Finite Group Databasebeta

← #11792769

Subgroups — #11792769

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
1C22×C3×#216C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic
3C2×#63827C211yesmaximal, characteristic
4C2×#63663C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6#10591070C211yesmaximal, characteristic
7#10589166C211yesmaximal, characteristic, complemented
8#72109—31nomaximal
9#72074—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#216—22no—
16C2×C3×#216—21no—
17C2×C3×#216—22no—
18C2×C3×#216—22no—
19C3×#942—22no—
20C3×#942—22no—
21C2×C3×#215—21no—
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C2×#3718—21no—
25#63827—22no—
26#63827—22no—
27C2×#3715—21no—
28#63663—22no—
29#63560—22no—
30(C2×D3)⋊(C3⋊C8)—22no—
31C2×C3×#137—31no—
32C22×C3×Q16—31no—
33C2×#951—31no—
34C22×#216D311yescharacteristic, complemented
35C2×#942—31no—
36C2×#942D311yescharacteristic
37#5234—31no—
38C2×#836—31no—
39C2×#836—31no—
40C2×#836—31no—
41C2×#836—61no—
42C2×#824—31no—
43#5110—31no—
44#4558—31no—
45#4552—31no—
46C2×C32×Q8—21no—
47C2×C32×Q8—21no—
48C2×C32×Q8—21no—
49C2×C32×Q8—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8D412yes—
52C22×C32×C4—21no—
53C22×C32×C4C2311yescharacteristic
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×Dic6D412yes—
68C2×C3×Dic6D411yescharacteristic
69#3748—21no—
70#3748—21no—
71#3748—21no—
72#3748D412yes—
73C2×C3⋊(C3⋊C8)—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C3×#216—24no—
76C3×#216—28no—
77C3×#215—24no—
78C3×#208—21no—
79C3×#208D412yes—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8D412yes—
84#3718—24no—
85#3715—24no—
86#2020—91noSylow
87C2×C3×#41—31no—
88C22×C3×Q8—21no—
89C22×C3×Q8—31no—
90C22×C3×Q8C2×D311yescharacteristic
91C22×C3×Q8C2×D311yescharacteristic
92C2×#235C2×D311yescharacteristic
93C22×Dic6C2×D311yescharacteristic
94C3×#137—62no—
95C3×#137—62no—
96C2×C3×Q16—62no—
97C2×C3×Q16—62no—
98C2×C3×Q16—61no—
99C2×C3×Q16—62no—
100C2×C3×SD16—61no—
101C2×C3×M16—31no—
102C22×C3×C8—31no—
103#951—62no—
104#951—62no—
105C2×#216—22no—
106C2×#216—21no—
107C2×#216—22no—
108C2×#216—22no—
109C2×#216—61no—
110C2×#216—61no—
111#942—22no—
112#942—22no—
113#942—62no—
114#942—62no—
115C2×#215—21no—
116C2×#215—61no—
117C2×#214—61no—
118C2×#208—31no—
119C2×#208—31no—
120C2×#208—31no—
121C2×#208—31no—
122C2×#208—61no—
123C2×#208—61no—
124C2×#208C2×D311yescharacteristic
125C22×C3⋊C8C2×D311yescharacteristic
126#836—62no—
127#836—62no—
128#836—64no—
129#836—62no—
130#830—62no—
131#824—62no—
132(C2×D3)⋊C8—62no—
133C23×C32#4111yescharacteristic
134C22×C3×D3—21no—
135C32×Q8—22no—
136C32×Q8—22no—
137C32×Q8—41no—
138C32×Q8—41no—
139C2×C32×C4—22no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
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—
164C3⋊(C3⋊C8)—22no—
165C3⋊(C3⋊C8)—41no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—22no—
168C2×#137—31no—
169C2×#137—91no—
170C22×Q16—31no—
171C2×#101—91no—
172C2×#101—91no—
173#408—91no—
174#341—91no—
175C3×#41—61no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—61no—
179C2×C3×Q8—22no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—22no—
191C2×C3×Q8—32no—
192C2×C3×Q8—31no—
193C2×C3×Q8—41no—
194C2×C3×Q8—41no—
195C2×C3×Q8—41no—
196C2×C3×Q8—41no—
197C2×C3×Q8—61no—
198C2×C3×Q8—61no—
199C2×C3×Q8—61no—
200C2×C3×Q8—61no—
201C2×C3×Q8—61no—
202C2×C3×Q8#6712yes—
203C2×C3×Q8#6712yes—
204C2×C3×D4—31no—
205C2×C3×D4—61no—
206C22×C3×C4—22no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—61no—
211C22×C3×C4—61no—
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C2×#67—21no—
215C22×Dic3—21no—
216#235—21no—
217#235—21no—
218#235—21no—
219#235—21no—
220C2×D12#6711yescharacteristic
221C2×C4×D3—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6#6711yescharacteristic
228C2×Dic6#6712yes—
229C3×Q16—68no—
230C3×Q16—64no—
231C3×SD16—64no—
232C3×M16—32no—
233C3×M16—61no—
234C2×C3×C8—32no—
235C2×C3×C8—61no—
236C2×C3×C8—61no—
237C2×C3×C8—61no—
238#216—24no—
239#216—28no—
240#216—64no—
241#216—64no—
242#215—24no—
243#215—64no—
244#214—64no—
245#208—21no—
246#208—32no—
247#208—32no—
248#208—62no—
249#208—61no—
250#208—61no—
251#208—62no—
252#208—61no—
253#208—61no—
254#208—61no—
255#208—61no—
256#208—121no—
257#208—121no—
258#208#6712yes—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—61no—
263C2×C3⋊C8—61no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8—61no—
266C2×C3⋊C8—121no—
267C2×C3⋊C8—121no—
268C2×C3⋊C8#6712yes—
269C22×C32—22no—
270C22×C32#12111yescharacteristic
271C22×C32#12111yescharacteristic
272C22×C32#12111yescharacteristic
273C2×C3×D3—41no—
274C2×C3×D3—42no—
275C32×C4—42no—
276C32×C4#12011yescharacteristic
277C32×C4#12011yescharacteristic
278C32×C4#12011yescharacteristic
279C32×C4#12711yescharacteristic
280C3×Dic3—42no—
281C3×Dic3—41no—
282C2×#41—31no—
283C22×Q8—31no—
284C22×Q8D3211yescharacteristic
285#137—62no—
286#137—62no—
287#137—182no—
288#137—182no—
289C2×Q16—62no—
290C2×Q16—62no—
291C2×Q16—61no—
292C2×Q16—62no—
293C2×Q16—181no—
294C2×SD16—61no—
295C2×SD16—181no—
296C2×M16—31no—
297C2×M16—91no—
298C2×M16—91no—
299C22×C8—31no—
300#101—182no—
301#101—182no—
302C22⋊C8—182no—
303C23×C3—21no—
304C23×C3—61no—
305C23×C3#23711yescharacteristic
306C23×C3#23711yescharacteristic
307C22×D3—21no—
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—42no—
316C3×Q8—41no—
317C3×Q8—41no—
318C3×Q8—41no—
319C3×Q8—41no—
320C3×Q8—62no—
321C3×Q8—61no—
322C3×Q8—62no—
323C3×Q8—61no—
324C3×Q8—121no—
325C3×Q8—121no—
326C3×D4—61no—
327C3×D4—61no—
328C3×D4—121no—
329C3×D4—121no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4—121no—
350C2×C3×C4—121no—
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358C2×C3×C4C2×#6711yescharacteristic
359C2×C3×C4C2×#6711yescharacteristic
360C2×C3×C4C2×#6711yescharacteristic
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363#67—41no—
364#67—41no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369D12—21no—
370D12—21no—
371C4×D3—41no—
372C4×D3—41no—
373Dic6—22no—
374Dic6—21no—
375Dic6—22no—
376Dic6—21no—
377Dic6—41no—
378Dic6—41no—
379C3×C8—62no—
380C3×C8—62no—
381C3⋊C8—22no—
382C3⋊C8—22no—
383C3⋊C8—62no—
384C3⋊C8—62no—
385C3⋊C8—122no—
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—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—31no—
404C2×Q8—32no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8#62512yes—
411C2×D4—31no—
412C2×D4—61no—
413C22×C4—21no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4C2×D3211yescharacteristic
417Q16—68no—
418Q16—64no—
419Q16—184no—
420SD16—64no—
421SD16—184no—
422M16—32no—
423M16—61no—
424M16—92no—
425M16—181no—
426M16—181no—
427M16—181no—
428C2×C8—32no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—181no—
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#93811yescharacteristic
443C22×C3#93811yescharacteristic
444C22×C3#93911yescharacteristic
445C22×C3#93911yescharacteristic
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—44no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4—121no—
458C3×C4—122no—
459C3×C4#93711yescharacteristic
460C3×C4#93711yescharacteristic
461C3×C4#93711yescharacteristic
462C3×C4#93711yescharacteristic
463C3×C4#94011yescharacteristic
464C3×C4#94011yescharacteristic
465C3×C4#95311yescharacteristic
466C3×C4#95311yescharacteristic
467Dic3—42no—
468Dic3—41no—
469C32#202011yescharacteristic, complemented, Sylow
470C23—61no—
471C23#380511yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476Q8—62no—
477Q8—61no—
478Q8—62no—
479Q8—61no—
480Q8—121no—
481Q8—121no—
482D4—61no—
483D4—61no—
484D4—121no—
485D4—121no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—62no—
490C2×C4—61no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4C2×#62511yescharacteristic
496C2×C4D3×#6711yescharacteristic
497C2×C4D3×#6711yescharacteristic
498C2×C4D3×#6711yescharacteristic
499C2×C4D3×#6711yescharacteristic
500C2×C4#381111yescharacteristic
501C8—62no—
502C8—62no—
503C8—182no—
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#519212yes—
515C2×C3#520412yes—
516C2×C3#522311yescharacteristic
517C2×C3#522311yescharacteristic
518D3—81no—
519C22—22no—
520C22—122no—
521C22—121no—
522C22#6397011yescharacteristic
523C22#6397011yescharacteristic
524C22#6398011yescharacteristic
525C4—42no—
526C4—121no—
527C4—122no—
528C4#6397611yescharacteristic
529C4#6397611yescharacteristic
530C4#6397911yescharacteristic
531C4#6398311yescharacteristic
532C3—21no—
533C3#7207411yescharacteristic, complemented
534C3#7210911yescharacteristic, complemented
535C2—21no—
536C2—21no—
537C2—241no—
538C2#1059164112yes—
539C2#1059172311yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC22×Q8D32
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#216C2semidirect
4. C2×#63663C2semidirect
5. C2×#63560C2semidirect
7. #10589166C2semidirect
34. C22×#216D3semidirect
469. C32#2020semidirect
533. C3#72074semidirect
534. C3#72109semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2162
C2×C3×#9422
C2×#638272
C2×#636632
C2×#635602
#105910702
#105891662
#721093
#720743