Finite Group Databasebeta

← #11859124

Subgroups — #11859124

Order 1152.

863 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594954C211yesmaximal, characteristic
2#10592828C211yesmaximal, characteristic
3C3×#5122C211yesmaximal, characteristic, complemented
4C3×#5118C211yesmaximal, characteristic, complemented
5C2×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10591073C211yesmaximal, characteristic
7#10591057C211yesmaximal, characteristic, complemented
8#10590665C212yesmaximal, complemented
9#10590648C212yesmaximal, complemented
10#10590647C212yesmaximal, complemented
11#10590643C212yesmaximal, complemented
12#79040—31nomaximal
13#78552—31nomaximal
14C2×#3828C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16C22×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#926C2211yescharacteristic
18C2×C3×C4×Dic3C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21#63952C412yes—
22#63952C412yes—
23#63894C411yescharacteristic
24#63894C411yescharacteristic
25C4×#624C2212yes—
26#63874C2212yes—
27#63844C411yescharacteristic
28#63844C411yescharacteristic
29#63647C2212yes—
30#63647C2212yes—
31C3×#815C2212yes—
32C3×Dic3⋊C8C2212yes—
33C3×Dic3⋊C8C2212yes—
34C3×C8×Dic3C2212yes—
35Dic3⋊(C3⋊C8)C2211yescharacteristic
36Dic3⋊(C3⋊C8)C2211yescharacteristic
37Dic3⋊(C3⋊C8)C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2211yescharacteristic
39#63557C2212yes—
40#63557C2212yes—
41#63554C2212yes—
42Dic3×C3⋊C8C2212yes—
43#5522—31no—
44C4×#235—31no—
45C3×#423—31no—
46C3×#405—31no—
47#5129—31no—
48#5129—31no—
49#5129—31no—
50#5129—61no—
51#5122—31no—
52#5122D311yescharacteristic, complemented
53#5118D311yescharacteristic, complemented
54#5110—31no—
55C2×Dic3⋊C8—31no—
56#4997—31no—
57C2×C4⋊(C3⋊C8)—31no—
58#4840—32no—
59#4840—32no—
60#4835—32no—
61#4833—32no—
62#4709—32no—
63#4705—32no—
64#4690—32no—
65#4685—32no—
66C22×C32×C4C2311yescharacteristic
67C2×#638C2×C412yes—
68#3828—22no—
69#3828—22no—
70C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
71C2×C4×C3⋊D3C2312yes—
72C2×#634C2×C411yescharacteristic
73C22×C3×Dic3C2×C411yescharacteristic
74C22×C3×Dic3C2×C411yescharacteristic
75C2×#624C2×C412yes—
76C2×#624C2×C412yes—
77C32×M16—22no—
78C2×C32×C8C2312yes—
79C2×C3⋊(C3⋊C8)—22no—
80C2×C3⋊(C3⋊C8)C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C3×#217C2×C412yes—
83C3×C4⋊Dic3C2×C411yescharacteristic
84C3×C4⋊Dic3C2×C411yescharacteristic
85C3×C4×Dic3C2×C411yescharacteristic
86C3×C4×Dic3C2×C411yescharacteristic
87C3×C4×Dic3C2311yescharacteristic
88C3×C4×Dic3C2311yescharacteristic
89C3×C4×Dic3C2312yes—
90C3×#208—22no—
91C2×C3×C3⋊C8—22no—
92C2×C3×C3⋊C8C2311yescharacteristic
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2312yes—
95#3724C2×C412yes—
96#3724C2×C412yes—
97#3722C2×C412yes—
98#3722C2×C412yes—
99Dic32C2×C412yes—
100Dic32C2×C412yes—
101#2925—91noSylow
102C2×#235—31no—
103C2×#235—31no—
104C2×#235—61no—
105C2×C3×M16—21no—
106C2×C3×M16—31no—
107C2×C3×M16C2×D311yescharacteristic
108C2×C3×M16C2×D311yescharacteristic
109C22×C3×C8—31no—
110C3×#117—31no—
111C2×C3×C42—31no—
112C4×#67—32no—
113C4×#67—32no—
114#926C2×D311yescharacteristic
115C2×C4×Dic3C2×D311yescharacteristic
116C2×#208C2×D311yescharacteristic
117C22×C3⋊C8—31no—
118C22×C3⋊C8—31no—
119C22×C3⋊C8—61no—
120C22×C3⋊C8C2×D311yescharacteristic
121#893—31no—
122#893—31no—
123#887—31no—
124#887—31no—
125#67⋊C4—32no—
126#67⋊C4—32no—
127C4×D12—31no—
128C4×D12—31no—
129#872—32no—
130#872—32no—
131C42×D3—32no—
132C42×D3—32no—
133C4×Dic6—31no—
134C4×Dic6—31no—
135C3×C4⋊C8—32no—
136C3×C4⋊C8—32no—
137C3×#97—32no—
138C3×C4×C8—32no—
139(C2×D3)⋊C8—32no—
140(C2×D3)⋊C8—32no—
141(C2×D3)⋊C8—32no—
142(C2×D3)⋊C8—32no—
143(C2×D3)⋊C8—32no—
144(C2×D3)⋊C8—32no—
145(C2×D3)⋊C8—32no—
146(C2×D3)⋊C8—32no—
147(C2×D3)⋊C8—62no—
148(C2×D3)⋊C8—62no—
149#815—32no—
150#815C2×D312yes—
151Dic3⋊C8—31no—
152Dic3⋊C8—31no—
153Dic3⋊C8—31no—
154Dic3⋊C8—31no—
155Dic3⋊C8C2×D312yes—
156Dic3⋊C8C2×D312yes—
157C8×Dic3—32no—
158C8×Dic3C2×D312yes—
159C4⋊(C3⋊C8)—31no—
160C4⋊(C3⋊C8)—31no—
161C4⋊(C3⋊C8)—31no—
162C4⋊(C3⋊C8)—31no—
163#803—32no—
164C4×C3⋊C8—32no—
165C23×C32C22×C411yescharacteristic
166C22×C3⋊D3C22×C412yes—
167C2×C32×C4C22×C411yescharacteristic
168C2×C32×C4#4112yes—
169C2×C32×C4#4112yes—
170C2×C32×C4C2411yescharacteristic
171#638—24no—
172C2×C3⋊Dic3C22×C412yes—
173C4⋊(C3⋊D3)—22no—
174C4×C3⋊D3—22no—
175C4×C3⋊D3—22no—
176#634—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3C22×C411yescharacteristic
180C2×C3×Dic3C22×C412yes—
181C2×C3×Dic3C22×C412yes—
182C2×C3×Dic3C22×C411yescharacteristic
183C2×C3×Dic3C22×C411yescharacteristic
184C2×C3×Dic3C22×C411yescharacteristic
185#624—24no—
186#624—42no—
187C32×C8—22no—
188C3⋊(C3⋊C8)—21no—
189C3⋊(C3⋊C8)—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—22no—
193C4×#41—91no—
194#433—92no—
195#433—92no—
196#432—92no—
197#431—92no—
198#423—31no—
199C2×C4⋊C8—91no—
200#408—91no—
201#408—91no—
202#405—31no—
203#405—91no—
204C22×C3×C4—21no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—32no—
210C2×#67—32no—
211C2×#67—62no—
212C22×Dic3C4×D311yescharacteristic
213C22×Dic3C4×D311yescharacteristic
214#235—62no—
215#235—62no—
216#235—62no—
217#235—62no—
218#235—122no—
219#235—122no—
220C2×D12—31no—
221C2×D12—31no—
222C2×D12—61no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—32no—
229C2×C4×D3—62no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—61no—
233C3×M16—22no—
234C3×M16—22no—
235C3×M16—42no—
236C3×M16—62no—
237C2×C3×C8—22no—
238C2×C3×C8—32no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—62no—
242C2×C3×C8C22×D312yes—
243C2×C3×C8C22×D312yes—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C22⋊C4—32no—
247C3×C42—31no—
248C3×C42—31no—
249C3×C42—32no—
250C3×C42—31no—
251C3×C42—31no—
252#217C4×D312yes—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257C4⋊Dic3C4×D311yescharacteristic
258C4⋊Dic3C4×D311yescharacteristic
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262Dic3⋊C4—32no—
263C4×Dic3—32no—
264C4×Dic3—32no—
265C4×Dic3—32no—
266C4×Dic3—32no—
267C4×Dic3C4×D311yescharacteristic
268C4×Dic3C4×D311yescharacteristic
269C4×Dic3C22×D312yes—
270C4×Dic3C22×D311yescharacteristic
271C4×Dic3C22×D311yescharacteristic
272#208—22no—
273C2×C3⋊C8—22no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8—62no—
281C2×C3⋊C8—62no—
282C2×C3⋊C8—122no—
283C2×C3⋊C8C22×D312yes—
284C2×C3⋊C8C22×D311yescharacteristic
285C2×C3⋊C8C22×D311yescharacteristic
286C22×C32—21no—
287C22×C32—21no—
288C22×C32#11712yes—
289C22×C32C23×C411yescharacteristic
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C32×C4#11712yes—
294C32×C4C2×#4111yescharacteristic
295C32×C4C2×#4111yescharacteristic
296C3⋊Dic3—22no—
297C3×Dic3—22no—
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—22no—
301C3×Dic3—21no—
302C3×Dic3#13112yes—
303C2×#41—91no—
304C2×M16—31no—
305C2×M16D3211yescharacteristic
306C22×C8—31no—
307C22×C8—91no—
308C4×Q8—91no—
309C4×Q8—91no—
310C4×D4—91no—
311C4×D4—91no—
312C4×D4—92no—
313C4×D4—92no—
314#117—31no—
315#117—92no—
316C2×C42—31no—
317C2×C42—92no—
318C4⋊C8—32no—
319C4⋊C8—32no—
320C4⋊C8—91no—
321C4⋊C8—91no—
322C4⋊C8—91no—
323C4⋊C8—91no—
324C22⋊C8—92no—
325C22⋊C8—92no—
326C22⋊C8—92no—
327C22⋊C8—92no—
328#97—32no—
329#97—92no—
330C4×C8—32no—
331C4×C8—92no—
332C23×C3—21no—
333C23×C3C2×C4×D311yescharacteristic
334C23×C3C2×C4×D311yescharacteristic
335C22×D3—32no—
336C22×D3—32no—
337C22×D3—62no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—62no—
349C2×C3×C4—121no—
350C2×C3×C4C2×C4×D311yescharacteristic
351C2×C3×C4C2×C4×D311yescharacteristic
352C2×C3×C4#23512yes—
353C2×C3×C4#23512yes—
354C2×C3×C4#23712yes—
355C2×C3×C4#23912yes—
356C2×C3×C4C23×D311yescharacteristic
357C2×C3×C4C23×D311yescharacteristic
358#67—64no—
359#67—64no—
360#67—124no—
361C2×Dic3—22no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—41no—
365C2×Dic3—62no—
366C2×Dic3C2×C4×D311yescharacteristic
367C2×Dic3C2×C4×D312yes—
368C2×Dic3C2×C4×D312yes—
369C2×Dic3C2×C4×D311yescharacteristic
370C2×Dic3C2×C4×D311yescharacteristic
371C2×Dic3C2×C4×D311yescharacteristic
372D12—62no—
373D12—62no—
374D12—122no—
375C4×D3—64no—
376C4×D3—62no—
377C4×D3—62no—
378C4×D3—62no—
379C4×D3—62no—
380C4×D3—64no—
381C4×D3—122no—
382C4×D3—122no—
383C4×D3—122no—
384C4×D3—122no—
385Dic6—62no—
386Dic6—62no—
387Dic6—122no—
388C3×C8—22no—
389C3×C8—22no—
390C3×C8—42no—
391C3×C8—62no—
392C3×C8—61no—
393C3×C8—61no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—22no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C3⋊C8—121no—
402C3⋊C8—121no—
403C2×C32—22no—
404C2×C32C2×#11711yescharacteristic
405C2×C32C2×#13111yescharacteristic
406C2×C32#56811yescharacteristic
407C3⋊D3—42no—
408#41—182no—
409#41—182no—
410C2×Q8—91no—
411C2×D4—92no—
412C2×D4—91no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—92no—
416C22×C4—92no—
417C22×C4—92no—
418C22×C4C2×D3211yescharacteristic
419M16—22no—
420M16—62no—
421C2×C8—31no—
422C2×C8—32no—
423C2×C8—31no—
424C2×C8—62no—
425C2×C8—91no—
426C2×C8—91no—
427C2×C8—182no—
428C2×C8C2×D3212yes—
429C4⋊C4—31no—
430C4⋊C4—31no—
431C4⋊C4—92no—
432C4⋊C4—92no—
433C22⋊C4—32no—
434C22⋊C4—92no—
435C22⋊C4—92no—
436C42—32no—
437C42—31no—
438C42—31no—
439C42—31no—
440C42—31no—
441C42—92no—
442C42—92no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3#87212yes—
452C22×C3#89212yes—
453C22×C3C22×C4×D311yescharacteristic
454C22×C3C22×C4×D311yescharacteristic
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—122no—
462C2×D3—122no—
463C2×D3—122no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—32no—
468C3×C4—62no—
469C3×C4—62no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4#87212yes—
474C3×C4#89212yes—
475C3×C4C2×#23511yescharacteristic
476C3×C4C2×#23511yescharacteristic
477C3×C4C2×#23711yescharacteristic
478C3×C4C2×#23911yescharacteristic
479Dic3—21no—
480Dic3—21no—
481Dic3—21no—
482Dic3—22no—
483Dic3—22no—
484Dic3—62no—
485Dic3—62no—
486Dic3—122no—
487Dic3#90712yes—
488C32#292511yescharacteristic, complemented, Sylow
489C23—92no—
490C23C4×D3211yescharacteristic
491Q8—182no—
492D4—184no—
493D4—182no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—62no—
501C2×C4—92no—
502C2×C4—121no—
503C2×C4—184no—
504C2×C4—182no—
505C2×C4—182no—
506C2×C4—362no—
507C2×C4#379912yes—
508C2×C4C4×D3211yescharacteristic
509C2×C4#380412yes—
510C2×C4C22×D3211yescharacteristic
511C8—22no—
512C8—61no—
513C8—61no—
514C8—62no—
515C8—181no—
516C8—181no—
517C2×C3—22no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3C2×#87211yescharacteristic
524C2×C3C2×#89211yescharacteristic
525C2×C3C2×#90711yescharacteristic
526C2×C3#574511yescharacteristic
527C2×C3D3×#13111yescharacteristic
528C2×C3#575011yescharacteristic
529D3—122no—
530D3—122no—
531D3—242no—
532C22—21no—
533C22—21no—
534C22—182no—
535C22—182no—
536C22—182no—
537C22#6383612yes—
538C22C2×C4×D3211yescharacteristic
539C4—32no—
540C4—61no—
541C4—62no—
542C4—61no—
543C4—61no—
544C4—62no—
545C4—182no—
546C4#6383612yes—
547C4C2×#379911yescharacteristic
548C4C2×#380411yescharacteristic
549C3—21no—
550C3#7855211yescharacteristic, complemented
551C3#7904011yescharacteristic, complemented
552C2—22no—
553C2—362no—
554C2D3×#90711yescharacteristic
555C2#1059470611yescharacteristic
556C2C2×#6383611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×M16C22
prime radicalC2×#3828C22
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2925
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C3×#5122C2semidirect
4. C3×#5118C2semidirect
5. C2×Dic3⋊(C3⋊C8)C2semidirect
7. #10591057C2semidirect
8. #10590665C2semidirect
9. #10590648C2semidirect
10. #10590647C2semidirect
11. #10590643C2semidirect
52. #5122D3semidirect
53. #5118D3semidirect
488. C32#2925semidirect
550. C3#78552semidirect
551. C3#79040semidirect

Maximal subgroups

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

subgroupindex
#105949542
#105928282
C3×#51222
C3×#51182
C2×Dic3⋊(C3⋊C8)2
#105910732
#105910572
#105906652
#105906482
#105906472
#105906432
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