Finite Group Databasebeta

← #11792773

Subgroups — #11792773

Order 1152.

757 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#909C211yesmaximal, characteristic
2C22×#3719C211yesmaximal, characteristic, complemented
3C2×#63831C211yesmaximal, characteristic
4C2×#63651C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6#10591070C211yesmaximal, characteristic
7C3×#4558C211yesmaximal, characteristic, complemented
8#72109—31nomaximal
9#68944—31nomaximal
10C22×#634C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#909—22no—
18C3×#909—22no—
19C2×C3×Dic12—21no—
20C2×C3×#204—21no—
21C2×#3719—22no—
22C2×#3719—21no—
23C2×#3719—22no—
24C2×#3719—22no—
25#63831—22no—
26#63831—22no—
27C2×#3716—21no—
28#63651—22no—
29#63560—22no—
30(C2×D3)⋊(C3⋊C8)—22no—
31C2×C3×#137—31no—
32C2×#951—31no—
33C22×#216—31no—
34C2×#909—31no—
35C2×#909D311yescharacteristic
36C22×Dic12—31no—
37#5234—31no—
38C2×#836—31no—
39C2×#824—31no—
40C2×#824—31no—
41C2×#824—31no—
42C2×#824—61no—
43#5110—31no—
44C3×#341—31no—
45#4558D311yescharacteristic, complemented
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×#634—21no—
49C2×#634—21no—
50C2×#634—21no—
51C2×#634—21no—
52C2×#634—21no—
53C2×#634D412yes—
54C2×C3×#67—21no—
55C22×C3×Dic3—21no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C2×C3×D12D411yescharacteristic
61C2×C3×C4×D3—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6D411yescharacteristic
68C2×C3×Dic6D412yes—
69C32×M16—21no—
70C32×M16D412yes—
71C2×C32×C8—21no—
72#3748—21no—
73#3748—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C3×#208—21no—
76C3×#208D412yes—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8D412yes—
81C3×Dic12—24no—
82C3×#204—24no—
83#3719—24no—
84#3719—28no—
85#3716—24no—
86#2020—91noSylow
87C2×C3×#41—31no—
88C22×C3×Q8—31no—
89C2×#235C2×D311yescharacteristic
90C22×Dic6—31no—
91C22×Dic6—31no—
92C22×Dic6—61no—
93C22×Dic6C2×D311yescharacteristic
94C3×#137—62no—
95C3×#137—62no—
96C2×C3×Q16—61no—
97C2×C3×SD16—61no—
98C2×C3×M16—21no—
99C2×C3×M16—31no—
100C2×C3×M16C2×D311yescharacteristic
101C2×C3×M16C2×D311yescharacteristic
102C22×C3×C8—31no—
103#951—62no—
104#951—62no—
105C2×#216—62no—
106C2×#216—62no—
107C2×#216—61no—
108C2×#216—62no—
109C2×#214—61no—
110C2×#208—31no—
111C2×#208—31no—
112C2×#208—61no—
113C2×#208C2×D311yescharacteristic
114C22×C3⋊C8C2×D311yescharacteristic
115#909—22no—
116#909—22no—
117#909—62no—
118#909—62no—
119C2×Dic12—21no—
120C2×Dic12—62no—
121C2×Dic12—62no—
122C2×Dic12—62no—
123C2×Dic12—61no—
124C2×#204—21no—
125C2×#204—61no—
126#836—62no—
127#830—62no—
128#824—64no—
129#824—62no—
130#824—62no—
131#824—62no—
132(C2×D3)⋊C8—62no—
133C23×C32#4111yescharacteristic
134C22×C3×D3—21no—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—41no—
143C2×C3⋊Dic3—41no—
144#634—22no—
145#634—22no—
146#634—41no—
147#634—41no—
148C3×#67—41no—
149C3×#67—41no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—41no—
153C2×C3×Dic3—41no—
154C3×D12—21no—
155C3×D12—21no—
156C3×C4×D3—41no—
157C3×C4×D3—41no—
158C3×Dic6—22no—
159C3×Dic6—21no—
160C3×Dic6—22no—
161C3×Dic6—21no—
162C3×Dic6—41no—
163C3×Dic6—41no—
164C32×C8—22no—
165C3⋊(C3⋊C8)—41no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—22no—
168C2×#137—31no—
169C2×#137—91no—
170C22×Q16—91no—
171C2×#101—91no—
172C2×#101—91no—
173#408—91no—
174#341—31no—
175C3×#41—61no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—61no—
179C2×C3×Q8—31no—
180C2×C3×Q8—32no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C2×C3×Q8—61no—
185C2×C3×Q8—61no—
186C2×C3×D4—31no—
187C2×C3×D4—61no—
188C22×C3×C4—21no—
189C22×C3×C4—61no—
190C22×C3×C4—61no—
191C22×C3×C4C22×D311yescharacteristic
192C22×C3×C4C22×D311yescharacteristic
193C2×#67—21no—
194C22×Dic3—21no—
195C22×Dic3—62no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198#235—21no—
199#235—21no—
200#235—21no—
201#235—21no—
202C2×D12#6711yescharacteristic
203C2×C4×D3—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—32no—
210C2×Dic6—32no—
211C2×Dic6—62no—
212C2×Dic6—61no—
213C2×Dic6—61no—
214C2×Dic6—61no—
215C2×Dic6—61no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×Dic6—61no—
219C2×Dic6—61no—
220C2×Dic6—62no—
221C2×Dic6—61no—
222C2×Dic6—61no—
223C2×Dic6—121no—
224C2×Dic6—121no—
225C2×Dic6—121no—
226C2×Dic6—121no—
227C2×Dic6#6711yescharacteristic
228C2×Dic6#6712yes—
229C3×Q16—64no—
230C3×SD16—64no—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—22no—
234C3×M16—32no—
235C3×M16—41no—
236C3×M16—61no—
237C3×M16D1212yes—
238C3×M16#6712yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—32no—
242C2×C3×C8—41no—
243C2×C3×C8—61no—
244C2×C3×C8—61no—
245C2×C3×C8—61no—
246#216—68no—
247#216—64no—
248#214—64no—
249#208—21no—
250#208—62no—
251#208—61no—
252#208—61no—
253#208—61no—
254#208—61no—
255#208—121no—
256#208D1212yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—121no—
263C2×C3⋊C8D1212yes—
264Dic12—24no—
265Dic12—68no—
266Dic12—64no—
267#204—24no—
268#204—64no—
269C22×C32—22no—
270C22×C32#12111yescharacteristic
271C22×C32#12111yescharacteristic
272C22×C32#12111yescharacteristic
273C2×C3×D3—41no—
274C2×C3×D3—42no—
275C32×C4#12011yescharacteristic
276C32×C4#12011yescharacteristic
277C32×C4#12011yescharacteristic
278C32×C4#12711yescharacteristic
279C3⋊Dic3—42no—
280C3×Dic3—42no—
281C3×Dic3—41no—
282C2×#41—31no—
283C22×Q8—31no—
284C22×Q8—91no—
285#137—62no—
286#137—62no—
287#137—182no—
288#137—182no—
289C2×Q16—61no—
290C2×Q16—182no—
291C2×Q16—182no—
292C2×Q16—181no—
293C2×Q16—182no—
294C2×SD16—61no—
295C2×SD16—181no—
296C2×M16—31no—
297C2×M16—91no—
298C2×M16D3211yescharacteristic
299C22×C8—31no—
300#101—182no—
301#101—182no—
302C22⋊C8—182no—
303C23×C3—21no—
304C23×C3—61no—
305C23×C3#23711yescharacteristic
306C23×C3#23911yescharacteristic
307C22×D3—21no—
308C3×Q8—62no—
309C3×Q8—61no—
310C3×Q8—62no—
311C3×Q8—61no—
312C3×Q8—121no—
313C3×Q8—121no—
314C3×D4—61no—
315C3×D4—61no—
316C3×D4—121no—
317C3×D4—121no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—61no—
325C2×C3×C4—62no—
326C2×C3×C4—121no—
327C2×C3×C4—121no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4C2×D1211yescharacteristic
331C2×C3×C4C2×D1211yescharacteristic
332C2×C3×C4C2×D1211yescharacteristic
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4D3×D411yescharacteristic
335C2×C3×C4D3×D411yescharacteristic
336C2×C3×C4D3×D411yescharacteristic
337C2×C3×C4D3×D411yescharacteristic
338C2×C3×C4D3×D411yescharacteristic
339C2×C3×C4C2×#6711yescharacteristic
340C2×C3×C4C2×#6711yescharacteristic
341C2×C3×C4C2×#6711yescharacteristic
342#67—41no—
343#67—41no—
344C2×Dic3—22no—
345C2×Dic3—21no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—64no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—122no—
352C2×Dic3—122no—
353C2×Dic3—121no—
354C2×Dic3—121no—
355C2×Dic3—121no—
356C2×Dic3—121no—
357D12—21no—
358D12—21no—
359C4×D3—41no—
360C4×D3—41no—
361Dic6—22no—
362Dic6—21no—
363Dic6—22no—
364Dic6—21no—
365Dic6—41no—
366Dic6—41no—
367Dic6—62no—
368Dic6—62no—
369Dic6—62no—
370Dic6—62no—
371Dic6—122no—
372Dic6—122no—
373Dic6—122no—
374Dic6—122no—
375Dic6—121no—
376Dic6—121no—
377Dic6—121no—
378Dic6—121no—
379C3×C8—22no—
380C3×C8—22no—
381C3×C8—42no—
382C3×C8—62no—
383C3×C8—62no—
384C3⋊C8—22no—
385C3⋊C8—22no—
386C3⋊C8—122no—
387C3⋊C8—121no—
388C3⋊C8—121no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32#39211yescharacteristic
392C2×C32#47312yes—
393C3×D3—81no—
394#41—61no—
395#41—61no—
396#41—61no—
397#41—61no—
398C2×Q8—31no—
399C2×Q8—32no—
400C2×Q8—61no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—92no—
406C2×Q8—181no—
407C2×Q8—181no—
408C2×Q8—181no—
409C2×Q8—181no—
410C2×Q8—181no—
411C2×D4—31no—
412C2×D4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—181no—
416C22×C4C2×D3211yescharacteristic
417Q16—64no—
418Q16—188no—
419Q16—184no—
420SD16—64no—
421SD16—184no—
422M16—21no—
423M16—32no—
424M16—61no—
425M16—181no—
426M16—181no—
427M16#62612yes—
428C2×C8—21no—
429C2×C8—32no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—181no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—42no—
440C22×C3—122no—
441C22×C3—121no—
442C22×C3#89511yescharacteristic
443C22×C3#89511yescharacteristic
444C22×C3#89511yescharacteristic
445C22×C3#93811yescharacteristic
446C22×C3#93911yescharacteristic
447C22×C3#93911yescharacteristic
448C2×D3—42no—
449C2×D3—41no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—121no—
455C3×C4—122no—
456C3×C4#87411yescharacteristic
457C3×C4#88211yescharacteristic
458C3×C4#88211yescharacteristic
459C3×C4#88211yescharacteristic
460C3×C4#93711yescharacteristic
461C3×C4#93711yescharacteristic
462C3×C4#94011yescharacteristic
463C3×C4#95311yescharacteristic
464Dic3—42no—
465Dic3—41no—
466Dic3—124no—
467Dic3—122no—
468Dic3—122no—
469C32#202011yescharacteristic, complemented, Sylow
470C23—61no—
471C23#379611yescharacteristic
472Q8—62no—
473Q8—61no—
474Q8—62no—
475Q8—61no—
476Q8—121no—
477Q8—121no—
478Q8—182no—
479Q8—182no—
480Q8—361no—
481Q8—361no—
482D4—61no—
483D4—61no—
484D4—121no—
485D4—121no—
486C2×C4—62no—
487C2×C4—61no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—182no—
493C2×C4—361no—
494C2×C4—361no—
495C2×C4D3×D1211yescharacteristic
496C2×C4D3×D1211yescharacteristic
497C2×C4#380211yescharacteristic
498C2×C4C2×#62611yescharacteristic
499C2×C4D3×#6711yescharacteristic
500C2×C4D3×#6711yescharacteristic
501C8—22no—
502C8—62no—
503C8—62no—
504C8—361no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—241no—
514C2×C3#467111yescharacteristic
515C2×C3#489612yes—
516C2×C3#520412yes—
517C2×C3#522311yescharacteristic
518D3—81no—
519C22—22no—
520C22—122no—
521C22—121no—
522C22#6388611yescharacteristic
523C22#6388611yescharacteristic
524C22#6391611yescharacteristic
525C4—121no—
526C4—122no—
527C4—362no—
528C4#6391111yescharacteristic
529C4#6392211yescharacteristic
530C4#6392211yescharacteristic
531C4#6397711yescharacteristic
532C3—21no—
533C3#6894411yescharacteristic, complemented
534C3#7210911yescharacteristic, complemented
535C2—21no—
536C2—21no—
537C2—241no—
538C2#1059098312yes—
539C2#1059139111yescharacteristic
subgroupG/H
centerC22#63886
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC2×M16D32
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C22×#3719C2semidirect
4. C2×#63651C2semidirect
5. C2×#63560C2semidirect
7. C3×#4558C2semidirect
45. #4558D3semidirect
469. C32#2020semidirect
533. C3#68944semidirect
534. C3#72109semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9092
C22×#37192
C2×#638312
C2×#636512
C2×#635602
#105910702
C3×#45582
#721093
#689443