Finite Group Databasebeta

← #11859488

Subgroups — #11859488

Order 1152.

1079 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic, complemented
2#10595029C211yesmaximal, characteristic
3C2×#63817C211yesmaximal, characteristic, complemented
4#10592828C211yesmaximal, characteristic, complemented
5C3×#5118C211yesmaximal, characteristic, complemented
6#10591071C211yesmaximal, characteristic, complemented
7C2×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8#10590731C214yesmaximal, complemented
9#10590720C214yesmaximal, complemented
10#83786—31nomaximal
11#79041—31nomaximal
12C2×#3828C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C22×C3×C3⋊C8C2211yescharacteristic
19C3×#907—24no—
20C2×C3×ΓL(1,𝔽25)C2212yes—
21C2×C3×D3×C8C2212yes—
22#63978C414yes—
23#63911C411yescharacteristic
24C4×#626C2214yes—
25#63866C411yescharacteristic
26#63864C411yescharacteristic
27#63860C411yescharacteristic
28C2×#3712C2212yes—
29#63817—24no—
30C2×#3710C2212yes—
31#63647C2214yes—
32C3×Dic3⋊C8C2214yes—
33(C2×D3)⋊(C3⋊C8)C2214yes—
34Dic3×C3⋊C8C2212yes—
35Dic3×C3⋊C8C2212yes—
36C2×C3×#131—31no—
37#5832—31no—
38C2×#907—31no—
39C2×#907D311yescharacteristic, complemented
40C2×#901—31no—
41#5477—31no—
42C3×#423—31no—
43#5234—31no—
44#5129—31no—
45#5129—31no—
46#5129—31no—
47#5129—61no—
48#5128—34no—
49#5118D311yescharacteristic, complemented
50C8×#67—34no—
51C2×C8×Dic3—31no—
52C2×C4×C3⋊C8—31no—
53#4838—34no—
54#4834—34no—
55C22×C32×C4C2311yescharacteristic
56C2×#638C2×C412yes—
57#3828—22no—
58#3828—22no—
59C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
60C2×C4×C3⋊D3C2312yes—
61C2×#634C2×C411yescharacteristic
62C2×C3×#67C2×C412yes—
63C22×C3×Dic3C2×C412yes—
64C3×#235—22no—
65C3×#235—22no—
66C2×C3×D12C2×C411yescharacteristic
67C2×C3×C4×D3C2312yes—
68C2×C3×Dic6C2×C411yescharacteristic
69C2×#626C2×C414yes—
70C32×M16—22no—
71C2×C32×C8C2312yes—
72C2×C3⋊(C3⋊C8)—22no—
73C2×C3⋊(C3⋊C8)C2312yes—
74C3×C4×Dic3C2×C411yescharacteristic
75C3×C4×Dic3C2×C411yescharacteristic
76C3×C4×Dic3C2312yes—
77C3×#208—22no—
78C2×C3×C3⋊C8D414yescomplemented
79C2×C3×C3⋊C8C2312yes—
80C2×C3×C3⋊C8C2311yescharacteristic
81C2×C3×C3⋊C8C2311yescharacteristic
82C3×ΓL(1,𝔽25)—24no—
83C3×D3×C8—24no—
84Dic3⋊Dic3C2×C414yes—
85#3722C2×C414yes—
86(C2×D3)⋊Dic3C2×C414yes—
87#3712—24no—
88#3710—24no—
89#2931—91noSylow
90C2×C3×#41—31no—
91C2×#235—31no—
92C2×#235—31no—
93C2×#235—61no—
94C2×#235C2×D311yescharacteristic
95C3×#131—64no—
96C2×C3×M16—21no—
97C2×C3×M16—31no—
98C2×C3×M16—32no—
99C2×C3×M16C2×D311yescharacteristic
100C2×C3×M16C2×D311yescharacteristic
101C22×C3×C8—31no—
102C22×C3×C8—32no—
103C2×C3×C42—31no—
104#947—31no—
105#944—31no—
106#940—31no—
107#939—34no—
108#936—31no—
109#930—34no—
110C4×#67—34no—
111C2×C4×Dic3C2×D311yescharacteristic
112C2×#208C2×D311yescharacteristic
113C22×C3⋊C8—31no—
114C22×C3⋊C8—31no—
115C22×C3⋊C8—61no—
116C22×C3⋊C8C2×D311yescharacteristic
117#907—24no—
118#907—64no—
119#901—64no—
120C2×ΓL(1,𝔽25)—32no—
121C2×ΓL(1,𝔽25)—32no—
122C2×ΓL(1,𝔽25)C2×D312yes—
123C2×D3×C8—32no—
124C2×D3×C8—32no—
125C2×D3×C8C2×D312yes—
126#875—31no—
127#875—31no—
128#874—31no—
129C4×D12—34no—
130#869—31no—
131C3×C4⋊C8—34no—
132#830—34no—
133(C2×D3)⋊C8—34no—
134(C2×D3)⋊C8—34no—
135(C2×D3)⋊C8—34no—
136(C2×D3)⋊C8—64no—
137Dic3⋊C8C2×D314yes—
138C8×Dic3—32no—
139C8×Dic3—32no—
140C4×C3⋊C8—32no—
141C4×C3⋊C8—32no—
142C23×C32C22×C411yescharacteristic
143C22×C3⋊D3C22×C412yes—
144C22×C3×D3C22×C412yes—
145C2×C32×C4C22×C411yescharacteristic
146C2×C32×C4C2×D412yes—
147C2×C32×C4#4112yes—
148C2×C32×C4C2411yescharacteristic
149#638—24no—
150C2×C3⋊Dic3C22×C412yes—
151C4⋊(C3⋊D3)—22no—
152C4×C3⋊D3—22no—
153C4×C3⋊D3—22no—
154#634—22no—
155C3×#67—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3C22×C412yes—
158C2×C3×Dic3C22×C412yes—
159C2×C3×Dic3C22×C412yes—
160C3×D12—22no—
161C3×C4×D3—22no—
162C3×C4×D3—22no—
163C3×Dic6—22no—
164#626—28no—
165C32×C8—22no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8C2×D412yes—
169C3×C3⋊C8C2×D412yes—
170C2×#131—31no—
171C2×#131—91no—
172#532—91no—
173#436—94no—
174C8×D4—94no—
175#423—31no—
176#408—91no—
177#408—91no—
178C2×C4×C8—91no—
179C3×#41—62no—
180C3×#41—62no—
181C2×C3×Q8—31no—
182C2×C3×D4—32no—
183C2×C3×D4—31no—
184C22×C3×C4—21no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4C22×D311yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C2×#67—32no—
190C2×#67—32no—
191C2×#67—34no—
192C2×#67—62no—
193C2×#67C4×D312yes—
194C22×Dic3C4×D312yes—
195#235—22no—
196#235—22no—
197#235—62no—
198#235—62no—
199#235—62no—
200#235—62no—
201#235—122no—
202#235—122no—
203C2×D12—34no—
204C2×D12—31no—
205C2×D12—31no—
206C2×D12—61no—
207C2×D12C4×D311yescharacteristic
208C2×C4×D3—32no—
209C2×C4×D3—32no—
210C2×C4×D3—62no—
211C2×C4×D3C22×D312yes—
212C2×Dic6—31no—
213C2×Dic6—31no—
214C2×Dic6—61no—
215C2×Dic6C4×D311yescharacteristic
216C3×M16—22no—
217C3×M16—22no—
218C3×M16—42no—
219C3×M16—64no—
220C3×M16—62no—
221C2×C3×C8—22no—
222C2×C3×C8—34no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—32no—
226C2×C3×C8—64no—
227C2×C3×C8C22×D312yes—
228C2×C3×C8C22×D312yes—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42—32no—
232#217—34no—
233(C2×D3)⋊C4—34no—
234(C2×D3)⋊C4—34no—
235(C2×D3)⋊C4—34no—
236C4⋊Dic3—34no—
237Dic3⋊C4—34no—
238C4×Dic3C4×D311yescharacteristic
239C4×Dic3C4×D311yescharacteristic
240C4×Dic3C22×D312yes—
241#208—22no—
242C2×C3⋊C8—32no—
243C2×C3⋊C8—32no—
244C2×C3⋊C8—62no—
245C2×C3⋊C8—62no—
246C2×C3⋊C8—62no—
247C2×C3⋊C8—122no—
248C2×C3⋊C8#6714yescomplemented
249C2×C3⋊C8C22×D312yes—
250C2×C3⋊C8C22×D311yescharacteristic
251C2×C3⋊C8C22×D311yescharacteristic
252ΓL(1,𝔽25)—24no—
253ΓL(1,𝔽25)—64no—
254ΓL(1,𝔽25)—64no—
255D3×C8—24no—
256D3×C8—64no—
257D3×C8—64no—
258C22×C32—22no—
259C22×C32C4×D412yes—
260C22×C32C23×C411yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—24no—
263C2×C3×D3—22no—
264C2×C3×D3—24no—
265C32×C4C4×D412yes—
266C32×C4C22×D411yescharacteristic
267C32×C4C2×#4111yescharacteristic
268C3⋊Dic3—22no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3#13114yescomplemented
272C2×#41—31no—
273C2×#41—91no—
274#131—64no—
275#131—184no—
276C2×M16—31no—
277C2×M16—32no—
278C2×M16—92no—
279C2×M16D3211yescharacteristic
280C22×C8—31no—
281C22×C8—32no—
282C22×C8—92no—
283C22×C8—91no—
284C4⋊Q8—91no—
285#127—91no—
286#124—91no—
287#124—91no—
288#121—94no—
289C4×D4—94no—
290C2×C42—31no—
291C4⋊C8—34no—
292C22⋊C8—94no—
293C22⋊C8—94no—
294C4×C8—92no—
295C4×C8—92no—
296C23×C3—21no—
297C23×C3—32no—
298C23×C3C2×C4×D311yescharacteristic
299C23×C3C2×C4×D311yescharacteristic
300C22×D3—32no—
301C22×D3—32no—
302C22×D3—62no—
303C22×D3C2×C4×D312yes—
304C3×Q8—62no—
305C3×D4—64no—
306C3×D4—62no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—32no—
312C2×C3×C4—32no—
313C2×C3×C4—32no—
314C2×C3×C4—64no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4C2×C4×D311yescharacteristic
318C2×C3×C4C2×C4×D311yescharacteristic
319C2×C3×C4#23512yes—
320C2×C3×C4D3×D412yes—
321C2×C3×C4#23912yes—
322C2×C3×C4C2×#6712yes—
323C2×C3×C4C23×D311yescharacteristic
324C2×C3×C4C23×D311yescharacteristic
325#67—24no—
326#67—64no—
327#67—64no—
328#67—68no—
329#67—124no—
330C2×Dic3—24no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—62no—
334C2×Dic3C2×C4×D312yes—
335C2×Dic3C2×C4×D312yes—
336C2×Dic3C2×C4×D312yes—
337D12—22no—
338D12—68no—
339D12—62no—
340D12—62no—
341D12—122no—
342C4×D3—22no—
343C4×D3—22no—
344C4×D3—62no—
345C4×D3—62no—
346C4×D3—62no—
347C4×D3—62no—
348C4×D3—122no—
349C4×D3—122no—
350Dic6—22no—
351Dic6—62no—
352Dic6—62no—
353Dic6—122no—
354C3×C8—22no—
355C3×C8—22no—
356C3×C8—32no—
357C3×C8—32no—
358C3×C8—42no—
359C3×C8—62no—
360C3⋊C8—22no—
361C3⋊C8—62no—
362C3⋊C8—62no—
363C3⋊C8—122no—
364C3⋊C8C2×#6712yes—
365C3⋊C8C2×#6712yes—
366C2×C32—22no—
367C2×C32C2×C4×D411yescharacteristic
368C2×C32C2×#13112yes—
369C3⋊D3—42no—
370C3×D3—42no—
371#41—62no—
372#41—62no—
373#41—182no—
374#41—182no—
375C2×Q8—31no—
376C2×Q8—91no—
377C2×D4—32no—
378C2×D4—31no—
379C2×D4—94no—
380C2×D4—92no—
381C2×D4—91no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—92no—
385C22×C4C2×D3211yescharacteristic
386M16—22no—
387M16—64no—
388M16—62no—
389M16—184no—
390C2×C8—31no—
391C2×C8—31no—
392C2×C8—34no—
393C2×C8—32no—
394C2×C8—64no—
395C2×C8—92no—
396C2×C8—184no—
397C2×C8—182no—
398C2×C8C2×D3212yes—
399C4⋊C4—94no—
400C22⋊C4—94no—
401C22⋊C4—94no—
402C42—32no—
403C42—31no—
404C42—31no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—42no—
410C22×C3—64no—
411C22×C3—62no—
412C22×C3#89312yes—
413C22×C3C4×#6712yes—
414C22×C3C22×C4×D311yescharacteristic
415C22×C3C22×C4×D311yescharacteristic
416C2×D3—24no—
417C2×D3—22no—
418C2×D3—64no—
419C2×D3—64no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—122no—
423C2×D3—124no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—34no—
428C3×C4—62no—
429C3×C4—62no—
430C3×C4#89312yes—
431C3×C4C4×#6712yes—
432C3×C4C2×#23511yescharacteristic
433C3×C4C2×D3×D411yescharacteristic
434C3×C4C2×#23911yescharacteristic
435C3×C4C22×#6711yescharacteristic
436Dic3—22no—
437Dic3—22no—
438Dic3—62no—
439Dic3—62no—
440Dic3—122no—
441Dic3#90714yescomplemented
442C32#293111yescharacteristic, complemented, Sylow
443C23—32no—
444C23—92no—
445C23C4×D3211yescharacteristic
446Q8—62no—
447Q8—182no—
448D4—64no—
449D4—62no—
450D4—188no—
451D4—184no—
452D4—182no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—92no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4#379912yes—
463C2×C4C4×D3211yescharacteristic
464C2×C4D3×#6712yes—
465C2×C4C22×D3211yescharacteristic
466C8—22no—
467C8—32no—
468C8—32no—
469C8—62no—
470C8—182no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—122no—
477C2×C3C2×#89311yescharacteristic
478C2×C3C2×#90712yes—
479C2×C3D3×#13112yes—
480C2×C3C2×C4×#6711yescharacteristic
481D3—42no—
482D3—122no—
483D3—122no—
484D3—242no—
485C22—22no—
486C22—64no—
487C22—62no—
488C22—184no—
489C22—182no—
490C22#6390412yes—
491C22C2×C4×D3211yescharacteristic
492C4—34no—
493C4—62no—
494C4—62no—
495C4—182no—
496C4#6390412yes—
497C4C2×#379911yescharacteristic
498C4C2×D3×#6711yescharacteristic
499C3—21no—
500C3#7904111yescharacteristic, complemented
501C3#8378611yescharacteristic, complemented
502C2—22no—
503C2—122no—
504C2—362no—
505C2D3×#90712yes—
506C2C2×#6390411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×M16C22
prime radical#10595029C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2931
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#907C2semidirect
3. C2×#63817C2semidirect
4. #10592828C2semidirect
5. C3×#5118C2semidirect
6. #10591071C2semidirect
7. C2×Dic3×C3⋊C8C2semidirect
8. #10590731C2semidirect
9. #10590720C2semidirect
39. C2×#907D3semidirect
49. #5118D3semidirect
78. C2×C3×C3⋊C8D4semidirect
248. C2×C3⋊C8#67semidirect
271. C3×Dic3#131semidirect
441. Dic3#907semidirect
442. C32#2931semidirect
500. C3#79041semidirect
501. C3#83786semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
#105950292
C2×#638172
#105928282
C3×#51182
#105910712
C2×Dic3×C3⋊C82
#105907312
#105907202
#837863
#790413