Finite Group Databasebeta

← #27081301

Subgroups — #27081301

Order 12032.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10625350C211yesmaximal, characteristic
2#20252489C213yesmaximal, complemented
3#20252486C213yesmaximal, complemented
4#20252485C213yesmaximal, complemented
5#20252484C213yesmaximal, complemented
6#20252476C211yesmaximal, characteristic, complemented
7#20252469C211yesmaximal, characteristic, complemented
8#20252467C211yesmaximal, characteristic, complemented
9Q8×#10625324C211yesmaximal, characteristic, complemented
10#20252336C213yesmaximal, complemented
11#20252319C213yesmaximal, complemented
12C8×Q8×D47C211yesmaximal, characteristic, complemented
13#20252055C213yesmaximal
14#20252027C213yesmaximal
15C47×#2984C211yesmaximal, characteristic, complemented
16Q8×Dic94C411yescharacteristic
17#15689851C413yes—
18#15689833C413yes—
19#15689828C413yes—
20#15689828C413yes—
21Q8×D188C411yescharacteristic
22Q8×C22⋊D47C411yescharacteristic
23Q8×C22⋊D47C411yescharacteristic
24#15689680C2213yes—
25#15689677C2213yes—
26#15689677C2213yes—
27C4×#10625350C2213yes—
28C4×Q8×D47C2211yescharacteristic
29C4×Q8×D47C2211yescharacteristic
30#15689253C2213yes—
31#15689253C2213yes—
32#15689251C2216yes—
33#15689251C2216yes—
34#15689250C2213yes—
35#15689250C2216yes—
36#15689250C2213yes—
37Q8×C47⋊C8C2211yescharacteristic
38Q8×C47⋊C8C2211yescharacteristic
39Q8×C47⋊C8C2213yes—
40Q8×C47⋊C8C2213yes—
41C8×Dic94C2213yes—
42C8×Dic94C2213yes—
43#15689247C2213yes—
44#15689247C2213yes—
45#15689247C2216yes—
46#15689185C2216yes—
47#15689183C2213yes—
48#15689183C2213yes—
49#15689181C2213yes—
50#15689181C2216yes—
51#15689181C2213yes—
52#15689179C2213yes—
53#15689176C2213yes—
54#15689176C2213yes—
55#15689175C2213yes—
56#15689174C2213yes—
57#15689174C2213yes—
58#15689148C2213yes—
59#15689148C2213yes—
60#15689145C2213yes—
61C4⋊C8×D47C2213yes—
62#15689088C2211yescharacteristic
63#15689088C2213yes—
64#15689080C2216yes—
65C4×#10625324C2213yes—
66C4×C8×D47C2213yes—
67C2×C4×Q8×C47C2211yescharacteristic
68C47×#446C2213yes—
69C47×#446C2213yes—
70C8×Q8×C47C2211yescharacteristic
71C8×Q8×C47C2211yescharacteristic
72C47×#432C2213yes—
73C47×#431C2213yes—
74C2×#10625350C2311yescharacteristic
75C2×Q8×D47C2×C411yescharacteristic
76C2×Q8×D47C2×C411yescharacteristic
77C2×#10625324C2311yescharacteristic
78C2×#10625324C2313yes—
79C2×C8×D47C2311yescharacteristic
80C2×C8×D47C2313yes—
81#13094024C2×C413yes—
82#13094024C2×C413yes—
83#13094023C2×C413yes—
84#13094023C2×C413yes—
85#13094022C2×C413yes—
86#13094022C2×C413yes—
87#13094016C2×C416yes—
88#13094014C2×C416yes—
89#13094014C2×C416yes—
90#13094004C2×C413yes—
91#13094004C2×C413yes—
92#13094003C2×C413yes—
93#13094003C2×C413yes—
94#13094003C2×C416yes—
95#13094002C2×C416yes—
96#13094002C2×C416yes—
97#13094000C2×C413yes—
98#13094000C2×C413yes—
99Q8×Dic47C2×C411yescharacteristic
100Q8×Dic47C2×C411yescharacteristic
101C4×Dic94C2×C413yes—
102C4×Dic94C2×C413yes—
103C4×Dic94C2313yes—
104C4×Dic94C2313yes—
105C4×Dic94C2316yes—
106#13093989C2×C413yes—
107#13093989C2×C413yes—
108C4×D188C2×C413yes—
109C4×D188C2×C413yes—
110C4×C22⋊D47C2×C413yes—
111C4×C22⋊D47C2×C413yes—
112C4×C22⋊D47C2×C413yes—
113C4×C22⋊D47C2×C413yes—
114#13093983C2313yes—
115#13093981C2313yes—
116#13093981C2313yes—
117C4⋊C4×D47C2×C413yes—
118C4⋊C4×D47C2×C413yes—
119C42×D47C2313yes—
120C42×D47C2313yes—
121C4⋊(C47⋊C8)C2313yes—
122C4⋊(C47⋊C8)C2313yes—
123C4⋊(C47⋊C8)C2313yes—
124C4⋊(C47⋊C8)C2313yes—
125Dic47⋊C8C2313yes—
126Dic47⋊C8C2313yes—
127Dic47⋊C8C2316yes—
128Dic47⋊C8C2316yes—
129Dic47⋊C8C2313yes—
130Dic47⋊C8C2313yes—
131C22⋊(C47⋊C8)C2311yescharacteristic
132C22⋊(C47⋊C8)C2313yes—
133(C2×D47)⋊C8C2311yescharacteristic
134(C2×D47)⋊C8C2311yescharacteristic
135(C2×D47)⋊C8C2313yes—
136(C2×D47)⋊C8C2313yes—
137#13093934C2313yes—
138#13093934C2316yes—
139#13093934C2313yes—
140C4×C47⋊C8C2313yes—
141C4×C47⋊C8C2316yes—
142C4×C47⋊C8C2313yes—
143C8×Dic47C2313yes—
144C8×Dic47C2316yes—
145C8×Dic47C2313yes—
146C22×Q8×C47C2×C411yescharacteristic
147C4×Q8×C47C2×C411yescharacteristic
148C4×Q8×C47C2×C413yes—
149C4×Q8×C47C2311yescharacteristic
150C4×Q8×C47C2313yes—
151C2×C4⋊C4×C47C2×C413yes—
152C2×C42×C47C2313yes—
153C4⋊C8×C47C2313yes—
154C4⋊C8×C47C2313yes—
155C4⋊C8×C47C2313yes—
156C4⋊C8×C47C2313yes—
157C22⋊C8×C47C2311yescharacteristic
158C22⋊C8×C47C2313yes—
159#97×C47C2316yes—
160C4×C8×C47C2313yes—
161C4×C8×C47C2313yes—
162C2×Q8×C47—24no—
163C2×Q8×C47C22×C413yes—
164C2×Q8×C47C22×C411yescharacteristic
165C22×C4×C47C22×C413yes—
166C22×C4×C47C22×C413yes—
167C22×C4×C47C2411yescharacteristic
168C2×C22⋊D47C22×C411yescharacteristic
169C2×C22⋊D47C22×C411yescharacteristic
170Q8×D47—24no—
171Q8×D47—24no—
172#10625350—22no—
173#10625350—22no—
174C2×D188C22×C411yescharacteristic
175C2×C4×D47C22×C413yes—
176C2×C4×D47C22×C413yes—
177C2×C4×D47C22×C413yes—
178C2×C4×D47C22×C413yes—
179C2×C4×D47C2411yescharacteristic
180C2×C4×D47C2411yescharacteristic
181C2×Dic94C22×C413yes—
182C2×Dic94C22×C413yes—
183C2×Dic94C22×C411yescharacteristic
184C2×Dic94C22×C416yes—
185C2×C8×C47C2411yescharacteristic
186C2×C8×C47C2413yes—
187C2×C8×C47C2413yes—
188C2×C8×C47C2411yescharacteristic
189C4⋊C4×C47C22×C413yes—
190C4⋊C4×C47C22×C416yes—
191C4⋊C4×C47C22×C413yes—
192C42×C47C22×C413yes—
193C42×C47C22×C413yes—
194C42×C47C2413yes—
195C42×C47C2413yes—
196C22⋊Dic47C22×C413yes—
197C22⋊Dic47C22×C413yes—
198(C2×D47)⋊C4C22×C413yes—
199(C2×D47)⋊C4C22×C413yes—
200(C2×D47)⋊C4C22×C413yes—
201(C2×D47)⋊C4C22×C413yes—
202C4⋊Dic47C22×C413yes—
203C4⋊Dic47C22×C413yes—
204C4⋊Dic47C22×C413yes—
205C4⋊Dic47C22×C413yes—
206C4⋊Dic47C22×C416yes—
207Dic47⋊C4C22×C416yes—
208Dic47⋊C4C22×C416yes—
209Dic47⋊C4C22×C413yes—
210Dic47⋊C4C22×C413yes—
211Dic47⋊C4C22×C413yes—
212Dic47⋊C4C22×C413yes—
213Dic47⋊C4C22×C416yes—
214Dic47⋊C4C22×C416yes—
215C4×Dic47C22×C413yes—
216C4×Dic47C22×C413yes—
217C4×Dic47C22×C413yes—
218C4×Dic47C22×C413yes—
219C4×Dic47C2413yes—
220C4×Dic47C2413yes—
221C2×C47⋊C8C2413yes—
222C2×C47⋊C8C2411yescharacteristic
223C2×C47⋊C8C2413yes—
224C2×C47⋊C8C2411yescharacteristic
225#10625324—22no—
226#10625324—26no—
227C8×D47—26no—
228C8×D47—22no—
229C23×C47C23×C411yescharacteristic
230C22×D47C23×C411yescharacteristic
231C22×D47C23×C411yescharacteristic
232Q8×C47—26no—
233Q8×C47#13114yes—
234C2×C4×C47—26no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—43no—
238C2×C4×C47C23×C413yes—
239C2×C4×C47C23×C413yes—
240C2×C4×C47C23×C413yes—
241C2×C4×C47C23×C411yescharacteristic
242C2×C4×C47C23×C413yes—
243C2×C4×C47C2511yescharacteristic
244C22⋊D47—22no—
245C22⋊D47—22no—
246C2×Dic47C23×C411yescharacteristic
247C2×Dic47C23×C413yes—
248C2×Dic47C23×C413yes—
249C2×Dic47C23×C413yes—
250C2×Dic47C23×C413yes—
251C2×Dic47C23×C411yescharacteristic
252D188—22no—
253C4×D47—26no—
254C4×D47—26no—
255C4×D47—21no—
256C4×D47—21no—
257C4×D47—43no—
258C4×D47—43no—
259C4×D472−1+412yes—
260C4×D472−1+412yes—
261Dic94—26no—
262Dic94—26no—
263Dic94—22no—
264Dic94—46no—
265C8×C47—21no—
266C8×C47—23no—
267C8×C47—23no—
268C8×C47—21no—
269C47⋊C8—23no—
270C47⋊C8—21no—
271C47⋊C8—21no—
272C47⋊C8—23no—
273#60253—471nomaximal, Sylow
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—21no—
277C22×C47C24×C411yescharacteristic
278C2×D47—21no—
279C2×D47—21no—
280C2×D47—21no—
281C2×D47—21no—
282C2×D47—21no—
283C2×D47#51912yes—
284C4×C47—21no—
285C4×C47—23no—
286C4×C47—23no—
287C4×C47—23no—
288C4×C47C2×#13116yes—
289C4×C47C2×2−1+411yescharacteristic
290C4×C47C2×2−1+411yescharacteristic
291Dic47—21no—
292Dic47—23no—
293Dic47—23no—
294Dic47—23no—
295Dic47—23no—
296Dic47#51912yes—
297Q8×#41—471no—
298#2986—473no—
299#2986—473no—
300#2985—473no—
301#2985—473no—
302#2984—471no—
303#2984D4711yescharacteristic, complemented
304#2982—471no—
305#2982—471no—
306Q8×M16—471no—
307#2962—473no—
308#2960—473no—
309C2×C8×Q8—471no—
310#2925—473no—
311#2922—473no—
312C2×C47—41no—
313C2×C47C2×#51911yescharacteristic
314C2×C47C22×#13111yescharacteristic
315C2×C47#357311yescharacteristic
316D47—22no—
317D47—41no—
318Q82—471no—
319#557—473no—
320#554—473no—
321#554—473no—
322#554—473no—
323D4×Q8—471no—
324D4×Q8—471no—
325D4×Q8—471no—
326#533—473no—
327#533—473no—
328#533—473no—
329C4×#41—473no—
330C2×C4×Q8—471no—
331C2×C4×Q8—471no—
332C2×C4×Q8C2×D4711yescharacteristic
333#446—473no—
334#446—476no—
335#446—473no—
336#446—476no—
337#446—476no—
338#446—473no—
339#446—473no—
340#446C2×D4713yes—
341#446C2×D4713yes—
342C8×Q8—471no—
343C8×Q8—473no—
344C8×Q8—473no—
345C8×Q8—471no—
346C8×Q8—473no—
347C8×Q8—473no—
348C8×Q8—476no—
349C8×Q8—473no—
350C8×Q8—473no—
351C8×Q8C2×D4711yescharacteristic
352C8×Q8C2×D4711yescharacteristic
353#433—473no—
354#433—473no—
355#433—473no—
356#433—473no—
357#433—476no—
358#433—476no—
359#432—473no—
360#432C2×D4713yes—
361#431—473no—
362#431—473no—
363#431—473no—
364#431—473no—
365#431—473no—
366#431C2×D4713yes—
367#424—473no—
368#424—473no—
369#423—473no—
370C2×C4⋊C8—473no—
371#408—471no—
372#408—473no—
373#405—476no—
374C4×M16—473no—
375C2×C4×C8—473no—
376C47#6025311yescharacteristic, complemented, Sylow
377C2×#41—471no—
378C22×Q8—471no—
379C22×Q8—471no—
380C22×Q8C4×D4711yescharacteristic
381C2×M16—471no—
382C2×M16—473no—
383C22×C8—471no—
384C22×C8—473no—
385C4⋊Q8—473no—
386C4⋊Q8—473no—
387C4⋊Q8—473no—
388C4⋊Q8—473no—
389C4⋊Q8—473no—
390C4⋊Q8—473no—
391#125—476no—
392#125—476no—
393#125—476no—
394C22⋊Q8—473no—
395C22⋊Q8—473no—
396C22⋊Q8—473no—
397C22⋊Q8—473no—
398C22⋊Q8—476no—
399C22⋊Q8—473no—
400C22⋊Q8—473no—
401C22⋊Q8—476no—
402C22⋊Q8—476no—
403C4×Q8—471no—
404C4×Q8—471no—
405C4×Q8—473no—
406C4×Q8—473no—
407C4×Q8—473no—
408C4×Q8—473no—
409C4×Q8—473no—
410C4×Q8—473no—
411C4×Q8—476no—
412C4×Q8C4×D4711yescharacteristic
413C4×Q8C4×D4713yes—
414C4×Q8C22×D4711yescharacteristic
415C4×Q8C22×D4713yes—
416C4×D4—473no—
417C4×D4—473no—
418C4×D4—473no—
419C4×D4—473no—
420C4×D4—473no—
421C4×D4—473no—
422#117—473no—
423#117—473no—
424#117—473no—
425C2×C4⋊C4—473no—
426C2×C4⋊C4—473no—
427C2×C4⋊C4C4×D4713yes—
428C2×C42—473no—
429C2×C42—473no—
430C2×C42C22×D4713yes—
431C4⋊C8—476no—
432C4⋊C8—476no—
433C4⋊C8—473no—
434C4⋊C8—473no—
435C4⋊C8—473no—
436C4⋊C8—473no—
437C4⋊C8—473no—
438C4⋊C8—473no—
439C4⋊C8—473no—
440C4⋊C8—473no—
441C4⋊C8C22×D4713yes—
442C4⋊C8C22×D4713yes—
443C4⋊C8C22×D4713yes—
444C4⋊C8C22×D4713yes—
445C22⋊C8—471no—
446C22⋊C8—473no—
447C22⋊C8—473no—
448C22⋊C8—471no—
449C22⋊C8—471no—
450C22⋊C8—473no—
451C22⋊C8C22×D4711yescharacteristic
452C22⋊C8C22×D4713yes—
453#97—473no—
454#97—473no—
455#97—476no—
456#97C22×D4716yes—
457C4×C8—473no—
458C4×C8—473no—
459C4×C8—476no—
460C4×C8—473no—
461C4×C8—473no—
462C4×C8—476no—
463C4×C8C22×D4713yes—
464C4×C8C22×D4713yes—
465#41—942no—
466#41—942no—
467C2×Q8—24no—
468C2×Q8—471no—
469C2×Q8—473no—
470C2×Q8—473no—
471C2×Q8—476no—
472C2×Q8—944no—
473C2×Q8—944no—
474C2×Q8C2×C4×D4713yes—
475C2×Q8C2×C4×D4711yescharacteristic
476C2×D4—471no—
477C2×D4—471no—
478C2×D4—471no—
479C22×C4—471no—
480C22×C4—473no—
481C22×C4—473no—
482C22×C4—471no—
483C22×C4—473no—
484C22×C4—473no—
485C22×C4C2×C4×D4713yes—
486C22×C4C2×C4×D4713yes—
487C22×C4C23×D4711yescharacteristic
488M16—942no—
489M16—946no—
490C2×C8—473no—
491C2×C8—471no—
492C2×C8—471no—
493C2×C8—473no—
494C2×C8—946no—
495C2×C8—942no—
496C2×C8C23×D4711yescharacteristic
497C2×C8C23×D4713yes—
498C2×C8C23×D4713yes—
499C2×C8C23×D4711yescharacteristic
500C4⋊C4—476no—
501C4⋊C4—476no—
502C4⋊C4—473no—
503C4⋊C4—473no—
504C4⋊C4—473no—
505C4⋊C4—473no—
506C4⋊C4—476no—
507C4⋊C4—476no—
508C4⋊C4—473no—
509C4⋊C4—473no—
510C4⋊C4—473no—
511C4⋊C4—473no—
512C4⋊C4—476no—
513C4⋊C4C2×C4×D4713yes—
514C4⋊C4C2×C4×D4716yes—
515C4⋊C4C2×C4×D4713yes—
516C22⋊C4—473no—
517C22⋊C4—473no—
518C22⋊C4—473no—
519C22⋊C4—473no—
520C22⋊C4—473no—
521C22⋊C4—473no—
522C42—473no—
523C42—473no—
524C42—473no—
525C42—473no—
526C42—473no—
527C42—473no—
528C42C2×C4×D4713yes—
529C42C2×C4×D4713yes—
530C42C23×D4713yes—
531C42C23×D4713yes—
532C23—471no—
533C23—471no—
534C23C22×C4×D4711yescharacteristic
535Q8—26no—
536Q8—942no—
537Q8—946no—
538Q8—946no—
539Q8—1886no—
540Q8#1309403214yes—
541D4—942no—
542D4—942no—
543D4—942no—
544C2×C4—26no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—43no—
548C2×C4—471no—
549C2×C4—472no—
550C2×C4—472no—
551C2×C4—471no—
552C2×C4—473no—
553C2×C4—473no—
554C2×C4—473no—
555C2×C4—473no—
556C2×C4—946no—
557C2×C4—946no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—1883no—
561C2×C4—1883no—
562C2×C4C22×C4×D4713yes—
563C2×C4C22×C4×D4711yescharacteristic
564C2×C4C22×C4×D4713yes—
565C2×C4C22×C4×D4713yes—
566C2×C4C22×C4×D4713yes—
567C2×C4C24×D4711yescharacteristic
568C8—21no—
569C8—23no—
570C8—23no—
571C8—21no—
572C8—943no—
573C8—941no—
574C8—941no—
575C8—943no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—472no—
580C22—941no—
581C22—941no—
582C22—941no—
583C22—941no—
584C22—941no—
585C22C23×C4×D4711yescharacteristic
586C4—23no—
587C4—23no—
588C4—23no—
589C4—21no—
590C4—472no—
591C4—943no—
592C4—943no—
593C4—943no—
594C4—943no—
595C4—941no—
596C4C2×#1309403216yes—
597C42−1+4×D4711yescharacteristic
598C42−1+4×D4711yescharacteristic
599C2—41no—
600C2—942no—
601C2—1881no—
602C2#519×D4711yescharacteristic
603C2C22×#1309403211yescharacteristic
604C2#2025809911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC47×#2984C2
prime radicalC2×#10625350C23
cosocleC2×C4×C47C25
hypercenter#2984D47
hypocenterC47#60253
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #20252489C2semidirect
3. #20252486C2semidirect
4. #20252485C2semidirect
5. #20252484C2semidirect
6. #20252476C2semidirect
7. #20252469C2semidirect
8. #20252467C2semidirect
9. Q8×#10625324C2semidirect
10. #20252336C2semidirect
11. #20252319C2semidirect
12. C8×Q8×D47C2semidirect
15. C47×#2984C2semidirect
303. #2984D47semidirect
376. C47#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#106253502
#202524892
#202524862
#202524852
#202524842
#202524762
#202524692
#202524672
Q8×#106253242
#202523362
#202523192
C8×Q8×D472
#202520552
#202520272
C47×#29842
#6025347