Finite Group Databasebeta

← #11792213

Subgroups — #11792213

Order 1152.

809 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#869C211yesmaximal, characteristic, complemented
2#10594866C211yesmaximal, characteristic
3C2×#63833C211yesmaximal, characteristic
4C2×#63632C211yesmaximal, characteristic
5#10591099C211yesmaximal, characteristic, complemented
6C2×#63560C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8#71795—31nomaximal
9#68768—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×C4⋊Dic3—22no—
17C2×C3×#208C2211yescharacteristic
18C3×#869—22no—
19C3×#869—22no—
20#63966—22no—
21#63871—22no—
22#63866—21no—
23#63864—21no—
24C2×#3719—21no—
25#63833—22no—
26#63833—22no—
27C2×#3717—21no—
28#63632—22no—
29#63565—22no—
30#63560—22no—
31C2×C3×C4⋊Q8—31no—
32#5816—31no—
33C2×#942—31no—
34C2×#909—31no—
35#5526—31no—
36C2×#869D311yescharacteristic, complemented
37C2×#836—31no—
38#5225—31no—
39C2×#824—31no—
40#5004—31no—
41C2×#805—31no—
42C2×#805—31no—
43C2×#805—61no—
44C3×#339—31no—
45#4523D311yescharacteristic, complemented
46C22×C32×C4—21no—
47C22×C32×C4C2311yescharacteristic
48C2×#638—21no—
49#3828—21no—
50#3828—21no—
51#3828—21no—
52#3828—21no—
53C2×C4⋊(C3⋊D3)D411yescharacteristic
54C2×C4×C3⋊D3—21no—
55C2×#634D411yescharacteristic
56C22×C3×Dic3—21no—
57C22×C3×Dic3D412yes—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6D412yes—
70C32×C42—21no—
71C32×C42D412yes—
72#3748—21no—
73#3748—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C3×#217—21no—
76C3×C4⋊Dic3—28no—
77C3×C4⋊Dic3D412yes—
78C3×C4×Dic3—21no—
79C3×#208—21no—
80C3×#208D412yes—
81C2×C3×C3⋊C8—21no—
82Dic3⋊Dic3—24no—
83#3722—24no—
84#3719—24no—
85#3717—24no—
86#2005—91noSylow
87C22×C3×Q8—31no—
88C22×C3×Q8—31no—
89C2×#235—31no—
90C2×#235—31no—
91C2×#235—61no—
92C22×Dic6C2×D311yescharacteristic
93C22×Dic6C2×D311yescharacteristic
94C2×C3×M16—31no—
95C3×C4⋊Q8—62no—
96C3×C4⋊Q8—62no—
97C3×#117—31no—
98C2×C3×C4⋊C4—62no—
99C2×C3×C42—21no—
100C2×C3×C42C2×D311yescharacteristic
101C2×C3×C42C2×D311yescharacteristic
102#947—61no—
103#946—62no—
104#944—61no—
105C2×#216—61no—
106#942—62no—
107#942—62no—
108C2×#215—61no—
109#929—62no—
110#926C2×D311yescharacteristic
111C2×C4⋊Dic3—22no—
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—61no—
115C2×#208C2×D311yescharacteristic
116#909—62no—
117#909—62no—
118C2×Dic12—61no—
119C2×#204—61no—
120#897—62no—
121#886—62no—
122#875—61no—
123#869—22no—
124#869—22no—
125#869—61no—
126#836—62no—
127#835—62no—
128#824—62no—
129#810—62no—
130#805—64no—
131#805—62no—
132#805—62no—
133C23×C32C2×D411yescharacteristic
134C22×C3⋊D3—21no—
135C2×C32×C4—24no—
136C2×C32×C4—22no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4#4111yescharacteristic
143#638—41no—
144#638—41no—
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C4×C3⋊D3—41no—
149C4×C3⋊D3—41no—
150#634—21no—
151#634—21no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—41no—
158C2×C3×Dic3—41no—
159C3×Dic6—22no—
160C3×Dic6—22no—
161C3×Dic6—41no—
162C3×Dic6—41no—
163C3×Dic6—41no—
164C3×Dic6—41no—
165C3×Dic6—41no—
166C3×Dic6—41no—
167C3⋊(C3⋊C8)—41no—
168C3×C3⋊C8—22no—
169C2×#137—91no—
170#536—91no—
171C2×C4⋊Q8—31no—
172C2×#104—91no—
173#419—91no—
174C2×#101—91no—
175#339—31no—
176C2×C3×Q8—32no—
177C2×C3×Q8—61no—
178C2×C3×Q8—61no—
179C2×C3×Q8—61no—
180C2×C3×Q8—61no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C2×C3×Q8—61no—
185C2×C3×Q8—61no—
186C2×C3×Q8—61no—
187C2×C3×Q8—61no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—32no—
192C22×C3×C4—41no—
193C22×C3×C4—61no—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D311yescharacteristic
196C2×#67—61no—
197C2×#67—61no—
198C2×#67—121no—
199C22×Dic3—21no—
200C22×Dic3#6712yes—
201#235—62no—
202#235—62no—
203#235—61no—
204#235—61no—
205#235—61no—
206#235—61no—
207#235—61no—
208#235—61no—
209#235—61no—
210#235—61no—
211#235—121no—
212#235—121no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—61no—
216C2×C4×D3—61no—
217C2×C4×D3—61no—
218C2×C4×D3—121no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—61no—
233C2×Dic6#6712yes—
234C3×M16—32no—
235C3×M16—61no—
236C2×C3×C8—61no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—68no—
239C3×C22⋊C4—61no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—22no—
243C3×C42—41no—
244C3×C42—61no—
245C3×C42D1212yes—
246C3×C42#6712yes—
247#217—21no—
248#216—64no—
249#215—64no—
250(C2×D3)⋊C4—64no—
251(C2×D3)⋊C4—64no—
252C4⋊Dic3—28no—
253C4⋊Dic3—64no—
254C4⋊Dic3D1212yes—
255Dic3⋊C4—64no—
256C4×Dic3—21no—
257#208—21no—
258#208—62no—
259#208—61no—
260#208—61no—
261#208—61no—
262#208—61no—
263#208—121no—
264#208D1212yes—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—121no—
269Dic12—64no—
270#204—64no—
271C22×C32—22no—
272C22×C32#12011yescharacteristic
273C22×C32#12011yescharacteristic
274C22×C32#12111yescharacteristic
275C2×C3⋊D3—41no—
276C2×C3⋊D3—42no—
277C32×C4—24no—
278C32×C4#12011yescharacteristic
279C32×C4#12111yescharacteristic
280C32×C4#12111yescharacteristic
281C32×C4#12711yescharacteristic
282C3⋊Dic3—41no—
283C3×Dic3—42no—
284C3×Dic3—42no—
285C3×Dic3—42no—
286C2×#41—91no—
287C22×Q8—31no—
288C22×Q8—31no—
289#137—182no—
290#137—182no—
291C2×Q16—181no—
292C2×SD16—181no—
293C2×M16—31no—
294C2×M16—91no—
295C4⋊Q8—62no—
296C4⋊Q8—62no—
297C4⋊Q8—181no—
298#124—181no—
299#123—182no—
300C22⋊Q8—182no—
301#117—31no—
302C2×C4⋊C4—62no—
303C2×C42D3211yescharacteristic
304#104—182no—
305#103—182no—
306#101—182no—
307C23×C3—21no—
308C23×C3D3×D411yescharacteristic
309C23×C3C2×#6711yescharacteristic
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—121no—
313C3×Q8—62no—
314C3×Q8—62no—
315C3×Q8—121no—
316C3×Q8—121no—
317C3×Q8—121no—
318C3×Q8—121no—
319C3×Q8—121no—
320C3×Q8—121no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4C2×D1211yescharacteristic
342C2×C3×C4C2×D1211yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4#23711yescharacteristic
349C2×C3×C4#23911yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352#67—122no—
353#67—122no—
354#67—121no—
355#67—121no—
356#67—121no—
357#67—121no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—121no—
368D12—62no—
369D12—61no—
370D12—61no—
371D12—61no—
372D12—61no—
373D12—121no—
374C4×D3—122no—
375C4×D3—122no—
376C4×D3—121no—
377C4×D3—121no—
378C4×D3—121no—
379C4×D3—121no—
380Dic6—22no—
381Dic6—22no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386Dic6—41no—
387Dic6—41no—
388Dic6—62no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—121no—
394C3×C8—62no—
395C3⋊C8—22no—
396C3⋊C8—122no—
397C3⋊C8—121no—
398C3⋊C8—121no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32#39211yescharacteristic
402C2×C32#45612yes—
403C3⋊D3—81no—
404#41—181no—
405#41—181no—
406#41—181no—
407#41—181no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—91no—
421C2×D4—91no—
422C2×D4—181no—
423C22×C4—21no—
424C22×C4—32no—
425C22×C4—61no—
426C22×C4—181no—
427C22×C4C2×D3211yescharacteristic
428Q16—184no—
429SD16—184no—
430M16—32no—
431M16—61no—
432M16—181no—
433M16—181no—
434C2×C8—61no—
435C2×C8—181no—
436C4⋊C4—32no—
437C4⋊C4—68no—
438C4⋊C4—184no—
439C22⋊C4—61no—
440C22⋊C4—184no—
441C42—21no—
442C42—61no—
443C42#62612yes—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—42no—
450C22×C3#88211yescharacteristic
451C22×C3#88211yescharacteristic
452C22×C3#89511yescharacteristic
453C22×C3#93711yescharacteristic
454C22×C3#93911yescharacteristic
455C22×C3#95311yescharacteristic
456C2×D3—122no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—122no—
460C2×D3—122no—
461C2×D3—242no—
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—44no—
469C3×C4—122no—
470C3×C4—122no—
471C3×C4—122no—
472C3×C4#87411yescharacteristic
473C3×C4#88211yescharacteristic
474C3×C4#89511yescharacteristic
475C3×C4#89511yescharacteristic
476C3×C4#93711yescharacteristic
477C3×C4#93811yescharacteristic
478C3×C4#93911yescharacteristic
479C3×C4#94011yescharacteristic
480Dic3—42no—
481Dic3—42no—
482Dic3—42no—
483Dic3—122no—
484Dic3—121no—
485Dic3—121no—
486C32#200511yescharacteristic, complemented, Sylow
487C23—181no—
488C23D3×#6711yescharacteristic
489Q8—62no—
490Q8—62no—
491Q8—121no—
492Q8—121no—
493Q8—121no—
494Q8—121no—
495Q8—121no—
496Q8—121no—
497Q8—181no—
498Q8—181no—
499D4—181no—
500D4—181no—
501D4—361no—
502D4—361no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—181no—
513C2×C4—361no—
514C2×C4—361no—
515C2×C4#379611yescharacteristic
516C2×C4D3×D1211yescharacteristic
517C2×C4D3×D1211yescharacteristic
518C2×C4#380211yescharacteristic
519C2×C4C2×#62611yescharacteristic
520C2×C4D3×#6711yescharacteristic
521C8—62no—
522C8—361no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—22no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3#467111yescharacteristic
532C2×C3#482612yes—
533C2×C3#509512yes—
534C2×C3#522311yescharacteristic
535D3—242no—
536D3—241no—
537D3—241no—
538C22—22no—
539C22—361no—
540C22—362no—
541C22#6388611yescharacteristic
542C22#6392211yescharacteristic
543C22#6397711yescharacteristic
544C4—24no—
545C4—122no—
546C4—122no—
547C4—122no—
548C4—361no—
549C4#6388611yescharacteristic
550C4#6391111yescharacteristic
551C4#6391611yescharacteristic
552C4#6392211yescharacteristic
553C3—21no—
554C3#6876811yescharacteristic, complemented
555C3#7179511yescharacteristic, complemented
556C2—21no—
557C2—21no—
558C2—721no—
559C2#1059039912yes—
560C2#1059139111yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC22×C32#120
prime radicalC2×#3828C22
hypercenterC2×C42D32
hypocenterC32#2005
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C3×#869C2semidirect
5. #10591099C2semidirect
6. C2×#63560C2semidirect
7. C3×#4523C2semidirect
36. C2×#869D3semidirect
45. #4523D3semidirect
486. C32#2005semidirect
554. C3#68768semidirect
555. C3#71795semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8692
#105948662
C2×#638332
C2×#636322
#105910992
C2×#635602
C3×#45232
#717953
#687683