Finite Group Databasebeta

← #32038756

Subgroups — #32038756

Order 18176.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986279C211yesmaximal, characteristic, complemented
2C2×#17986056C211yesmaximal, characteristic, complemented
3#22596692C211yesmaximal, characteristic, complemented
4#22589771C212yesmaximal, complemented
5#22589526C212yesmaximal
6#22589292C211yesmaximal, characteristic, complemented
7#22589103C211yesmaximal, characteristic, complemented
8#22589080C212yesmaximal, complemented
9#22589040C212yesmaximal, complemented
10#22588340C211yesmaximal, characteristic, complemented
11C4×#121×C71C211yesmaximal, characteristic
12C2×#14556128C2211yescharacteristic
13C2×#14556107—21no—
14#17986279—24no—
15#17986056—24no—
16C2×#14556061C2212yes—
17C2×#14556059C2212yes—
18C2×C4⋊(C71⋊C8)C2211yescharacteristic
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556049C2212yes—
21C2×#14556049C2212yes—
22C2×C22⋊(C71⋊C8)C2212yes—
23C2×#14556039C2212yes—
24#17985905—21no—
25#17985897—21no—
26#17985897C2212yes—
27#17985869C2211yescharacteristic
28#17985859—21no—
29#17985859C2212yes—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34C2×#121×C71C2211yescharacteristic
35C2×C4×D4×C71—21no—
36C2×C4×D4×C71C2211yescharacteristic
37C22×C42×C71C2211yescharacteristic
38#390×C71C2211yescharacteristic
39#389×C71C2211yescharacteristic
40#388×C71—21no—
41#386×C71—21no—
42#380×C71C2211yescharacteristic
43C4×C4⋊C4×C71C2211yescharacteristic
44C4×C22⋊C4×C71—21no—
45C22×Dic142—21no—
46C22×C71⋊C8C2312yes—
47C22×C71⋊C8C2312yes—
48#14556128—22no—
49#14556128—22no—
50#14556107—44no—
51C2×C4⋊Dic71—21no—
52C2×C4⋊Dic71—21no—
53C2×C4⋊Dic71—21no—
54C2×C4⋊Dic71C2312yes—
55C2×C4⋊Dic71C2312yes—
56C2×Dic71⋊C4—22no—
57C2×C22⋊Dic71—22no—
58#14556061—24no—
59#14556059—24no—
60C4⋊(C71⋊C8)—22no—
61C4⋊(C71⋊C8)D414yescomplemented
62#14556049—24no—
63#14556049—24no—
64C22⋊(C71⋊C8)—24no—
65#14556039—24no—
66C71⋊#95—22no—
67C71⋊#95—22no—
68C22×D4×C71—21no—
69C22×D4×C71C2311yescharacteristic
70C23×C4×C71—21no—
71C23×C4×C71D411yescharacteristic
72C23×C4×C71D411yescharacteristic
73C23×C4×C71C2311yescharacteristic
74#121×C71—22no—
75#121×C71—22no—
76C4×D4×C71—22no—
77C4×D4×C71—22no—
78C4×D4×C71—44no—
79C2×C4⋊C4×C71—21no—
80C2×C4⋊C4×C71C2311yescharacteristic
81C2×C4⋊C4×C71C2311yescharacteristic
82C2×C4⋊C4×C71C2311yescharacteristic
83C2×C22⋊C4×C71—21no—
84C2×C22⋊C4×C71—21no—
85C2×C22⋊C4×C71—21no—
86C2×C22⋊C4×C71—21no—
87C2×C22⋊C4×C71—21no—
88C2×C42×C71—22no—
89C2×C42×C71—22no—
90C2×C42×C71—21no—
91C2×C42×C71D411yescharacteristic
92C2×C42×C71D411yescharacteristic
93C2×C42×C71C2311yescharacteristic
94C2×C42×C71C2311yescharacteristic
95#95×C71—21no—
96#95×C71—21no—
97#95×C71—21no—
98C2×Dic142—42no—
99C2×Dic142—42no—
100C2×Dic142—42no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C2×C71⋊C8—22no—
104C2×C71⋊C8—22no—
105C2×C71⋊C8—22no—
106C2×C71⋊C8—22no—
107C2×C71⋊C8—22no—
108C2×C71⋊C8C2×D414yes—
109C4⋊Dic71—22no—
110C4⋊Dic71—22no—
111C4⋊Dic71—22no—
112C4⋊Dic71—22no—
113C4⋊Dic71—42no—
114C4⋊Dic71—42no—
115C4⋊Dic71—42no—
116Dic71⋊C4—44no—
117C22⋊Dic71—44no—
118C24×C71—21no—
119C24×C71C2×D411yescharacteristic
120C2×D4×C71—24no—
121C2×D4×C71—21no—
122C2×D4×C71—21no—
123C2×D4×C71—21no—
124C2×D4×C71—21no—
125C2×D4×C71—21no—
126C2×D4×C71—42no—
127C2×D4×C71—42no—
128C2×D4×C71#4112yes—
129C22×C4×C71—22no—
130C22×C4×C71—21no—
131C22×C4×C71—22no—
132C22×C4×C71—22no—
133C22×C4×C71—22no—
134C22×C4×C71—22no—
135C22×C4×C71—21no—
136C22×C4×C71—21no—
137C22×C4×C71—21no—
138C22×C4×C71—21no—
139C22×C4×C71—21no—
140C22×C4×C71—21no—
141C22×C4×C71—21no—
142C22×C4×C71—21no—
143C22×C4×C71—41no—
144C22×C4×C71—42no—
145C22×C4×C71—42no—
146C22×C4×C71—42no—
147C22×C4×C71C2×D411yescharacteristic
148C22×C4×C71C2×D411yescharacteristic
149C22×C4×C71C2×D411yescharacteristic
150C22×C4×C71C2×D411yescharacteristic
151C22×C4×C71C2×D411yescharacteristic
152C22×C4×C71C2411yescharacteristic
153C4⋊C4×C71—21no—
154C4⋊C4×C71—21no—
155C4⋊C4×C71—21no—
156C4⋊C4×C71—21no—
157C4⋊C4×C71—21no—
158C4⋊C4×C71—42no—
159C4⋊C4×C71#4112yes—
160C22⋊C4×C71—24no—
161C22⋊C4×C71—42no—
162C22⋊C4×C71—42no—
163C22⋊C4×C71—42no—
164C22⋊C4×C71—42no—
165C42×C71—22no—
166C42×C71—22no—
167C42×C71—21no—
168C42×C71—21no—
169C42×C71—42no—
170C42×C71C2×D412yes—
171C42×C71#4112yes—
172C23×C71—21no—
173C23×C71—21no—
174C23×C71—21no—
175C23×C71—21no—
176C23×C71—21no—
177C23×C71—21no—
178C23×C71—22no—
179C23×C71—41no—
180C23×C71—41no—
181C23×C71—41no—
182C23×C71—41no—
183C23×C71—41no—
184C23×C71—41no—
185C23×C71#13712yes—
186C23×C71C22×D411yescharacteristic
187D4×C71—22no—
188D4×C71—22no—
189D4×C71—48no—
190D4×C71—41no—
191D4×C71—41no—
192C2×C4×C71—21no—
193C2×C4×C71—21no—
194C2×C4×C71—21no—
195C2×C4×C71—21no—
196C2×C4×C71—21no—
197C2×C4×C71—21no—
198C2×C4×C71—22no—
199C2×C4×C71—21no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—44no—
205C2×C4×C71—41no—
206C2×C4×C71—41no—
207C2×C4×C71—41no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—44no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—84no—
219C2×C4×C71#12112yes—
220C2×C4×C71#12112yes—
221C2×C4×C71AGL(1,ℤ/8)12yes—
222C2×C4×C71AGL(1,ℤ/8)12yes—
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C2×#4111yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C712−1+411yescharacteristic
228C2×C4×C712−1+411yescharacteristic
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235Dic142—84no—
236C71⋊C8—24no—
237C71⋊C8—42no—
238C22×C71—22no—
239C22×C71—22no—
240C22×C71—22no—
241C22×C71—41no—
242C22×C71—41no—
243C22×C71—41no—
244C22×C71—41no—
245C22×C71—42no—
246C22×C71—42no—
247C22×C71—42no—
248C22×C71—81no—
249C22×C71—81no—
250C22×C71—81no—
251C22×C71—81no—
252C22×C71C2×#12111yescharacteristic
253C22×C71#52511yescharacteristic
254C22×C71#53611yescharacteristic
255C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C71C2×#13711yescharacteristic
258C22×C71#57511yescharacteristic
259C4×C71—22no—
260C4×C71—42no—
261C4×C71—41no—
262C4×C71—41no—
263C4×C71—41no—
264C4×C71—81no—
265C4×C71—81no—
266C4×C71#46914yes—
267C4×C71#54711yescharacteristic
268C4×C71#54811yescharacteristic
269C4×C71#55211yescharacteristic
270C4×C71#55611yescharacteristic
271Dic71—82no—
272Dic71—82no—
273#24709—711nomaximal, Sylow
274C2×C71—42no—
275C2×C71—82no—
276C2×C71#241511yescharacteristic
277C2×C71C2×#46911yescharacteristic
278C2×C71#305611yescharacteristic
279C2×C71#311011yescharacteristic
280C2×C71#311511yescharacteristic
281C2×C71#323511yescharacteristic
282C2×C71#323911yescharacteristic
283C2×#474—711no—
284C2×#432—711no—
285#2857—711no—
286C4×#121D7111yescharacteristic
287#2076—712no—
288#2034—712no—
289#1988—711no—
290#1948—711no—
291#1944—712no—
292#1936—712no—
293#1802—711no—
294C71#2470911yescharacteristic, complemented, Sylow
295C2×C4⋊Q8—711no—
296C2×C22⋊Q8—1421no—
297C2×#121C2×D7111yescharacteristic
298C2×C4×D4—21no—
299C2×C4×D4C2×D7111yescharacteristic
300C22×C42C2×D7111yescharacteristic
301#474—1424no—
302#432—1424no—
303C2×#107—712no—
304C2×#106—712no—
305C2×C4⋊C8—711no—
306C2×C4⋊C8—711no—
307C2×#102—712no—
308C2×#102—712no—
309C2×C22⋊C8—712no—
310C2×#97—712no—
311#391—1421no—
312#390C2×D7111yescharacteristic
313#389—712no—
314#389—1421no—
315#389C2×D7111yescharacteristic
316#388—21no—
317#386—21no—
318#384—711no—
319#380—712no—
320#380—1421no—
321#380C2×D7111yescharacteristic
322C4×C4⋊C4C2×D7111yescharacteristic
323C4×C22⋊C4—21no—
324#340—712no—
325#340—712no—
326#340—712no—
327#340—712no—
328C22×Q8—1421no—
329C22×D4—21no—
330C22×D4C22×D7111yescharacteristic
331C23×C4—21no—
332C23×C4C22⋊D7111yescharacteristic
333C23×C4C22⋊D7111yescharacteristic
334C23×C4C22×D7111yescharacteristic
335C22×C8—712no—
336C22×C8—712no—
337C4⋊Q8—1422no—
338C4⋊Q8—1422no—
339C22⋊Q8—2844no—
340#121—22no—
341#121—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—44no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—712no—
347C2×C4⋊C4—712no—
348C2×C4⋊C4—1421no—
349C2×C4⋊C4—1422no—
350C2×C4⋊C4—1421no—
351C2×C4⋊C4—1421no—
352C2×C4⋊C4C22×D7111yescharacteristic
353C2×C4⋊C4C22×D7111yescharacteristic
354C2×C4⋊C4C22×D7111yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—1422no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42C22⋊D7111yescharacteristic
365C2×C42C22⋊D7111yescharacteristic
366C2×C42C22×D7111yescharacteristic
367C2×C42C22×D7111yescharacteristic
368#107—1424no—
369#106—1424no—
370C4⋊C8—714no—
371C4⋊C8—1422no—
372#102—1424no—
373#102—1424no—
374C22⋊C8—1424no—
375#97—1424no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—1422no—
380#95—1422no—
381C24—21no—
382C24C2×C22⋊D7111yescharacteristic
383C2×Q8—2842no—
384C2×Q8—2842no—
385C2×Q8—2842no—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4#1176586512yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—1422no—
414C22×C4—1422no—
415C22×C4C2×C22⋊D7111yescharacteristic
416C22×C4C2×C22⋊D7111yescharacteristic
417C22×C4C2×C22⋊D7111yescharacteristic
418C22×C4C2×C22⋊D7111yescharacteristic
419C22×C4C2×C22⋊D7111yescharacteristic
420C22×C4C23×D7111yescharacteristic
421C2×C8—714no—
422C2×C8—1422no—
423C2×C8—1422no—
424C2×C8—1422no—
425C2×C8—1422no—
426C2×C8—1422no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—2844no—
434C4⋊C4—2842no—
435C4⋊C4—2842no—
436C4⋊C4—2842no—
437C4⋊C4—1422no—
438C4⋊C4—1422no—
439C4⋊C4—1422no—
440C4⋊C4—1422no—
441C4⋊C4#1176586512yes—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—2844no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—42no—
453C42D4×D7112yes—
454C42#1176586412yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#1455615812yes—
469C23C22×C22⋊D7111yescharacteristic
470Q8—5684no—
471D4—22no—
472D4—22no—
473D4—48no—
474D4—41no—
475D4—41no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—2842no—
504C2×C4—2842no—
505C2×C4—2842no—
506C2×C4—2842no—
507C2×C4—2842no—
508C2×C4—2842no—
509C2×C4#1455610012yes—
510C2×C4#1455610012yes—
511C2×C4#1455615312yes—
512C2×C4#1455615412yes—
513C2×C4C2×D4×D7111yescharacteristic
514C2×C4C2×#1176586411yescharacteristic
515C2×C4C2×#1176586511yescharacteristic
516C2×C4C2×#1176586511yescharacteristic
517C2×C4#1455617511yescharacteristic
518C2×C4#1455617511yescharacteristic
519C8—2842no—
520C8—1424no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×#1455610011yescharacteristic
536C22#1798651011yescharacteristic
537C22#1798657011yescharacteristic
538C22C2×#1455615311yescharacteristic
539C22C2×#1455615411yescharacteristic
540C22C2×#1455615811yescharacteristic
541C22#1798679711yescharacteristic
542C4—22no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—81no—
549C4—5682no—
550C4—5682no—
551C4#1798625014yes—
552C4#1798665411yescharacteristic
553C4#1798665811yescharacteristic
554C4#1798668911yescharacteristic
555C4#1798671911yescharacteristic
556C2—42no—
557C2—82no—
558C2#2259176911yescharacteristic
559C2C2×#1798625011yescharacteristic
560C2#2259797211yescharacteristic
561C2#2259843911yescharacteristic
562C2#2259849111yescharacteristic
563C2#2259953411yescharacteristic
564C2#2259956711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#121×C71C22
hypercenterC4×#121D71
hypocenterC71#24709
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#17986279C2semidirect
2. C2×#17986056C2semidirect
3. #22596692C2semidirect
4. #22589771C2semidirect
6. #22589292C2semidirect
7. #22589103C2semidirect
8. #22589080C2semidirect
9. #22589040C2semidirect
10. #22588340C2semidirect
61. C4⋊(C71⋊C8)D4semidirect
294. C71#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#179862792
C2×#179860562
#225966922
#225897712
#225895262
#225892922
#225891032
#225890802
#225890402
#225883402
C4×#121×C712
#2470971