Finite Group Databasebeta

← #32025025

Subgroups — #32025025

Order 18176.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556059C211yesmaximal, characteristic, complemented
2C2×#17986000C211yesmaximal, characteristic, complemented
3C2×#17985995C211yesmaximal, characteristic, complemented
4C2×#17985966C211yesmaximal, characteristic, complemented
5#22590932C211yesmaximal, characteristic, complemented
6#22589533C214yesmaximal, complemented
7#22589526C214yesmaximal, complemented
8#22588059C211yesmaximal, characteristic, complemented
9C2×#537×C71C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C23×C71⋊C8C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C2×#14556059—24no—
15C2×#14556059C2214yes—
16#17986000—24no—
17#17985995—24no—
18C2×#14556052C2212yes—
19C2×#14556052C2212yes—
20C2×#14556049C2212yes—
21C2×#14556049C2212yes—
22#17985966—24no—
23C2×C22⋊(C71⋊C8)C2214yes—
24#17985906—21no—
25#17985902—21no—
26#17985879—22no—
27#17985859C2214yes—
28#17985761C2214yes—
29C22×#41×C71C2211yescharacteristic
30C22×#41×C71C2211yescharacteristic
31#537×C71—24no—
32#537×C71—24no—
33C2×C22⋊Q8×C71—21no—
34C2×C22⋊Q8×C71C2212yes—
35C2×#121×C71—21no—
36C2×#121×C71—22no—
37C2×#121×C71C2212yes—
38C22×C4⋊C4×C71C2211yescharacteristic
39C23×Dic71—21no—
40C2×#11765849—22no—
41C2×#11765849—22no—
42C2×#11765849—22no—
43C22×C71⋊C8—22no—
44C22×C71⋊C8—22no—
45C22×C71⋊C8D414yescomplemented
46C22×C71⋊C8C2312yes—
47C22×C71⋊C8C2312yes—
48#14556087—22no—
49#14556087—22no—
50C2×C4⋊Dic71—22no—
51C2×C4⋊Dic71—22no—
52C2×C4⋊Dic71—21no—
53C2×C4⋊Dic71C2312yes—
54C2×C4⋊Dic71C2312yes—
55C2×C22⋊Dic71—21no—
56C2×C22⋊Dic71—24no—
57C2×C4×Dic71—21no—
58#14556059—28no—
59#14556052—24no—
60#14556052—24no—
61#14556049—24no—
62#14556049—24no—
63C22⋊(C71⋊C8)—28no—
64C71⋊#95—24no—
65C2×#41×C71—22no—
66C2×#41×C71—22no—
67C2×#41×C71—22no—
68C2×#41×C71—22no—
69C2×#41×C71—22no—
70C2×#41×C71—22no—
71C2×#41×C71—22no—
72C2×#41×C71—22no—
73C2×#41×C71—22no—
74C2×#41×C71—22no—
75C2×#41×C71—22no—
76C2×#41×C71D414yes—
77C22×Q8×C71D411yescharacteristic
78C22×Q8×C71C2311yescharacteristic
79C22×D4×C71—21no—
80C22×D4×C71—22no—
81C22×D4×C71D412yes—
82C22×D4×C71D411yescharacteristic
83C22×D4×C71C2311yescharacteristic
84C23×C4×C71—21no—
85C23×C4×C71—21no—
86C23×C4×C71C2312yes—
87C23×C4×C71C2311yescharacteristic
88C22⋊Q8×C71—28no—
89C22⋊Q8×C71—24no—
90C22⋊Q8×C71—24no—
91#121×C71—216no—
92#121×C71—28no—
93#121×C71—24no—
94#121×C71—24no—
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71D414yes—
99C2×C4⋊C4×C71C2312yes—
100C2×C22⋊C4×C71—22no—
101C2×C22⋊C4×C71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—42no—
105C22×Dic71—82no—
106#11765849—44no—
107C2×C71⋊C8—28no—
108C2×C71⋊C8—22no—
109C2×C71⋊C8—22no—
110C2×C71⋊C8—22no—
111C2×C71⋊C8—22no—
112C2×C71⋊C8—22no—
113C2×C71⋊C8C2×D414yes—
114C4⋊Dic71—22no—
115C4⋊Dic71—22no—
116C4⋊Dic71—22no—
117C4⋊Dic71—22no—
118C4⋊Dic71—42no—
119C22⋊Dic71—42no—
120C22⋊Dic71—84no—
121C4×Dic71—42no—
122C24×C71—21no—
123C24×C71C2×D412yes—
124C24×C71C2×D411yescharacteristic
125#41×C71—28no—
126#41×C71—28no—
127#41×C71—44no—
128#41×C71—44no—
129#41×C71—44no—
130#41×C71—44no—
131#41×C71—44no—
132#41×C71—44no—
133C2×Q8×C71—22no—
134C2×Q8×C71—22no—
135C2×Q8×C71—22no—
136C2×Q8×C71—21no—
137C2×Q8×C71—21no—
138C2×Q8×C71—21no—
139C2×Q8×C71—21no—
140C2×Q8×C71—21no—
141C2×Q8×C71C2×D412yes—
142C2×D4×C71—24no—
143C2×D4×C71—24no—
144C2×D4×C71—24no—
145C2×D4×C71—24no—
146C2×D4×C71—28no—
147C2×D4×C71—22no—
148C2×D4×C71—22no—
149C2×D4×C71—22no—
150C2×D4×C71—21no—
151C2×D4×C71—21no—
152C2×D4×C71—21no—
153C2×D4×C71—21no—
154C2×D4×C71—21no—
155C2×D4×C71—42no—
156C2×D4×C71—42no—
157C2×D4×C71—84no—
158C2×D4×C71C2×D412yes—
159C22×C4×C71—22no—
160C22×C4×C71—21no—
161C22×C4×C71—24no—
162C22×C4×C71—24no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—42no—
170C22×C4×C71—42no—
171C22×C4×C71—82no—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×D412yes—
177C22×C4×C71#4112yes—
178C22×C4×C71#4112yes—
179C22×C4×C71#4112yes—
180C22×C4×C71C2411yescharacteristic
181C4⋊C4×C71—28no—
182C4⋊C4×C71—24no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71C2×D414yes—
185C22⋊C4×C71—28no—
186C22⋊C4×C71—28no—
187C23×C71—22no—
188C23×C71—22no—
189C23×C71—22no—
190C23×C71—22no—
191C23×C71—22no—
192C23×C71—22no—
193C23×C71—24no—
194C23×C71—24no—
195C23×C71—41no—
196C23×C71—41no—
197C23×C71—42no—
198C23×C71—42no—
199C23×C71#12012yes—
200C23×C71#12112yes—
201C23×C71#12312yes—
202C23×C71C22×D411yescharacteristic
203Q8×C71—22no—
204Q8×C71—22no—
205Q8×C71—41no—
206Q8×C71—41no—
207Q8×C71—44no—
208D4×C71—22no—
209D4×C71—22no—
210D4×C71—48no—
211D4×C71—48no—
212D4×C71—41no—
213D4×C71—41no—
214D4×C71—44no—
215D4×C71—88no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—44no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—48no—
229C2×C4×C71—84no—
230C2×C4×C71#12012yes—
231C2×C4×C71#12114yes—
232C2×C4×C71#12114yes—
233C2×C4×C71#12112yes—
234C2×C4×C71#12314yes—
235C2×C4×C71#12312yes—
236C2×C4×C71#12414yes—
237C2×C4×C71C22×D411yescharacteristic
238C2×C4×C71C22×D411yescharacteristic
239C2×C4×C71C22×D411yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—44no—
247C2×Dic71—42no—
248C2×Dic71—84no—
249C71⋊C8—24no—
250C71⋊C8—42no—
251C22×C71—24no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—24no—
255C22×C71—44no—
256C22×C71—42no—
257C22×C71—42no—
258C22×C71—42no—
259C22×C71—42no—
260C22×C71—42no—
261C22×C71—42no—
262C22×C71—44no—
263C22×C71—82no—
264C22×C71—82no—
265C22×C71#39414yes—
266C22×C71C2×#12011yescharacteristic
267C22×C71C2×#12111yescharacteristic
268C22×C71C2×#12311yescharacteristic
269C22×C71#57712yes—
270C22×C71#57712yes—
271C4×C71—42no—
272C4×C71—42no—
273C4×C71—44no—
274C4×C71#39414yes—
275C4×C71C2×#12111yescharacteristic
276C4×C71C2×#12111yescharacteristic
277C4×C71C2×#12311yescharacteristic
278C4×C71C2×#12411yescharacteristic
279Dic71—82no—
280Dic71—82no—
281#23489—711nomaximal, Sylow
282C2×C71—24no—
283C2×C71—82no—
284C2×C71—82no—
285C2×C71C2×#39411yescharacteristic
286C2×C71#300911yescharacteristic
287C2×C71#301011yescharacteristic
288C2×C71#306012yes—
289C2×C71#309912yes—
290C2×#537D7111yescharacteristic
291C22×#106—711no—
292C2×#417—711no—
293C2×#416—711no—
294C2×#408—711no—
295#2311—711no—
296#2035—714no—
297#2034—714no—
298#1736—711no—
299C71#2348911yescharacteristic, complemented, Sylow
300C22×#41C2×D7111yescharacteristic
301C22×#41C2×D7111yescharacteristic
302C22×M16—711no—
303C23×C8—711no—
304#537—24no—
305#537—24no—
306C2×C22⋊Q8—21no—
307C2×C22⋊Q8C2×D7112yes—
308C2×#121—21no—
309C2×#121—22no—
310C2×#121C2×D7112yes—
311C2×#117—711no—
312C22×C4⋊C4—711no—
313C22×C4⋊C4C2×D7111yescharacteristic
314C2×#106—714no—
315C2×#106—1424no—
316#417—1424no—
317#416—1424no—
318C2×#103—712no—
319C2×#103—712no—
320C2×#102—712no—
321C2×#102—712no—
322#408—1424no—
323C2×C22⋊C8—714no—
324#391—1421no—
325#390—1421no—
326#386—1422no—
327#380—714no—
328#340—714no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41C22⋊D7114yes—
341C22×Q8C22⋊D7111yescharacteristic
342C22×Q8C22×D7111yescharacteristic
343C22×D4—21no—
344C22×D4—22no—
345C22×D4C22⋊D7112yes—
346C22×D4C22⋊D7111yescharacteristic
347C22×D4C22×D7111yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1421no—
351C23×C4C22×D7112yes—
352C23×C4C22×D7111yescharacteristic
353C2×M16—1422no—
354C2×M16—1422no—
355C2×M16—1422no—
356C22×C8—714no—
357C22×C8—712no—
358C22×C8—712no—
359C22×C8—1422no—
360C22×C8—1422no—
361C22⋊Q8—28no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364#121—216no—
365#121—28no—
366#121—24no—
367#121—24no—
368#117—1422no—
369#117—1422no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—712no—
374C2×C4⋊C4—712no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1422no—
377C2×C4⋊C4—1422no—
378C2×C4⋊C4C22⋊D7114yes—
379C2×C4⋊C4C22×D7112yes—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1424no—
384C2×C42—1421no—
385#106—1428no—
386#103—1424no—
387#103—1424no—
388#102—1424no—
389#102—1424no—
390C22⋊C8—1428no—
391#95—1424no—
392C24—21no—
393C24C2×C22⋊D7112yes—
394C24C2×C22⋊D7111yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401#41—44no—
402#41—44no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×C22⋊D7112yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—84no—
428C2×D4C2×C22⋊D7112yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—2842no—
443C22×C4—5682no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4D4×D7112yes—
447C22×C4C2×C22⋊D7112yes—
448C22×C4C2×C22⋊D7112yes—
449C22×C4C2×C22⋊D7112yes—
450C22×C4C2×C22⋊D7111yescharacteristic
451C22×C4#1176586412yes—
452C22×C4#1176586412yes—
453C22×C4#1176586412yes—
454C22×C4C23×D7111yescharacteristic
455M16—2844no—
456C2×C8—714no—
457C2×C8—1422no—
458C2×C8—1422no—
459C2×C8—1422no—
460C2×C8—1422no—
461C2×C8—1422no—
462C2×C8—1428no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—2842no—
467C4⋊C4—1422no—
468C4⋊C4—1422no—
469C4⋊C4—1422no—
470C4⋊C4—1422no—
471C4⋊C4C2×C22⋊D7114yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—2842no—
475C22⋊C4—5684no—
476C42—2842no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23#1455609712yes—
490C23#1455610012yes—
491C23#1455610912yes—
492C23C22×C22⋊D7111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—44no—
496Q8—41no—
497Q8—41no—
498D4—22no—
499D4—22no—
500D4—48no—
501D4—48no—
502D4—44no—
503D4—41no—
504D4—41no—
505D4—88no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—2842no—
521C2×C4—2842no—
522C2×C4—2842no—
523C2×C4—2844no—
524C2×C4—2842no—
525C2×C4—5684no—
526C2×C4#1455609712yes—
527C2×C4#1455609914yes—
528C2×C4#1455610012yes—
529C2×C4#1455610014yes—
530C2×C4#1455610912yes—
531C2×C4#1455610914yes—
532C2×C4#1455611514yes—
533C2×C4C2×D4×D7111yescharacteristic
534C2×C4C22×C22⋊D7111yescharacteristic
535C2×C4C22×C22⋊D7111yescharacteristic
536C2×C4C2×#1176586411yescharacteristic
537C2×C4C2×#1176586411yescharacteristic
538C2×C4C2×#1176586411yescharacteristic
539C8—2842no—
540C8—1424no—
541C22—24no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22#1798591314yes—
556C22C2×#1455609711yescharacteristic
557C22C2×#1455610011yescharacteristic
558C22C2×#1455610911yescharacteristic
559C22#1798680712yes—
560C22#1798681012yes—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—5682no—
565C4—5682no—
566C4#1798591314yes—
567C4C2×#1455609911yescharacteristic
568C4C2×#1455610011yescharacteristic
569C4C2×#1455610911yescharacteristic
570C4C2×#1455611511yescharacteristic
571C2—24no—
572C2—82no—
573C2—82no—
574C2C2×#1798591311yescharacteristic
575C2#2259761711yescharacteristic
576C2#2259762611yescharacteristic
577C2#2259799912yes—
578C2#2259832512yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
Fitting, prime radicalC2×#537×C71C2
socleC23×C71C22×D4
hypercenterC2×#537D71
hypocenterC71#23489
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. C22×#14556059C2semidirect
2. C2×#17986000C2semidirect
3. C2×#17985995C2semidirect
4. C2×#17985966C2semidirect
5. #22590932C2semidirect
6. #22589533C2semidirect
7. #22589526C2semidirect
8. #22588059C2semidirect
45. C22×C71⋊C8D4semidirect
299. C71#23489semidirect

Maximal subgroups

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

subgroupindex
C22×#145560592
C2×#179860002
C2×#179859952
C2×#179859662
#225909322
#225895332
#225895262
#225880592
C2×#537×C712
#2348971