Finite Group Databasebeta

← C3×#72109 #11792754

Subgroups — C3×#72109 #11792754

Order 1152.

813 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic
2C2×C3×#951C211yesmaximal, characteristic
3C22×C3×#216C211yesmaximal, characteristic, complemented
4C3×#5234C211yesmaximal, characteristic
5C2×C3×#836C211yesmaximal, characteristic, complemented
6C2×C3×#824C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8C3×#2020—31nomaximal
9#72109C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×Q8C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C32×#137—22no—
14C32×#137—22no—
15C2×C32×Q16—21no—
16C2×C32×SD16—21no—
17C2×C32×M16C2211yescharacteristic
18C3×#951—22no—
19C3×#951—22no—
20C2×C3×#216—22no—
21C2×C3×#216—21no—
22C2×C3×#216—22no—
23C2×C3×#216—22no—
24C2×C3×#214—21no—
25C2×C3×#208C2211yescharacteristic
26C2×C3×#208C2211yescharacteristic
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#836—22no—
29C3×#830—22no—
30C3×#824—22no—
31C2×C3×#137—21no—
32C2×C3×#137—31no—
33C2×C3×#137D311yescharacteristic
34C2×C3×#137C2×C311yescharacteristic
35C22×C3×Q16—31no—
36C2×#951C2×C311yescharacteristic
37C22×#216C2×C311yescharacteristic, complemented
38C2×C3×#101—31no—
39C2×C3×#101—31no—
40C3×#408—31no—
41#5234C2×C311yescharacteristic
42C2×#836C2×C311yescharacteristic, complemented
43C2×#824C2×C311yescharacteristic, complemented
44C3×#341—31no—
45#4558C2×C311yescharacteristic, complemented
46C32×#41—21no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—21no—
54C2×C32×Q8—21no—
55C2×C32×Q8D412yes—
56C2×C32×Q8D411yescharacteristic
57C2×C32×D4—21no—
58C2×C32×D4D411yescharacteristic
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6D412yes—
69C32×Q16—24no—
70C32×SD16—24no—
71C32×M16—21no—
72C32×M16D412yes—
73C2×C32×C8—21no—
74C3×#216—28no—
75C3×#216—24no—
76C3×#214—24no—
77C3×#208—21no—
78C3×#208—21no—
79C3×#208—21no—
80C3×#208D412yes—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8—21no—
84C2×C3×C3⋊C8—21no—
85C2×C3×C3⋊C8D412yes—
86#2020—31noSylow
87C2×C3×#41—21no—
88C2×C3×#41C2×D311yescharacteristic
89C2×C3×#41C22×C311yescharacteristic
90C22×C3×Q8—21no—
91C22×C3×Q8—31no—
92C22×C3×Q8C2×D311yescharacteristic
93C22×C3×Q8C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C3×#137—24no—
96C3×#137—24no—
97C3×#137—22no—
98C3×#137—22no—
99C3×#137—22no—
100C3×#137—22no—
101C3×#137—62no—
102C3×#137—62no—
103C2×C3×Q16—22no—
104C2×C3×Q16—21no—
105C2×C3×Q16—21no—
106C2×C3×Q16—62no—
107C2×C3×Q16—62no—
108C2×C3×Q16—62no—
109C2×C3×Q16—61no—
110C2×C3×SD16—22no—
111C2×C3×SD16—21no—
112C2×C3×SD16—21no—
113C2×C3×SD16—61no—
114C2×C3×M16—21no—
115C2×C3×M16—31no—
116C2×C3×M16—31no—
117C2×C3×M16C2×D311yescharacteristic
118C2×C3×M16C22×C311yescharacteristic
119C22×C3×C8—31no—
120#951—22no—
121#951—22no—
122C2×#216—22no—
123C2×#216—21no—
124C2×#216—22no—
125C2×#216—22no—
126C2×#214—21no—
127C2×#208C22×C311yescharacteristic
128C2×#208C22×C311yescharacteristic
129C22×C3⋊C8C22×C311yescharacteristic
130C3×#101—62no—
131C3×#101—62no—
132C3×C22⋊C8—62no—
133#836—22no—
134#830—22no—
135#824—22no—
136C23×C32—21no—
137C23×C32#4111yescharacteristic
138C32×Q8—22no—
139C32×Q8—21no—
140C32×Q8—22no—
141C32×Q8—21no—
142C32×Q8—41no—
143C32×Q8—41no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—41no—
147C32×D4—41no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4—41no—
153C2×C32×C4—41no—
154C2×C32×C4C2×D411yescharacteristic
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4C2×D411yescharacteristic
158C2×C32×C4C2×D411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C3×Dic3—22no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—41no—
163C3×Dic6—22no—
164C3×Dic6—22no—
165C3×Dic6—41no—
166C3×Dic6—41no—
167C32×C8—22no—
168C3×C3⋊C8—22no—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—41no—
171C2×#137—31no—
172C2×#137C3×D311yescharacteristic
173C22×Q16—31no—
174C2×#101—31no—
175C2×#101—31no—
176#408—31no—
177#341—31no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C3×#41—21no—
185C3×#41—21no—
186C3×#41—21no—
187C3×#41—21no—
188C3×#41—41no—
189C3×#41—41no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—22no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—22no—
204C2×C3×Q8—32no—
205C2×C3×Q8—41no—
206C2×C3×Q8—41no—
207C2×C3×Q8—41no—
208C2×C3×Q8—61no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×Q8#6711yescharacteristic
214C2×C3×Q8#6712yes—
215C2×C3×Q8C3×D411yescharacteristic
216C2×C3×Q8C3×D412yes—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—41no—
221C2×C3×D4#6711yescharacteristic
222C2×C3×D4C3×D411yescharacteristic
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—41no—
229C22×C3×C4—41no—
230C22×C3×C4—61no—
231C22×C3×C4C22×D311yescharacteristic
232C22×C3×C4C23×C311yescharacteristic
233C22×Dic3—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6C3×D412yes—
240C3×Q16—28no—
241C3×Q16—24no—
242C3×Q16—24no—
243C3×Q16—68no—
244C3×Q16—64no—
245C3×SD16—28no—
246C3×SD16—24no—
247C3×SD16—24no—
248C3×SD16—64no—
249C3×M16—22no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—22no—
253C3×M16—32no—
254C3×M16—61no—
255C3×M16—61no—
256C3×M16—61no—
257C3×M16#6712yes—
258C3×M16C3×D412yes—
259C2×C3×C8—22no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—32no—
263C2×C3×C8—61no—
264C2×C3×C8—61no—
265C2×C3×C8—61no—
266C2×C3×C8—61no—
267#216—28no—
268#216—24no—
269#214—24no—
270#208—21no—
271#208—21no—
272#208—21no—
273#208C3×D412yes—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8C3×D412yes—
279C22×C32—22no—
280C22×C32—41no—
281C22×C32—42no—
282C22×C32#12111yescharacteristic
283C22×C32#12111yescharacteristic
284C22×C32#12111yescharacteristic
285C32×C4—41no—
286C32×C4—42no—
287C32×C4#12011yescharacteristic
288C32×C4#12011yescharacteristic
289C32×C4#12011yescharacteristic
290C32×C4#12711yescharacteristic
291C3×Dic3—42no—
292C2×#41C2×C3×D311yescharacteristic
293C22×Q8—31no—
294C22×Q8C2×C3×D311yescharacteristic
295#137—22no—
296#137—22no—
297#137—62no—
298#137—62no—
299C2×Q16—21no—
300C2×Q16—62no—
301C2×Q16—62no—
302C2×Q16—62no—
303C2×Q16—61no—
304C2×SD16—21no—
305C2×SD16—61no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16C2×C3×D311yescharacteristic
309C22×C8—31no—
310#101—62no—
311#101—62no—
312C22⋊C8—62no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—41no—
317C23×C3#23711yescharacteristic
318C23×C3C3×#4111yescharacteristic
319C3×Q8—24no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—21no—
323C3×Q8—22no—
324C3×Q8—21no—
325C3×Q8—22no—
326C3×Q8—21no—
327C3×Q8—22no—
328C3×Q8—21no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×Q8—62no—
338C3×Q8—62no—
339C3×Q8—121no—
340C3×Q8—121no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—21no—
346C3×D4—41no—
347C3×D4—42no—
348C3×D4—42no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—62no—
378C2×C3×C4—121no—
379C2×C3×C4—121no—
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4D3×D411yescharacteristic
382C2×C3×C4D3×D411yescharacteristic
383C2×C3×C4C2×#6711yescharacteristic
384C2×C3×C4C2×#6711yescharacteristic
385C2×C3×C4C2×#6711yescharacteristic
386C2×C3×C4C2×C3×D411yescharacteristic
387C2×C3×C4C2×C3×D411yescharacteristic
388C2×C3×C4C2×C3×D411yescharacteristic
389C2×C3×C4C2×C3×D411yescharacteristic
390C2×C3×C4C2×C3×D411yescharacteristic
391C2×C3×C4C2×C3×D411yescharacteristic
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395Dic6—22no—
396Dic6—22no—
397Dic6—41no—
398Dic6—41no—
399C3×C8—24no—
400C3×C8—22no—
401C3×C8—22no—
402C3×C8—62no—
403C3×C8—62no—
404C3×C8—121no—
405C3⋊C8—22no—
406C3⋊C8—22no—
407C3⋊C8—41no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—81no—
411C2×C32#39211yescharacteristic
412C2×C32#47312yes—
413#41—21no—
414#41—21no—
415#41—21no—
416#41—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8C3×#6711yescharacteristic
429C2×Q8C3×#6712yes—
430C2×D4—21no—
431C2×D4C3×#6711yescharacteristic
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—61no—
435C22×C4C22×C3×D311yescharacteristic
436Q16—24no—
437Q16—68no—
438Q16—64no—
439SD16—24no—
440SD16—64no—
441M16—21no—
442M16—32no—
443M16—61no—
444M16—61no—
445M16—61no—
446M16C3×#6712yes—
447C2×C8—21no—
448C2×C8—32no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—82no—
465C22×C3#93811yescharacteristic
466C22×C3#93911yescharacteristic
467C22×C3#93911yescharacteristic
468C22×C3C3×#12111yescharacteristic
469C22×C3C3×#12111yescharacteristic
470C22×C3C3×#12111yescharacteristic
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—42no—
476C3×C4—44no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—122no—
482C3×C4#93711yescharacteristic
483C3×C4#93711yescharacteristic
484C3×C4#94011yescharacteristic
485C3×C4#95311yescharacteristic
486C3×C4C3×#12011yescharacteristic
487C3×C4C3×#12011yescharacteristic
488C3×C4C3×#12011yescharacteristic
489C3×C4C3×#12711yescharacteristic
490Dic3—42no—
491C32#202011yescharacteristic, complemented, Sylow
492C23—21no—
493C23C3×#23711yescharacteristic
494Q8—22no—
495Q8—21no—
496Q8—22no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—62no—
501Q8—62no—
502Q8—121no—
503Q8—121no—
504D4—21no—
505D4—21no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—62no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4C3×D3×D411yescharacteristic
518C2×C4C3×D3×D411yescharacteristic
519C2×C4C3×D3×D411yescharacteristic
520C2×C4C2×C3×#6711yescharacteristic
521C2×C4C2×C3×#6711yescharacteristic
522C2×C4C2×C3×#6711yescharacteristic
523C8—22no—
524C8—62no—
525C8—62no—
526C8—121no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—22no—
532C2×C3—21no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3—82no—
536C2×C3—81no—
537C2×C3—81no—
538C2×C3#520412yes—
539C2×C3#522311yescharacteristic
540C2×C3C3×#39211yescharacteristic
541C2×C3C3×#47312yes—
542C22—22no—
543C22—42no—
544C22—41no—
545C22C3×#93811yescharacteristic
546C22C3×#93911yescharacteristic
547C22C3×#93911yescharacteristic
548C4—41no—
549C4—42no—
550C4—122no—
551C4C3×#93711yescharacteristic
552C4C3×#93711yescharacteristic
553C4C3×#94011yescharacteristic
554C4C3×#95311yescharacteristic
555C3—21no—
556C3#7210911yescharacteristic, direct product factor
557C3C3×#202011yescharacteristic, complemented
558C2—21no—
559C2—21no—
560C2—81no—
561C2C3×#520412yes—
562C2C3×#522311yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#137C2
socleC22×C32#121
prime radicalC2×C32×#41C22
hypercenterC2×C3×#137D3
hypocenterC3C3×#2020
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
3. C22×C3×#216C2semidirect
5. C2×C3×#836C2semidirect
6. C2×C3×#824C2semidirect
7. C3×#4558C2semidirect
9. #72109C3direct (N × H)
37. C22×#216C2×C3semidirect
42. C2×#836C2×C3semidirect
43. C2×#824C2×C3semidirect
45. #4558C2×C3semidirect
491. C32#2020semidirect
556. C3#72109direct (N × H)
557. C3C3×#2020semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C2×C3×#9512
C22×C3×#2162
C3×#52342
C2×C3×#8362
C2×C3×#8242
C3×#45582
C3×#20203
#721093