Finite Group Databasebeta

← #30545027

Subgroups — #30545027

Order 15616.

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×#11758222C211yesmaximal, characteristic
2#21454920C213yesmaximal, complemented
3#21454919C213yesmaximal, complemented
4#21454913C213yesmaximal, complemented
5#21454910C213yesmaximal, complemented
6#21454908C211yesmaximal, characteristic, complemented
7#21454907C211yesmaximal, characteristic, complemented
8#21454898C211yesmaximal, characteristic, complemented
9Q8×#11758189C211yesmaximal, characteristic, complemented
10#21454761C213yesmaximal, complemented
11#21454748C213yesmaximal, complemented
12C8×Q8×D61C211yesmaximal, characteristic, complemented
13#21454480C213yesmaximal
14#21454458C213yesmaximal
15C61×#2982C211yesmaximal, characteristic, complemented
16Q8×Dic122C411yescharacteristic
17#16840490C413yes—
18#16840472C413yes—
19#16840467C413yes—
20#16840467C413yes—
21Q8×D244C411yescharacteristic
22Q8×C22⋊D61C411yescharacteristic
23Q8×C22⋊D61C411yescharacteristic
24#16840319C2213yes—
25#16840316C2213yes—
26#16840316C2213yes—
27C4×#11758222C2213yes—
28C4×Q8×D61C2211yescharacteristic
29C4×Q8×D61C2211yescharacteristic
30#16839893C2213yes—
31#16839893C2213yes—
32#16839892C2213yes—
33#16839892C2216yes—
34#16839892C2213yes—
35#16839891C2216yes—
36#16839891C2216yes—
37#16839889C2213yes—
38#16839889C2213yes—
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41C8×Dic122C2213yes—
42C8×Dic122C2213yes—
43C8×Dic122C2216yes—
44#16839886C2213yes—
45#16839886C2213yes—
46#16839822C2213yes—
47#16839822C2213yes—
48#16839821C2216yes—
49#16839820C2213yes—
50#16839820C2216yes—
51#16839820C2213yes—
52#16839819C2213yes—
53#16839818C2213yes—
54#16839815C2213yes—
55#16839814C2213yes—
56#16839813C2213yes—
57#16839813C2213yes—
58#16839787C2213yes—
59#16839787C2213yes—
60#16839784C2213yes—
61C4⋊C8×D61C2213yes—
62#16839727C2211yescharacteristic
63#16839727C2213yes—
64#16839716C2216yes—
65C4×#11758189C2213yes—
66C4×C8×D61C2213yes—
67C2×C4×Q8×C61C2211yescharacteristic
68C8×Q8×C61C2211yescharacteristic
69C8×Q8×C61C2211yescharacteristic
70C8×Q8×C61C2213yes—
71C8×Q8×C61C2213yes—
72C61×#431C2213yes—
73C61×#431C2213yes—
74C2×#11758222C2311yescharacteristic
75C2×Q8×D61C2×C411yescharacteristic
76C2×Q8×D61C2×C411yescharacteristic
77C2×#11758189C2311yescharacteristic
78C2×#11758189C2313yes—
79C2×C8×D61C2311yescharacteristic
80C2×C8×D61C2313yes—
81#14519777C2×C413yes—
82#14519777C2×C413yes—
83#14519776C2×C413yes—
84#14519776C2×C413yes—
85#14519775C2×C413yes—
86#14519775C2×C413yes—
87#14519769C2×C416yes—
88#14519767C2×C416yes—
89#14519767C2×C416yes—
90#14519757C2×C413yes—
91#14519757C2×C413yes—
92#14519756C2×C416yes—
93#14519756C2×C413yes—
94#14519756C2×C413yes—
95#14519755C2×C416yes—
96#14519755C2×C416yes—
97#14519753C2×C413yes—
98#14519753C2×C413yes—
99Q8×Dic61C2×C411yescharacteristic
100Q8×Dic61C2×C411yescharacteristic
101C4×Dic122C2×C413yes—
102C4×Dic122C2×C413yes—
103C4×Dic122C2313yes—
104C4×Dic122C2313yes—
105C4×Dic122C2316yes—
106#14519742C2×C413yes—
107#14519742C2×C413yes—
108C4×D244C2×C413yes—
109C4×D244C2×C413yes—
110C4×C22⋊D61C2×C413yes—
111C4×C22⋊D61C2×C413yes—
112C4×C22⋊D61C2×C413yes—
113C4×C22⋊D61C2×C413yes—
114#14519736C2313yes—
115#14519734C2313yes—
116#14519734C2313yes—
117C4⋊C4×D61C2×C413yes—
118C4⋊C4×D61C2×C413yes—
119C42×D61C2313yes—
120C42×D61C2313yes—
121C4⋊#92418C2313yes—
122C4⋊#92418C2313yes—
123C4⋊#92418C2313yes—
124C4⋊#92418C2313yes—
125#14519706C2316yes—
126#14519706C2316yes—
127#14519706C2313yes—
128#14519706C2313yes—
129#14519706C2313yes—
130#14519706C2313yes—
131C22⋊#92418C2311yescharacteristic
132C22⋊#92418C2313yes—
133#14519689C2311yescharacteristic
134#14519689C2311yescharacteristic
135#14519689C2313yes—
136#14519689C2313yes—
137#14519688C2316yes—
138#14519687C2313yes—
139#14519687C2313yes—
140#14519687C2316yes—
141C4×#92418C2313yes—
142C4×#92418C2313yes—
143C8×Dic61C2313yes—
144C8×Dic61C2313yes—
145C8×Dic61C2316yes—
146C22×Q8×C61C2×C411yescharacteristic
147C4×Q8×C61C2×C413yes—
148C4×Q8×C61C2×C411yescharacteristic
149C4×Q8×C61C2311yescharacteristic
150C4×Q8×C61C2313yes—
151C2×C4⋊C4×C61C2×C413yes—
152C2×C42×C61C2313yes—
153C4⋊C8×C61C2313yes—
154C4⋊C8×C61C2313yes—
155C4⋊C8×C61C2313yes—
156C4⋊C8×C61C2313yes—
157C22⋊C8×C61C2311yescharacteristic
158C22⋊C8×C61C2313yes—
159C4×C8×C61C2313yes—
160C4×C8×C61C2313yes—
161C4×C8×C61C2316yes—
162C2×Q8×C61—24no—
163C2×Q8×C61C22×C413yes—
164C2×Q8×C61C22×C411yescharacteristic
165C22×C4×C61C22×C413yes—
166C22×C4×C61C22×C413yes—
167C22×C4×C61C2411yescharacteristic
168C2×C22⋊D61C22×C411yescharacteristic
169C2×C22⋊D61C22×C411yescharacteristic
170Q8×D61—24no—
171Q8×D61—24no—
172#11758222—22no—
173#11758222—22no—
174C2×D244C22×C411yescharacteristic
175C2×C4×D61C22×C413yes—
176C2×C4×D61C22×C413yes—
177C2×C4×D61C22×C413yes—
178C2×C4×D61C22×C413yes—
179C2×C4×D61C2411yescharacteristic
180C2×C4×D61C2411yescharacteristic
181C2×Dic122C22×C413yes—
182C2×Dic122C22×C413yes—
183C2×Dic122C22×C411yescharacteristic
184C2×Dic122C22×C416yes—
185C2×C8×C61C2411yescharacteristic
186C2×C8×C61C2413yes—
187C2×C8×C61C2411yescharacteristic
188C2×C8×C61C2413yes—
189C4⋊C4×C61C22×C413yes—
190C4⋊C4×C61C22×C416yes—
191C4⋊C4×C61C22×C413yes—
192C42×C61C22×C413yes—
193C42×C61C22×C413yes—
194C42×C61C2413yes—
195C42×C61C2413yes—
196C22⋊Dic61C22×C413yes—
197C22⋊Dic61C22×C413yes—
198#11758198C22×C413yes—
199#11758198C22×C413yes—
200#11758198C22×C413yes—
201#11758198C22×C413yes—
202C4⋊Dic61C22×C413yes—
203C4⋊Dic61C22×C413yes—
204C4⋊Dic61C22×C413yes—
205C4⋊Dic61C22×C413yes—
206C4⋊Dic61C22×C416yes—
207#11758196C22×C416yes—
208#11758196C22×C416yes—
209#11758196C22×C413yes—
210#11758196C22×C413yes—
211#11758196C22×C413yes—
212#11758196C22×C413yes—
213#11758196C22×C416yes—
214#11758196C22×C416yes—
215C4×Dic61C22×C413yes—
216C4×Dic61C22×C413yes—
217C4×Dic61C22×C413yes—
218C4×Dic61C22×C413yes—
219C4×Dic61C2413yes—
220C4×Dic61C2413yes—
221C2×#92418C2411yescharacteristic
222C2×#92418C2413yes—
223C2×#92418C2413yes—
224C2×#92418C2411yescharacteristic
225#11758189—22no—
226#11758189—26no—
227C8×D61—26no—
228C8×D61—22no—
229C23×C61C23×C411yescharacteristic
230C22×D61C23×C411yescharacteristic
231C22×D61C23×C411yescharacteristic
232Q8×C61—26no—
233Q8×C61#13114yes—
234C2×C4×C61—26no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—43no—
238C2×C4×C61C23×C413yes—
239C2×C4×C61C23×C413yes—
240C2×C4×C61C23×C413yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C23×C413yes—
243C2×C4×C61C2511yescharacteristic
244C22⋊D61—22no—
245C22⋊D61—22no—
246C2×Dic61C23×C411yescharacteristic
247C2×Dic61C23×C413yes—
248C2×Dic61C23×C413yes—
249C2×Dic61C23×C413yes—
250C2×Dic61C23×C413yes—
251C2×Dic61C23×C411yescharacteristic
252D244—22no—
253C4×D61—26no—
254C4×D61—26no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—43no—
258C4×D61—43no—
259C4×D612−1+412yes—
260C4×D612−1+412yes—
261Dic122—26no—
262Dic122—26no—
263Dic122—22no—
264Dic122—46no—
265C8×C61—23no—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—23no—
269#92418—21no—
270#92418—23no—
271#92418—23no—
272#92418—21no—
273#60253—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C24×C411yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61#51912yes—
284C4×C61—21no—
285C4×C61—23no—
286C4×C61—23no—
287C4×C61—23no—
288C4×C61C2×#13116yes—
289C4×C61C2×2−1+411yescharacteristic
290C4×C61C2×2−1+411yescharacteristic
291Dic61—21no—
292Dic61—23no—
293Dic61—23no—
294Dic61—23no—
295Dic61—23no—
296Dic61#51912yes—
297Q8×#41—611no—
298#2986—613no—
299#2986—613no—
300#2985—613no—
301#2985—613no—
302#2984—611no—
303#2984—611no—
304#2982—611no—
305#2982D6111yescharacteristic, complemented
306Q8×M16—611no—
307#2962—613no—
308#2960—613no—
309C2×C8×Q8—611no—
310#2925—613no—
311#2922—613no—
312C2×C61—41no—
313C2×C61C2×#51911yescharacteristic
314C2×C61C22×#13111yescharacteristic
315C2×C61#357311yescharacteristic
316D61—22no—
317D61—41no—
318Q82—611no—
319#557—613no—
320#554—613no—
321#554—613no—
322#554—613no—
323D4×Q8—611no—
324D4×Q8—611no—
325D4×Q8—611no—
326#533—613no—
327#533—613no—
328#533—613no—
329C4×#41—613no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8C2×D6111yescharacteristic
333#446—613no—
334#446—613no—
335#446—613no—
336#446—616no—
337#446—616no—
338#446—613no—
339#446—613no—
340#446—616no—
341#446—613no—
342C8×Q8—611no—
343C8×Q8—611no—
344C8×Q8—613no—
345C8×Q8—613no—
346C8×Q8—613no—
347C8×Q8—613no—
348C8×Q8—616no—
349C8×Q8C2×D6111yescharacteristic
350C8×Q8C2×D6113yes—
351C8×Q8C2×D6113yes—
352C8×Q8C2×D6111yescharacteristic
353#433—613no—
354#433—613no—
355#433—613no—
356#433—613no—
357#433—616no—
358#433—616no—
359#432—613no—
360#432—613no—
361#431—613no—
362#431—613no—
363#431—613no—
364#431—613no—
365#431C2×D6113yes—
366#431C2×D6113yes—
367#424—613no—
368#424—613no—
369#423—613no—
370C2×C4⋊C8—613no—
371#408—611no—
372#408—613no—
373#405—616no—
374C4×M16—613no—
375C2×C4×C8—613no—
376C61#6025311yescharacteristic, complemented, Sylow
377C2×#41—611no—
378C22×Q8—611no—
379C22×Q8—611no—
380C22×Q8C4×D6111yescharacteristic
381C2×M16—611no—
382C2×M16—613no—
383C22×C8—611no—
384C22×C8—613no—
385C4⋊Q8—613no—
386C4⋊Q8—613no—
387C4⋊Q8—613no—
388C4⋊Q8—613no—
389C4⋊Q8—613no—
390C4⋊Q8—613no—
391#125—616no—
392#125—616no—
393#125—616no—
394C22⋊Q8—613no—
395C22⋊Q8—613no—
396C22⋊Q8—613no—
397C22⋊Q8—613no—
398C22⋊Q8—616no—
399C22⋊Q8—613no—
400C22⋊Q8—613no—
401C22⋊Q8—616no—
402C22⋊Q8—616no—
403C4×Q8—611no—
404C4×Q8—611no—
405C4×Q8—613no—
406C4×Q8—613no—
407C4×Q8—613no—
408C4×Q8—613no—
409C4×Q8—613no—
410C4×Q8—613no—
411C4×Q8—616no—
412C4×Q8C4×D6111yescharacteristic
413C4×Q8C4×D6113yes—
414C4×Q8C22×D6111yescharacteristic
415C4×Q8C22×D6113yes—
416C4×D4—613no—
417C4×D4—613no—
418C4×D4—613no—
419C4×D4—613no—
420C4×D4—613no—
421C4×D4—613no—
422#117—613no—
423#117—613no—
424#117—613no—
425C2×C4⋊C4—613no—
426C2×C4⋊C4—613no—
427C2×C4⋊C4C4×D6113yes—
428C2×C42—613no—
429C2×C42—613no—
430C2×C42C22×D6113yes—
431C4⋊C8—616no—
432C4⋊C8—616no—
433C4⋊C8—613no—
434C4⋊C8—613no—
435C4⋊C8—613no—
436C4⋊C8—613no—
437C4⋊C8—613no—
438C4⋊C8—613no—
439C4⋊C8—613no—
440C4⋊C8—613no—
441C4⋊C8C22×D6113yes—
442C4⋊C8C22×D6113yes—
443C4⋊C8C22×D6113yes—
444C4⋊C8C22×D6113yes—
445C22⋊C8—611no—
446C22⋊C8—613no—
447C22⋊C8—611no—
448C22⋊C8—611no—
449C22⋊C8—613no—
450C22⋊C8—613no—
451C22⋊C8C22×D6111yescharacteristic
452C22⋊C8C22×D6113yes—
453#97—613no—
454#97—613no—
455#97—616no—
456#97—616no—
457C4×C8—613no—
458C4×C8—613no—
459C4×C8—616no—
460C4×C8—613no—
461C4×C8—613no—
462C4×C8C22×D6113yes—
463C4×C8C22×D6113yes—
464C4×C8C22×D6116yes—
465#41—1222no—
466#41—1222no—
467C2×Q8—24no—
468C2×Q8—611no—
469C2×Q8—613no—
470C2×Q8—613no—
471C2×Q8—616no—
472C2×Q8—1224no—
473C2×Q8—1224no—
474C2×Q8C2×C4×D6113yes—
475C2×Q8C2×C4×D6111yescharacteristic
476C2×D4—611no—
477C2×D4—611no—
478C2×D4—611no—
479C22×C4—611no—
480C22×C4—613no—
481C22×C4—613no—
482C22×C4—611no—
483C22×C4—613no—
484C22×C4—613no—
485C22×C4C2×C4×D6113yes—
486C22×C4C2×C4×D6113yes—
487C22×C4C23×D6111yescharacteristic
488M16—1222no—
489M16—1226no—
490C2×C8—611no—
491C2×C8—613no—
492C2×C8—611no—
493C2×C8—613no—
494C2×C8—1226no—
495C2×C8—1222no—
496C2×C8C23×D6113yes—
497C2×C8C23×D6111yescharacteristic
498C2×C8C23×D6111yescharacteristic
499C2×C8C23×D6113yes—
500C4⋊C4—616no—
501C4⋊C4—616no—
502C4⋊C4—613no—
503C4⋊C4—613no—
504C4⋊C4—613no—
505C4⋊C4—613no—
506C4⋊C4—613no—
507C4⋊C4—613no—
508C4⋊C4—616no—
509C4⋊C4—616no—
510C4⋊C4—613no—
511C4⋊C4—613no—
512C4⋊C4—616no—
513C4⋊C4C2×C4×D6113yes—
514C4⋊C4C2×C4×D6116yes—
515C4⋊C4C2×C4×D6113yes—
516C22⋊C4—613no—
517C22⋊C4—613no—
518C22⋊C4—613no—
519C22⋊C4—613no—
520C22⋊C4—613no—
521C22⋊C4—613no—
522C42—613no—
523C42—613no—
524C42—613no—
525C42—613no—
526C42—613no—
527C42—613no—
528C42C2×C4×D6113yes—
529C42C2×C4×D6113yes—
530C42C23×D6113yes—
531C42C23×D6113yes—
532C23—611no—
533C23—611no—
534C23C22×C4×D6111yescharacteristic
535Q8—26no—
536Q8—1222no—
537Q8—1226no—
538Q8—1226no—
539Q8—2446no—
540Q8#1451978314yes—
541D4—1222no—
542D4—1222no—
543D4—1222no—
544C2×C4—26no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—43no—
548C2×C4—611no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—613no—
553C2×C4—613no—
554C2×C4—613no—
555C2×C4—613no—
556C2×C4—1226no—
557C2×C4—1226no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—2443no—
561C2×C4—2443no—
562C2×C4C22×C4×D6113yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C22×C4×D6113yes—
565C2×C4C22×C4×D6113yes—
566C2×C4C22×C4×D6113yes—
567C2×C4C24×D6111yescharacteristic
568C8—23no—
569C8—21no—
570C8—21no—
571C8—23no—
572C8—1221no—
573C8—1223no—
574C8—1221no—
575C8—1223no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—612no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22C23×C4×D6111yescharacteristic
586C4—23no—
587C4—23no—
588C4—23no—
589C4—21no—
590C4—612no—
591C4—1223no—
592C4—1223no—
593C4—1223no—
594C4—1223no—
595C4—1221no—
596C4C2×#1451978316yes—
597C42−1+4×D6111yescharacteristic
598C42−1+4×D6111yescharacteristic
599C2—41no—
600C2—1222no—
601C2—2441no—
602C2#519×D6111yescharacteristic
603C2C22×#1451978311yescharacteristic
604C2#2146053211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D61
commutator, socleC22×C61C24×C4
FittingC61×#2982C2
prime radicalC2×#11758222C23
cosocleC2×C4×C61C25
hypercenter#2982D61
hypocenterC61#60253
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21454920C2semidirect
3. #21454919C2semidirect
4. #21454913C2semidirect
5. #21454910C2semidirect
6. #21454908C2semidirect
7. #21454907C2semidirect
8. #21454898C2semidirect
9. Q8×#11758189C2semidirect
10. #21454761C2semidirect
11. #21454748C2semidirect
12. C8×Q8×D61C2semidirect
15. C61×#2982C2semidirect
305. #2982D61semidirect
376. C61#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582222
#214549202
#214549192
#214549132
#214549102
#214549082
#214549072
#214548982
Q8×#117581892
#214547612
#214547482
C8×Q8×D612
#214544802
#214544582
C61×#29822
#6025361