Finite Group Databasebeta

← #11859493

Subgroups — #11859493

Order 1152.

1155 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596303C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic, complemented
3C2×#63817C211yesmaximal, characteristic, complemented
4C3×#5234C211yesmaximal, characteristic, complemented
5#10591148C214yesmaximal, complemented
6#10591120C211yesmaximal, characteristic, complemented
7#10591073C211yesmaximal, characteristic, complemented
8C2×#63553C211yesmaximal, characteristic, complemented
9#10591051C214yesmaximal, complemented
10#83786—31nomaximal
11#78657—31nomaximal
12C2×C32×#41C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14#64156C411yescharacteristic
15#64153C411yescharacteristic
16#64149C411yescharacteristic
17#64148C414yes—
18#64145C411yescharacteristic
19C4×#638C2214yes—
20C2×C4×C3⋊Dic3C2211yescharacteristic
21C3×#948—24no—
22C2×C3×#208C2212yes—
23C2×C3×#208C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C22×C3×C3⋊C8C2212yes—
27C22×C3×C3⋊C8C2211yescharacteristic
28C2×#3712C2212yes—
29#63817—24no—
30C2×#3710C2212yes—
31C3×#830C2214yes—
32#63572C2214yes—
33#63557C2214yes—
34#63553C2212yes—
35#63553C2212yes—
36C2×C3×#131—31no—
37#5832—31no—
38#5832—31no—
39#5832—61no—
40C2×#948D311yescharacteristic, complemented
41C2×#907—31no—
42C2×#901—31no—
43C3×#408—31no—
44#5234D311yescharacteristic, complemented
45#5129—31no—
46#5128—34no—
47#5118—31no—
48#5118—31no—
49#5110—31no—
50C8×#67—34no—
51C2×C8×Dic3—31no—
52C2×C8×Dic3—31no—
53#4729—34no—
54#67⋊C8—34no—
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2×C411yescharacteristic
58C2×C32×D4C2×C411yescharacteristic
59C2×C32×D4C2×C412yes—
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×#638C2×C412yes—
63C2×#638C2×C414yes—
64C22×C3⋊Dic3C2×C412yes—
65#3828—22no—
66#3828—22no—
67C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
68C2×C4×C3⋊D3C2312yes—
69C2×#634C2×C411yescharacteristic
70#3757C2×C414yes—
71#3752C2×C414yes—
72#3750C2×C414yes—
73C4×C3⋊Dic3C2×C411yescharacteristic
74C4×C3⋊Dic3C2×C411yescharacteristic
75C4×C3⋊Dic3C2312yes—
76C3×#208—22no—
77C3×#208—22no—
78C3×#208—24no—
79C2×C3×C3⋊C8—22no—
80C2×C3×C3⋊C8—24no—
81C2×C3×C3⋊C8D414yescomplemented
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2312yes—
84C2×C3×C3⋊C8C2312yes—
85C2×C3×C3⋊C8C2311yescharacteristic
86C2×C3×C3⋊C8C2311yescharacteristic
87#3712—24no—
88#3710—24no—
89#2931—91noSylow
90C2×C3×#41—21no—
91C2×C3×#41C2×D311yescharacteristic
92C2×C3×#41C2×D311yescharacteristic
93C2×#235—31no—
94C2×#235—31no—
95C2×#235—61no—
96C3×#131—64no—
97C2×C3×M16—32no—
98C2×C3×M16—31no—
99C2×C3×M16—31no—
100C22×C3×C8—31no—
101C22×C3×C8—32no—
102C22×C3×C8—31no—
103#948—24no—
104#947—31no—
105#947—31no—
106#947—61no—
107#944—31no—
108#944—31no—
109#944—61no—
110#940—31no—
111#940—31no—
112#940—61no—
113#939—34no—
114#939—34no—
115#939—64no—
116#936—31no—
117#936—31no—
118#936—61no—
119C4×#67—34no—
120C4×#67—34no—
121C4×#67—64no—
122C2×C4×Dic3—31no—
123C2×C4×Dic3—31no—
124C2×C4×Dic3—61no—
125C2×#208C2×D311yescharacteristic
126C2×#208C2×D312yes—
127C2×#208C2×D311yescharacteristic
128C22×C3⋊C8C2×D311yescharacteristic
129C22×C3⋊C8C2×D312yes—
130C22×C3⋊C8C2×D311yescharacteristic
131#907—64no—
132#901—64no—
133C2×ΓL(1,𝔽25)—32no—
134C2×ΓL(1,𝔽25)—32no—
135C2×D3×C8—32no—
136C2×D3×C8—32no—
137C3×C22⋊C8—34no—
138#830C2×D314yes—
139(C2×D3)⋊C8—34no—
140(C2×D3)⋊C8—34no—
141Dic3⋊C8—34no—
142Dic3⋊C8—34no—
143C8×Dic3—32no—
144C8×Dic3—32no—
145C8×Dic3—32no—
146C8×Dic3—32no—
147C23×C32C22×C412yes—
148C23×C32C22×C411yescharacteristic
149C22×C3⋊D3C22×C412yes—
150C32×Q8—22no—
151C32×D4—24no—
152C32×D4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4C22×C412yes—
156C2×C32×C4C22×C411yescharacteristic
157C2×C32×C4C2×D412yes—
158C2×C32×C4#4112yes—
159C2×C32×C4C2411yescharacteristic
160#638—24no—
161#638—28no—
162C2×C3⋊Dic3—24no—
163C2×C3⋊Dic3C22×C412yes—
164C2×C3⋊Dic3C22×C412yes—
165C2×C3⋊Dic3C22×C412yes—
166C4⋊(C3⋊D3)—22no—
167C4×C3⋊D3—22no—
168C4×C3⋊D3—22no—
169#634—22no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—22no—
172C3×C3⋊C8—22no—
173C3×C3⋊C8C2×D412yes—
174C3×C3⋊C8C2×D412yes—
175C2×#131—31no—
176C2×#131—91no—
177#532—91no—
178#436—94no—
179C8×D4—94no—
180#423—91no—
181#408—31no—
182#408—91no—
183C2×C4×C8—91no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—42no—
189C3×#41—42no—
190C2×C3×Q8—21no—
191C2×C3×Q8C4×D311yescharacteristic
192C2×C3×Q8C4×D311yescharacteristic
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4C4×D311yescharacteristic
196C2×C3×D4C4×D311yescharacteristic
197C2×C3×D4C4×D312yes—
198C2×C3×D4C4×D312yes—
199C22×C3×C4—21no—
200C22×C3×C4—22no—
201C22×C3×C4C22×D312yes—
202C22×C3×C4C22×D312yes—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C22×D311yescharacteristic
205C2×#67—34no—
206C2×#67—34no—
207C2×#67—32no—
208C2×#67—32no—
209C2×#67—64no—
210C2×#67—62no—
211C22×Dic3—32no—
212C22×Dic3—32no—
213C22×Dic3—62no—
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—62no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C3×M16—64no—
230C3×M16—62no—
231C3×M16—62no—
232C2×C3×C8—34no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—32no—
236C2×C3×C8—32no—
237C2×C3×C8—32no—
238C2×C3×C8—64no—
239C2×C3×C8—62no—
240#217—34no—
241#217—34no—
242#217—64no—
243(C2×D3)⋊C4—34no—
244(C2×D3)⋊C4—34no—
245(C2×D3)⋊C4—64no—
246Dic3⋊C4—34no—
247Dic3⋊C4—34no—
248Dic3⋊C4—64no—
249C4×Dic3—31no—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—31no—
253C4×Dic3—32no—
254C4×Dic3—32no—
255C4×Dic3—62no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258#208—22no—
259#208—22no—
260#208—24no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—24no—
263C2×C3⋊C8#6714yescomplemented
264C2×C3⋊C8C22×D312yes—
265C2×C3⋊C8C22×D312yes—
266C2×C3⋊C8C22×D312yes—
267C2×C3⋊C8C22×D311yescharacteristic
268C2×C3⋊C8C22×D311yescharacteristic
269ΓL(1,𝔽25)—64no—
270ΓL(1,𝔽25)—64no—
271D3×C8—64no—
272D3×C8—64no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—24no—
276C22×C32C4×D412yes—
277C22×C32C23×C411yescharacteristic
278C2×C3⋊D3—22no—
279C2×C3⋊D3—24no—
280C32×C4—22no—
281C32×C4C4×D412yes—
282C32×C4C22×D411yescharacteristic
283C32×C4C2×#4111yescharacteristic
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3#13114yescomplemented
287C2×#41—91no—
288C2×#41D3211yescharacteristic
289#131—64no—
290#131—184no—
291C2×M16—31no—
292C2×M16—32no—
293C2×M16—31no—
294C2×M16—92no—
295C22×C8—31no—
296C22×C8—32no—
297C22×C8—31no—
298C22×C8—92no—
299C4⋊Q8—91no—
300#127—91no—
301#124—91no—
302#124—91no—
303#121—94no—
304C4×D4—94no—
305C2×C42—91no—
306C4⋊C8—94no—
307C22⋊C8—34no—
308C22⋊C8—94no—
309C4×C8—92no—
310C4×C8—92no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3C2×C4×D312yes—
314C23×C3C2×C4×D312yes—
315C23×C3C2×C4×D311yescharacteristic
316C23×C3C2×C4×D311yescharacteristic
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—62no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—42no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4—42no—
328C3×D4—44no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4C2×C4×D312yes—
341C2×C3×C4C2×C4×D312yes—
342C2×C3×C4C2×C4×D311yescharacteristic
343C2×C3×C4C2×C4×D311yescharacteristic
344C2×C3×C4#23512yes—
345C2×C3×C4D3×D412yes—
346C2×C3×C4#23712yes—
347C2×C3×C4C2×#6712yes—
348C2×C3×C4C23×D311yescharacteristic
349C2×C3×C4C23×D311yescharacteristic
350#67—68no—
351#67—68no—
352#67—64no—
353#67—64no—
354#67—128no—
355#67—124no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—64no—
366C2×Dic3—64no—
367C2×Dic3—124no—
368D12—62no—
369D12—62no—
370D12—122no—
371C4×D3—62no—
372C4×D3—62no—
373C4×D3—62no—
374C4×D3—62no—
375C4×D3—122no—
376C4×D3—122no—
377Dic6—62no—
378Dic6—62no—
379Dic6—122no—
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—62no—
383C3×C8—62no—
384C3×C8—62no—
385C3⋊C8—22no—
386C3⋊C8—22no—
387C3⋊C8—22no—
388C3⋊C8C2×#6712yes—
389C3⋊C8C2×#6712yes—
390C2×C32—22no—
391C2×C32—42no—
392C2×C32C2×C4×D411yescharacteristic
393C2×C32C2×#13112yes—
394C3⋊D3—42no—
395#41—22no—
396#41—22no—
397#41—182no—
398#41—182no—
399C2×Q8—91no—
400C2×Q8#62411yescharacteristic
401C2×D4—94no—
402C2×D4—92no—
403C2×D4—91no—
404C2×D4#62412yes—
405C2×D4#62411yescharacteristic
406C22×C4—92no—
407C22×C4—92no—
408C22×C4C2×D3212yes—
409C22×C4C2×D3211yescharacteristic
410M16—64no—
411M16—62no—
412M16—62no—
413M16—184no—
414C2×C8—31no—
415C2×C8—31no—
416C2×C8—34no—
417C2×C8—32no—
418C2×C8—32no—
419C2×C8—32no—
420C2×C8—64no—
421C2×C8—62no—
422C2×C8—184no—
423C4⋊C4—94no—
424C22⋊C4—94no—
425C22⋊C4—94no—
426C42—92no—
427C42—91no—
428C42—91no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—42no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3#67⋊C412yes—
441C22×C3C4×#6712yes—
442C22×C3C22×C4×D311yescharacteristic
443C22×C3C22×C4×D311yescharacteristic
444C2×D3—62no—
445C2×D3—64no—
446C2×D3—64no—
447C2×D3—62no—
448C2×D3—122no—
449C2×D3—124no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4#67⋊C412yes—
457C3×C4C4×#6712yes—
458C3×C4C2×#23511yescharacteristic
459C3×C4C2×D3×D411yescharacteristic
460C3×C4C2×#23711yescharacteristic
461C3×C4C22×#6711yescharacteristic
462Dic3—34no—
463Dic3—34no—
464Dic3—64no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—62no—
469Dic3—122no—
470Dic3—122no—
471C32#293111yescharacteristic, complemented, Sylow
472C23—92no—
473C23C2×#62412yes—
474C23C2×#62411yescharacteristic
475Q8—22no—
476Q8—182no—
477D4—24no—
478D4—22no—
479D4—188no—
480D4—184no—
481D4—182no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—184no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4#380412yes—
491C2×C4C2×#62412yes—
492C2×C4C2×#62411yescharacteristic
493C2×C4D3×#6712yes—
494C2×C4C22×D3211yescharacteristic
495C8—32no—
496C8—32no—
497C8—62no—
498C8—62no—
499C8—62no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—42no—
505C2×C3—42no—
506C2×C3—42no—
507C2×C3—82no—
508C2×C3C2×#67⋊C411yescharacteristic
509C2×C3D3×#13112yes—
510C2×C3D3×#13112yes—
511C2×C3C2×C4×#6711yescharacteristic
512D3—122no—
513D3—122no—
514D3—242no—
515C22—22no—
516C22—24no—
517C22—22no—
518C22—184no—
519C22—182no—
520C22#6395212yes—
521C22C22×#62411yescharacteristic
522C4—22no—
523C4—94no—
524C4—182no—
525C4—182no—
526C4#6395212yes—
527C4C2×#380411yescharacteristic
528C4C2×D3×#6711yescharacteristic
529C3—21no—
530C3#7865711yescharacteristic, complemented
531C3#8378611yescharacteristic, complemented
532C2—22no—
533C2—42no—
534C2—362no—
535C2#1059544212yes—
536C2C2×#6395211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×#41C22
prime radical#10596303C2
cosocleC2×C32×C4C24
hypercenterC2×#41D32
hypocenterC32#2931
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#948C2semidirect
3. C2×#63817C2semidirect
4. C3×#5234C2semidirect
5. #10591148C2semidirect
6. #10591120C2semidirect
7. #10591073C2semidirect
8. C2×#63553C2semidirect
9. #10591051C2semidirect
40. C2×#948D3semidirect
44. #5234D3semidirect
81. C2×C3×C3⋊C8D4semidirect
263. C2×C3⋊C8#67semidirect
286. C3⋊Dic3#131semidirect
471. C32#2931semidirect
530. C3#78657semidirect
531. C3#83786semidirect

Maximal subgroups

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

subgroupindex
#105963032
C2×C3×#9482
C2×#638172
C3×#52342
#105911482
#105911202
#105910732
C2×#635532
#105910512
#837863
#786573