Finite Group Databasebeta

← #11788721

Subgroups — #11788721

Order 1152.

753 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596216C211yesmaximal, characteristic
2#10594866C211yesmaximal, characteristic
3C3×#5120C211yesmaximal, characteristic, complemented
4#10591232C211yesmaximal, characteristic
5#10591099C211yesmaximal, characteristic, complemented
6C3×#4557C211yesmaximal, characteristic, complemented
7#10589157C211yesmaximal, characteristic, complemented
8#70350—31nomaximal
9#68712—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14#64116—22no—
15M16×C3⋊D3—22no—
16#64110—22no—
17C3×#926C2211yescharacteristic
18C3×#926C2211yescharacteristic
19C3×#926C2211yescharacteristic
20#63966—22no—
21#63871—22no—
22#63866C412yescomplemented
23#63864C412yescomplemented
24C3×#816—22no—
25#63581—22no—
26#63565—22no—
27#5816—31no—
28#5745—31no—
29#5745—31no—
30#5745—61no—
31#5526—31no—
32C3×#419—31no—
33#5225—31no—
34#5120D311yescharacteristic, complemented
35#5006—31no—
36#5006—31no—
37#5004—31no—
38C3×H(3,ℤ/4)—31no—
39#4557D311yescharacteristic, complemented
40#4531—31no—
41#4531—31no—
42C22×C32×C4C2311yescharacteristic
43C2×#638D412yes—
44#3828—21no—
45#3828—21no—
46#3828—21no—
47#3828—21no—
48C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
49C2×C4×C3⋊D3D412yes—
50C2×#634C2×C411yescharacteristic
51C22×C3×Dic3—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6C2×C412yes—
58C32×M16—21no—
59C32×M16—21no—
60C2×C32×C8D412yes—
61#3748—21no—
62#3748—21no—
63C2×C3⋊(C3⋊C8)D412yes—
64#3743—24no—
65C8×C3⋊D3—24no—
66C3×#217—21no—
67C3×#217—21no—
68C3×#217—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3C2×C412yes—
75Dic3⋊Dic3—24no—
76#3722—24no—
77#1909—91noSylow
78C22×C3×Q8—31no—
79C2×#235—31no—
80C2×#235—31no—
81C2×#235—61no—
82C22×Dic6C2×D311yescharacteristic
83C2×C3×M16—21no—
84C2×C3×M16C2×D311yescharacteristic
85C2×C3×M16C2×D311yescharacteristic
86C3×#117—31no—
87C3×#117—31no—
88C3×#117—31no—
89#947—32no—
90#946—62no—
91#944—32no—
92#929—62no—
93#926C2×D311yescharacteristic
94#926C2×D311yescharacteristic
95#926C2×D311yescharacteristic
96C2×#208—31no—
97C2×#208—31no—
98C2×#208—61no—
99#907—62no—
100#907—62no—
101#907—64no—
102D3×M16—62no—
103D3×M16—62no—
104D3×M16—64no—
105#901—64no—
106#901—62no—
107#901—62no—
108#897—62no—
109#886—62no—
110#875—32no—
111#869—32no—
112C3×#103—62no—
113#835—62no—
114#816—22no—
115#810—62no—
116#806—62no—
117#806—62no—
118C23×C32C2×D411yescharacteristic
119C22×C3⋊D3C22⋊C412yes—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C2×C32×C4#4111yescharacteristic
126#638—22no—
127#638—22no—
128C2×C3⋊Dic3C22⋊C412yes—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4×C3⋊D3—22no—
132C4×C3⋊D3—22no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C3×Dic6—22no—
142C3×Dic6—22no—
143C3×Dic6—41no—
144C3×Dic6—41no—
145C32×C8—22no—
146C3⋊(C3⋊C8)—22no—
147#568—91no—
148#536—91no—
149#419—31no—
150#419—91no—
151#410—91no—
152H(3,ℤ/4)—31no—
153H(3,ℤ/4)—91no—
154C2×C3×Q8—32no—
155C2×C3×Q8—61no—
156C2×C3×Q8—61no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C22×C3×C4—21no—
161C22×C3×C4—61no—
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C22×D311yescharacteristic
164C2×#67—32no—
165C2×#67—32no—
166C2×#67—62no—
167C22×Dic3—21no—
168#235—62no—
169#235—62no—
170#235—62no—
171#235—62no—
172#235—61no—
173#235—61no—
174#235—61no—
175#235—61no—
176#235—61no—
177#235—61no—
178#235—61no—
179#235—61no—
180C2×D12—31no—
181C2×D12—31no—
182C2×D12—61no—
183C2×C4×D3—32no—
184C2×C4×D3—32no—
185C2×C4×D3—62no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—31no—
192C2×Dic6—31no—
193C2×Dic6—61no—
194C2×Dic6C4×D312yes—
195C3×M16—22no—
196C3×M16—22no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—21no—
201C2×C3×C8—22no—
202C2×C3×C8D1212yes—
203C2×C3×C8#6712yes—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C42—32no—
211C3×C42—61no—
212C3×C42—61no—
213#217—21no—
214#217—21no—
215#217—21no—
216(C2×D3)⋊C4—64no—
217(C2×D3)⋊C4—64no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—64no—
222Dic3⋊C4—64no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3C4×D312yes—
226#208—62no—
227#208—62no—
228#208—61no—
229#208—61no—
230#208—61no—
231#208—61no—
232C2×C3⋊C8—32no—
233C2×C3⋊C8—32no—
234C2×C3⋊C8—62no—
235ΓL(1,𝔽25)—68no—
236ΓL(1,𝔽25)—64no—
237ΓL(1,𝔽25)—64no—
238D3×C8—68no—
239D3×C8—64no—
240D3×C8—64no—
241C22×C32—41no—
242C22×C32C2×C22⋊C411yescharacteristic
243C22×C32#12111yescharacteristic
244C22×C32#12111yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—42no—
247C32×C4C4×D411yescharacteristic
248C32×C4C4×D411yescharacteristic
249C32×C4#12011yescharacteristic
250C32×C4#12311yescharacteristic
251C3⋊Dic3—22no—
252C3×Dic3—42no—
253C3×Dic3—42no—
254C3×Dic3—81no—
255C3×Dic3—81no—
256C2×#41—91no—
257C22×Q8—31no—
258#131—182no—
259#131—182no—
260C2×M16—91no—
261C2×M16—182no—
262C2×M16D3211yescharacteristic
263C4⋊Q8—92no—
264#124—92no—
265#123—182no—
266C22⋊Q8—182no—
267#117—31no—
268#117—31no—
269#117—31no—
270#103—62no—
271#103—182no—
272#99—182no—
273C23×C3—21no—
274C23×C3D3×D411yescharacteristic
275C23×C3D3×D411yescharacteristic
276C22×D3—32no—
277C22×D3—32no—
278C22×D3—62no—
279C3×Q8—62no—
280C3×Q8—62no—
281C3×Q8—121no—
282C3×Q8—121no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—121no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4—121no—
295C2×C3×C4C2×C4×D311yescharacteristic
296C2×C3×C4C2×C4×D311yescharacteristic
297C2×C3×C4C2×D1211yescharacteristic
298C2×C3×C4#23511yescharacteristic
299C2×C3×C4#23511yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4D3×D411yescharacteristic
304C2×C3×C4#23711yescharacteristic
305C2×C3×C4#23911yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307#67—64no—
308#67—64no—
309#67—62no—
310#67—62no—
311#67—62no—
312#67—62no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—62no—
322D12—62no—
323D12—62no—
324D12—61no—
325D12—61no—
326D12—61no—
327D12—61no—
328C4×D3—64no—
329C4×D3—64no—
330C4×D3—62no—
331C4×D3—62no—
332C4×D3—62no—
333C4×D3—62no—
334Dic6—22no—
335Dic6—22no—
336Dic6—41no—
337Dic6—41no—
338Dic6—62no—
339Dic6—62no—
340Dic6—61no—
341Dic6—61no—
342Dic6—61no—
343Dic6—61no—
344C3×C8—24no—
345C3×C8—22no—
346C3×C8—22no—
347C3⋊C8—64no—
348C3⋊C8—62no—
349C3⋊C8—62no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32#38611yescharacteristic
354C3⋊D3—42no—
355#41—181no—
356#41—181no—
357#41—181no—
358#41—181no—
359C2×Q8—32no—
360C2×Q8—61no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—91no—
366C2×D4—92no—
367C2×D4—91no—
368C22×C4—61no—
369C22×C4—92no—
370C22×C4C2×D3211yescharacteristic
371M16—21no—
372M16—21no—
373M16—184no—
374M16—181no—
375M16—181no—
376C2×C8—92no—
377C2×C8—184no—
378C2×C8#62612yes—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—184no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—61no—
386C22⋊C4—184no—
387C42—32no—
388C42—61no—
389C42—61no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—81no—
396C22×C3D3×C22⋊C411yescharacteristic
397C22×C3D3×C22⋊C411yescharacteristic
398C22×C3#88411yescharacteristic
399C22×C3#88411yescharacteristic
400C22×C3#89511yescharacteristic
401C22×C3#93811yescharacteristic
402C2×D3—64no—
403C2×D3—62no—
404C2×D3—62no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—242no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—122no—
413C3×C4—122no—
414C3×C4—241no—
415C3×C4—241no—
416C3×C4C4×D1211yescharacteristic
417C3×C4#67⋊C411yescharacteristic
418C3×C4#88211yescharacteristic
419C3×C4#88311yescharacteristic
420C3×C4#89311yescharacteristic
421C3×C4#89411yescharacteristic
422C3×C4C4×#6711yescharacteristic
423C3×C4#93711yescharacteristic
424Dic3—42no—
425Dic3—42no—
426Dic3—64no—
427Dic3—62no—
428Dic3—62no—
429Dic3—81no—
430Dic3—81no—
431C32#190911yescharacteristic, complemented, Sylow
432C23—92no—
433C23#380211yescharacteristic
434Q8—62no—
435Q8—62no—
436Q8—121no—
437Q8—121no—
438Q8—181no—
439Q8—181no—
440D4—182no—
441D4—182no—
442D4—181no—
443D4—181no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—92no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—182no—
452C2×C4—182no—
453C2×C4#379611yescharacteristic
454C2×C4#379811yescharacteristic
455C2×C4C4×D3211yescharacteristic
456C2×C4C2×#62611yescharacteristic
457C2×C4#381111yescharacteristic
458C2×C4#381111yescharacteristic
459C8—22no—
460C8—182no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3#466511yescharacteristic
472C2×C3#495611yescharacteristic
473D3—124no—
474D3—122no—
475D3—122no—
476C22—41no—
477C22—182no—
478C22—362no—
479C22#6391211yescharacteristic
480C22#6391611yescharacteristic
481C22#6392111yescharacteristic
482C4—122no—
483C4—122no—
484C4—182no—
485C4—241no—
486C4—241no—
487C4#6385311yescharacteristic
488C4#6388111yescharacteristic
489C4C4×#62611yescharacteristic
490C4#6398111yescharacteristic
491C3—21no—
492C3#6871211yescharacteristic, complemented
493C3#7035011yescharacteristic, complemented
494C2—21no—
495C2—21no—
496C2—21no—
497C2—362no—
498C2#1059138711yescharacteristic
subgroupG/H
centerC2#10591387
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#386
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1909
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C3×#5120C2semidirect
5. #10591099C2semidirect
6. C3×#4557C2semidirect
7. #10589157C2semidirect
22. #63866C4semidirect
23. #63864C4semidirect
34. #5120D3semidirect
39. #4557D3semidirect
431. C32#1909semidirect
492. C3#68712semidirect
493. C3#70350semidirect

Maximal subgroups

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

subgroupindex
#105962162
#105948662
C3×#51202
#105912322
#105910992
C3×#45572
#105891572
#703503
#687123