝔽7))⋊C4—24no—
55C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
56C25×C7C22×C311yescharacteristic
57C22×C22⋊D7—22no—
58C22×C22⋊D7C22×C312yes—
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4C22×C311yescharacteristic
61C2×C7×C22⋊C4C22×C311yescharacteristic
62#6516—24no—
63#6516—24no—
64C2×C22⋊Dic7—22no—
65C2×C22⋊Dic7C22×C312yes—
66#6493—22no—
67#6449—22no—
68#6445—22no—
69#6380—24no—
70C3×#534—71no—
71C2×C3×#120—72no—
72C2×C3×#99—72no—
73C3×#379—71no—
74C3×#342—71no—
75C23×ΓL(1,𝔽8)—24no—
76C23×ΓL(1,𝔽8)—28no—
77C23×ΓL(1,𝔽8)—24no—
78C23×ΓL(1,𝔽8)—22no—
79C23×ΓL(1,𝔽8)—44no—
80C23×ΓL(1,𝔽8)—42no—
81C23×ΓL(1,𝔽8)—44no—
82C23×ΓL(1,𝔽8)—42no—
83C23×ΓL(1,𝔽8)—48no—
84C23×ΓL(1,𝔽8)—44no—
85C23×ΓL(1,𝔽8)—41no—
86C23×ΓL(1,𝔽8)—42no—
87C23×ΓL(1,𝔽8)—41no—
88C23×ΓL(1,𝔽8)C2×D412yes—
89C23×ΓL(1,𝔽8)C2×D411yescharacteristic
90C23×ΓL(1,𝔽8)C2×D411yescharacteristic
91C23×ΓL(1,𝔽8)C2×D412yes—
92C23×ΓL(1,𝔽8)#4111yescharacteristic
93C22×AGL(1,𝔽7)—22no—
94C22×AGL(1,𝔽7)—22no—
95C22×AGL(1,𝔽7)—22no—
96C22×AGL(1,𝔽7)—44no—
97C2×C4×ΓL(1,𝔽8)—22no—
98C2×C4×ΓL(1,𝔽8)—42no—
99C2×C4×ΓL(1,𝔽8)—42no—
100C2×C4×ΓL(1,𝔽8)—41no—
101(C22⋊D7)⋊C3—416no—
102(C22⋊D7)⋊C3—42no—
103(C22⋊D7)⋊C3—42no—
104(C22⋊D7)⋊C3—42no—
105(C22⋊D7)⋊C3—42no—
106C2×ΓL(1,𝔽8)⋊C4—24no—
107C2×ΓL(1,𝔽8)⋊C4—22no—
108C2×ΓL(1,𝔽8)⋊C4—22no—
109C2×ΓL(1,𝔽8)⋊C4—22no—
110C2×ΓL(1,𝔽8)⋊C4—44no—
111C4⋊AGL(1,𝔽7)—82no—
112#2021—71noSylow
113C24×C7—22no—
114C24×C7—21no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—44no—
118C24×C7C3×D412yescomplemented
119C24×C7C23×C311yescharacteristic
120C23×D7C3×D412yescomplemented
121C22×C4×C7—21no—
122C22×C4×C7C3×D412yescomplemented
123C2×C22⋊D7—28no—
124C2×C22⋊D7—22no—
125C2×C22⋊D7—22no—
126C2×C22⋊D7—22no—
127C2×C22⋊D7—22no—
128C2×C22⋊D7—22no—
129C2×C22⋊D7—22no—
130C2×C22⋊D7—48no—
131C22×Dic7—22no—
132C22×Dic7C3×D412yescomplemented
133C2×D28—22no—
134C7×C22⋊C4—22no—
135C7×C22⋊C4—22no—
136C7×C22⋊C4—44no—
137C22⋊Dic7—28no—
138C22⋊Dic7—22no—
139C22⋊Dic7C3×D414yescomplemented
140(C2×D7)⋊C4—24no—
141C4⋊Dic7—22no—
142C25×C3—71no—
143C22×C3×D4—72no—
144C22×C3×D4—142no—
145C3×#123—142no—
146C3×#121—142no—
147C3×#120—144no—
148C3×#120—142no—
149C3×#120—144no—
150C2×C3×C22⋊C4—72no—
151C2×C3×C22⋊C4—71no—
152C2×C3×C22⋊C4—71no—
153C2×C3×C22⋊C4—142no—
154C2×C3×C22⋊C4—142no—
155C3×#99—144no—
156C22×ΓL(1,𝔽8)—24no—
157C22×ΓL(1,𝔽8)—22no—
158C22×ΓL(1,𝔽8)—28no—
159C22×ΓL(1,𝔽8)—24no—
160C22×ΓL(1,𝔽8)—41no—
161C22×ΓL(1,𝔽8)—42no—
162C22×ΓL(1,𝔽8)—41no—
163C22×ΓL(1,𝔽8)—44no—
164C22×ΓL(1,𝔽8)—44no—
165C22×ΓL(1,𝔽8)—48no—
166C22×ΓL(1,𝔽8)—42no—
167C22×ΓL(1,𝔽8)—42no—
168C22×ΓL(1,𝔽8)—44no—
169C22×ΓL(1,𝔽8)#12012yes—
170C22×ΓL(1,𝔽8)#12011yescharacteristic
171C22×ΓL(1,𝔽8)#12112yes—
172C22×ΓL(1,𝔽8)#12111yescharacteristic
173C22×ΓL(1,𝔽8)#12711yescharacteristic
174C2×AGL(1,𝔽7)—44no—
175C2×AGL(1,𝔽7)—42no—
176C2×AGL(1,𝔽7)—44no—
177C2×AGL(1,𝔽7)—82no—
178C4×ΓL(1,𝔽8)—42no—
179C4×ΓL(1,𝔽8)—81no—
180ΓL(1,𝔽8)⋊C4—44no—
181ΓL(1,𝔽8)⋊C4—42no—
182#534—71no—
183C2×#120—72no—
184C2×#99—72no—
185#379AGL(1,𝔽7)11yescharacteristic, complemented
186#342—71no—
187C23×C7—24no—
188C23×C7—28no—
189C23×C7—24no—
190C23×C7—22no—
191C23×C7—44no—
192C23×C7—42no—
193C23×C7—44no—
194C23×C7—42no—
195C23×C7—48no—
196C23×C7—44no—
197C23×C7—41no—
198C23×C7—42no—
199C23×C7—41no—
200C23×C7C2×C3×D411yescharacteristic
201C23×C7C2×C3×D412yes—
202C23×C7C2×C3×D412yes—
203C23×C7C2×C3×D411yescharacteristic
204C23×C7C3×#4111yescharacteristic
205C22×D7—22no—
206C22×D7—22no—
207C22×D7—22no—
208C22×D7—44no—
209C2×C4×C7—22no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C22⋊D7—416no—
214C22⋊D7—42no—
215C22⋊D7—42no—
216C22⋊D7—42no—
217C22⋊D7—42no—
218C2×Dic7—22no—
219C2×Dic7—22no—
220C2×Dic7—24no—
221C2×Dic7—22no—
222C2×Dic7—44no—
223D28—82no—
224C24×C3—72no—
225C24×C3—72no—
226C24×C3—71no—
227C24×C3—141no—
228C24×C3—142no—
229C24×C3—141no—
230C24×C3—142no—
231C24×C3—284no—
232C2×C3×D4—142no—
233C2×C3×D4—142no—
234C2×C3×D4—142no—
235C2×C3×D4—142no—
236C2×C3×D4—142no—
237C2×C3×D4—142no—
238C2×C3×D4—148no—
239C2×C3×D4—142no—
240C2×C3×D4—288no—
241C22×C3×C4—72no—
242C22×C3×C4—72no—
243C22×C3×C4—141no—
244C22×C3×C4—142no—
245C3×C4⋊C4—142no—
246C3×C22⋊C4—74no—
247C3×C22⋊C4—142no—
248C3×C22⋊C4—142no—
249C3×C22⋊C4—142no—
250C3×C22⋊C4—148no—
251C3×C22⋊C4—144no—
252C3×C22⋊C4—284no—
253C2×ΓL(1,𝔽8)—22no—
254C2×ΓL(1,𝔽8)—21no—
255C2×ΓL(1,𝔽8)—22no—
256C2×ΓL(1,𝔽8)—21no—
257C2×ΓL(1,𝔽8)—44no—
258C2×ΓL(1,𝔽8)#39211yescharacteristic
259C2×ΓL(1,𝔽8)B(4,𝔽2)12yes—
260AGL(1,𝔽7)—82no—
261C25C2×AGL(1,𝔽7)11yescharacteristic
262C22×D4—72no—
263C22×D4—142no—
264#123—142no—
265#121—142no—
266#120—142no—
267#120—144no—
268#120—144no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—72no—
271C2×C22⋊C4—142no—
272C2×C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic
273C2×C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic
274#99—144no—
275C22×C7—24no—
276C22×C7—22no—
277C22×C7—28no—
278C22×C7—24no—
279C22×C7—41no—
280C22×C7—42no—
281C22×C7—41no—
282C22×C7—44no—
283C22×C7—44no—
284C22×C7—48no—
285C22×C7—42no—
286C22×C7—42no—
287C22×C7—44no—
288C22×C7C3×#12012yes—
289C22×C7C3×#12011yescharacteristic
290C22×C7C3×#12112yes—
291C22×C7C3×#12111yescharacteristic
292C22×C7C3×#12711yescharacteristic
293C2×D7—44no—
294C2×D7—42no—
295C2×D7—44no—
296C2×D7—82no—
297C4×C7—42no—
298C4×C7—81no—
299Dic7—44no—
300Dic7—42no—
301C23×C3—71no—
302C23×C3—71no—
303C23×C3—72no—
304C23×C3—72no—
305C23×C3—71no—
306C23×C3—144no—
307C23×C3—142no—
308C23×C3—144no—
309C23×C3—148no—
310C23×C3—142no—
311C23×C3—142no—
312C23×C3—142no—
313C23×C3—282no—
314C23×C3—281no—
315C23×C3—281no—
316C23×C3—284no—
317C23×C3—282no—
318C23×C3—282no—
319C23×C3—284no—
320C23×C3—288no—
321C23×C3—284no—
322C23×C3—284no—
323C3×D4—282no—
324C3×D4—282no—
325C3×D4—282no—
326C3×D4—282no—
327C3×D4—2816no—
328C3×D4—562no—
329C2×C3×C4—142no—
330C2×C3×C4—144no—
331C2×C3×C4—142no—
332C2×C3×C4—142no—
333C2×C3×C4—142no—
334C2×C3×C4—282no—
335C2×C3×C4—281no—
336C2×C3×C4—282no—
337C2×C3×C4—284no—
338ΓL(1,𝔽8)#202111yescharacteristic, complemented, Hall
339C24—22no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—44no—
344C24—72no—
345C24C4⋊AGL(1,𝔽7)12yescomplemented
346C24C22×AGL(1,𝔽7)11yescharacteristic
347C2×D4—142no—
348C2×D4—148no—
349C2×D4—142no—
350C2×D4—142no—
351C2×D4—142no—
352C2×D4—142no—
353C2×D4—142no—
354C2×D4—142no—
355C2×D4—288no—
356C22×C4—21no—
357C22×C4—72no—
358C22×C4—142no—
359C22×C4(C22⋊D7)⋊C312yescomplemented
360C4⋊C4—142no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—44no—
364C22⋊C4—74no—
365C22⋊C4—144no—
366C22⋊C4—148no—
367C22⋊C4—142no—
368C2×C7—22no—
369C2×C7—21no—
370C2×C7—22no—
371C2×C7—21no—
372C2×C7—44no—
373C2×C7C3×#39211yescharacteristic
374C2×C7C3×B(4,𝔽2)12yes—
375D7—82no—
376C22×C3—72no—
377C22×C3—72no—
378C22×C3—71no—
379C22×C3—71no—
380C22×C3—71no—
381C22×C3—142no—
382C22×C3—144no—
383C22×C3—148no—
384C22×C3—144no—
385C22×C3—284no—
386C22×C3—282no—
387C22×C3—281no—
388C22×C3—281no—
389C22×C3—282no—
390C22×C3—284no—
391C22×C3—284no—
392C22×C3—282no—
393C22×C3—288no—
394C22×C3—282no—
395C22×C3—284no—
396C22×C3—284no—
397C22×C3—562no—
398C3×C4—282no—
399C3×C4—282no—
400C3×C4—284no—
401C3×C4—561no—
402C23—24no—
403C23—28no—
404C23—24no—
405C23—22no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—48no—
411C23—44no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—142no—
416C23—142no—
417C23—142no—
418C23—284no—
419C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
420C23#1157⋊C311yescharacteristic
421C23D4×AGL(1,𝔽7)12yes—
422C23D4×AGL(1,𝔽7)12yes—
423C23C2×(C22⋊D7)⋊C311yescharacteristic
424D4—2816no—
425D4—282no—
426D4—282no—
427D4—282no—
428D4—282no—
429D4—562no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—144no—
435C2×C4—142no—
436C2×C4—142no—
437C2×C4—142no—
438C2×C4—284no—
439C7C3×#202111yescharacteristic, complemented, Sylow
440C2×C3—72no—
441C2×C3—71no—
442C2×C3—142no—
443C2×C3—141no—
444C2×C3—141no—
445C2×C3—142no—
446C2×C3—284no—
447C2×C3—562no—
448C22—22no—
449C22—24no—
450C22—28no—
451C22—24no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—48no—
459C22—44no—
460C22—42no—
461C22—284no—
462C22—284no—
463C22—282no—
464C22—562no—
465C22#6445⋊C312yes—
466C22#6447⋊C312yes—
467C22#6493⋊C311yescharacteristic
468C22#6500⋊C311yescharacteristic
469C22#6503⋊C311yescharacteristic
470C4—42no—
471C4—81no—
472C4—284no—
473C4—282no—
474C3—71noSylow
475C2—21no—
476C2—22no—
477C2—21no—
478C2—22no—
479C2—44no—
480C2—562no—
481C2#89732⋊C311yescharacteristic
482C2#89804⋊C312yes—
subgroupG/H
centerC22#6500⋊C3
commutator, cosocleC24×C7C23×C3
FrattiniC24C22×AGL(1,𝔽7)
FittingC7×#379C2×C3
socleC22×C7C3×#120
hypercenter#379AGL(1,𝔽7)
hypocenterC7C3×#2021
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#10620789C2semidirect
2. #90184⋊C3C2semidirect
3. ΓL(1,𝔽8)×#379C2semidirect
4. C2×#6380⋊C3C2semidirect
5. #89321⋊C3C2semidirect
6. #11727254C3semidirect
21. C2×#6516C2×C3semidirect
22. #90184C2×C3semidirect
23. C7×#379C2×C3semidirect
24. C2×#6380C2×C3semidirect
25. #89321C2×C3semidirect
32. C24×ΓL(1,𝔽8)D4semidirect
34. C23×AGL(1,𝔽7)D4semidirect
36. C22×C4×ΓL(1,𝔽8)D4semidirect
46. C22×ΓL(1,𝔽8)⋊C4D4semidirect
53. (C22⋊Dic7)⋊C3D4semidirect
118. C24×C7C3×D4semidirect
120. C23×D7C3×D4semidirect
122. C22×C4×C7C3×D4semidirect
132. C22×Dic7C3×D4semidirect
139. C22⋊Dic7C3×D4semidirect
185. #379AGL(1,𝔽7)semidirect
338. ΓL(1,𝔽8)#2021semidirect
345. C24C4⋊AGL(1,𝔽7)semidirect
359. C22×C4(C22⋊D7)⋊C3semidirect
439. C7C3×#2021semidirect

Maximal subgroups

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

subgroupindex
C2×#106207892
#90184⋊C32
ΓL(1,𝔽8)×#3792
C2×#6380⋊C32
#89321⋊C32
#117272543
C3×#20217