Finite Group Databasebeta

← C3×#70125 #11792760

Subgroups — C3×#70125 #11792760

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×C3×#909C211yesmaximal, characteristic
4C2×C32×#101C211yesmaximal, characteristic, complemented
5C2×C3×#824C211yesmaximal, characteristic, complemented
6C3×#5129C211yesmaximal, characteristic
7C3×#4558C211yesmaximal, characteristic, complemented
8C3×#2020—31nomaximal
9#70125C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
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
24C3×#909—22no—
25C3×#909—22no—
26C2×C3×Dic12—21no—
27C2×C3×#204—21no—
28C32×#101—22no—
29C3×#824—22no—
30C3×(C2×D3)⋊C8—22no—
31C2×C3×#137—31no—
32C2×C3×#137—31no—
33C22×C3×Q16—31no—
34C22×#216C2×C311yescharacteristic, complemented
35C2×#942C2×C311yescharacteristic
36C2×#909C2×C311yescharacteristic
37C2×C3×#101—21no—
38C2×C3×#101—31no—
39C2×C3×#101D311yescharacteristic, complemented
40C2×C3×#101C2×C311yescharacteristic, complemented
41C3×#408—31no—
42C2×#824C2×C311yescharacteristic, complemented
43#5129C2×C311yescharacteristic
44C3×#341—31no—
45#4558C2×C311yescharacteristic, complemented
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×Dic6D411yescharacteristic
68C2×C3×Dic6D412yes—
69C32×M16—21no—
70C32×M16—21no—
71C32×M16—21no—
72C32×M16D412yes—
73C2×C32×C8—21no—
74C2×C32×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—
84C3×Dic12—24no—
85C3×#204—24no—
86#2020—31noSylow
87C2×C3×#41—31no—
88C22×C3×Q8—21no—
89C22×C3×Q8—31no—
90C22×C3×Q8C2×D311yescharacteristic
91C22×C3×Q8C22×C311yescharacteristic
92C2×#235C22×C311yescharacteristic
93C22×Dic6C22×C311yescharacteristic
94C3×#137—62no—
95C3×#137—62no—
96C3×#137—62no—
97C3×#137—62no—
98C2×C3×Q16—61no—
99C2×C3×Q16—62no—
100C2×C3×Q16—62no—
101C2×C3×Q16—61no—
102C2×C3×Q16—62no—
103C2×C3×SD16—61no—
104C2×C3×SD16—61no—
105C2×C3×M16—21no—
106C2×C3×M16—21no—
107C2×C3×M16—31no—
108C2×C3×M16C2×D311yescharacteristic
109C2×C3×M16C2×D311yescharacteristic
110C2×C3×M16C22×C311yescharacteristic
111C2×C3×M16C22×C311yescharacteristic
112C22×C3×C8—31no—
113C2×#216—22no—
114C2×#216—21no—
115C2×#216—22no—
116C2×#216—22no—
117#942—22no—
118#942—22no—
119C2×#215—21no—
120C2×#208C22×C311yescharacteristic
121C22×C3⋊C8C22×C311yescharacteristic
122#909—22no—
123#909—22no—
124C2×Dic12—21no—
125C2×#204—21no—
126C3×#101—24no—
127C3×#101—22no—
128C3×#101—22no—
129C3×#101—62no—
130C3×C22⋊C8—62no—
131#824—22no—
132(C2×D3)⋊C8—22no—
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—
164C32×C8—22no—
165C32×C8—41no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—22no—
168C2×#137—31no—
169C2×#137—31no—
170C22×Q16—31no—
171C2×#101—31no—
172C2×#101C3×D311yescharacteristic, complemented
173#408—31no—
174#341—31no—
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—31no—
192C2×C3×Q8—32no—
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×Q8D1212yes—
203C2×C3×Q8C3×D412yes—
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×C4C23×C311yescharacteristic
214C2×#67—21no—
215C22×Dic3—21no—
216#235—21no—
217#235—21no—
218#235—21no—
219#235—21no—
220C2×D12C3×D411yescharacteristic
221C2×C4×D3—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C3×D411yescharacteristic
228C2×Dic6C3×D412yes—
229C3×Q16—64no—
230C3×Q16—68no—
231C3×Q16—64no—
232C3×SD16—64no—
233C3×SD16—64no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—22no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—22no—
242C3×M16—32no—
243C3×M16—41no—
244C3×M16—41no—
245C3×M16—61no—
246C3×M16#6712yes—
247C3×M16C3×D412yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—32no—
253C2×C3×C8—41no—
254C2×C3×C8—41no—
255C2×C3×C8—61no—
256C2×C3×C8—61no—
257C2×C3×C8—61no—
258#216—24no—
259#216—28no—
260#215—24no—
261#208—21no—
262#208C3×D412yes—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C3×D412yes—
267Dic12—24no—
268#204—24no—
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×Q8C2×C3×D311yescharacteristic
285#137—62no—
286#137—62no—
287#137—62no—
288#137—62no—
289C2×Q16—61no—
290C2×Q16—62no—
291C2×Q16—62no—
292C2×Q16—61no—
293C2×Q16—62no—
294C2×SD16—61no—
295C2×SD16—61no—
296C2×M16—31no—
297C2×M16C2×C3×D311yescharacteristic
298C2×M16C2×C3×D311yescharacteristic
299C22×C8—31no—
300#101—22no—
301#101—62no—
302C22⋊C8—62no—
303C23×C3—21no—
304C23×C3—61no—
305C23×C3#23511yescharacteristic
306C23×C3C3×#4111yescharacteristic
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—62no—
346C2×C3×C4—61no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4—121no—
350C2×C3×C4—121no—
351C2×C3×C4C2×D1211yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C2×C3×D411yescharacteristic
358C2×C3×C4C2×C3×D411yescharacteristic
359C2×C3×C4C2×C3×D411yescharacteristic
360C2×C3×C4C2×C3×D411yescharacteristic
361C2×C3×C4C2×C3×D411yescharacteristic
362C2×C3×C4C2×C3×D411yescharacteristic
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—22no—
380C3×C8—22no—
381C3×C8—42no—
382C3×C8—42no—
383C3×C8—41no—
384C3×C8—41no—
385C3×C8—62no—
386C3×C8—62no—
387C3⋊C8—22no—
388C3⋊C8—22no—
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×Q8C3×D1212yes—
411C2×D4—31no—
412C2×D4—61no—
413C22×C4—21no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4C22×C3×D311yescharacteristic
417Q16—64no—
418Q16—68no—
419Q16—64no—
420SD16—64no—
421SD16—64no—
422M16—21no—
423M16—21no—
424M16—21no—
425M16—32no—
426M16—61no—
427M16C3×#6712yes—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—32no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—42no—
440C22×C3—121no—
441C22×C3—122no—
442C22×C3#88411yescharacteristic
443C22×C3#88411yescharacteristic
444C22×C3#93011yescharacteristic
445C22×C3C3×#12111yescharacteristic
446C22×C3C3×#12111yescharacteristic
447C22×C3C3×#12111yescharacteristic
448C2×D3—41no—
449C2×D3—42no—
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#88211yescharacteristic
460C3×C4#88211yescharacteristic
461C3×C4#93711yescharacteristic
462C3×C4#94011yescharacteristic
463C3×C4C3×#12011yescharacteristic
464C3×C4C3×#12011yescharacteristic
465C3×C4C3×#12011yescharacteristic
466C3×C4C3×#12711yescharacteristic
467Dic3—42no—
468Dic3—41no—
469C32#202011yescharacteristic, complemented, Sylow
470C23—61no—
471C23C3×#23511yescharacteristic
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—61no—
490C2×C4—62no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4C2×C3×D1211yescharacteristic
496C2×C4C3×D3×D411yescharacteristic
497C2×C4C3×D3×D411yescharacteristic
498C2×C4C3×D3×D411yescharacteristic
499C2×C4C3×D3×D411yescharacteristic
500C2×C4C2×C3×#6711yescharacteristic
501C8—22no—
502C8—41no—
503C8—62no—
504C8—62no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—241no—
514C2×C3#475512yes—
515C2×C3#496011yescharacteristic
516C2×C3C3×#39211yescharacteristic
517C2×C3C3×#47312yes—
518D3—81no—
519C22—22no—
520C22—121no—
521C22—122no—
522C22C3×#88411yescharacteristic
523C22C3×#88411yescharacteristic
524C22C3×#93011yescharacteristic
525C4—42no—
526C4—121no—
527C4—122no—
528C4C3×#88211yescharacteristic
529C4C3×#88211yescharacteristic
530C4C3×#93711yescharacteristic
531C4C3×#94011yescharacteristic
532C3—21no—
533C3#7012511yescharacteristic, direct product factor
534C3C3×#202011yescharacteristic, complemented
535C2—21no—
536C2—21no—
537C2—241no—
538C2C3×#475512yes—
539C2C3×#496011yescharacteristic
subgroupG/H
centerC22×C3#884
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#101C2
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC2×C3×#101D3
hypocenterC3C3×#2020
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C22×C3×#216C2semidirect
4. C2×C32×#101C2semidirect
5. C2×C3×#824C2semidirect
7. C3×#4558C2semidirect
9. #70125C3direct (N × H)
34. C22×#216C2×C3semidirect
39. C2×C3×#101D3semidirect
40. C2×C3×#101C2×C3semidirect
42. C2×#824C2×C3semidirect
45. #4558C2×C3semidirect
172. C2×#101C3×D3semidirect
469. C32#2020semidirect
533. C3#70125direct (N × H)
534. C3C3×#2020semidirect

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×C3×#9092
C2×C32×#1012
C2×C3×#8242
C3×#51292
C3×#45582
C3×#20203
#701253