Finite Group Databasebeta

← #13147849

Subgroups — #13147849

Order 1728.

826 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
1C4×#6229C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4#11721145C212yesmaximal, complemented
5#11721141C211yesmaximal, characteristic
6#11721139C212yesmaximal, complemented
7#11721116C211yesmaximal, characteristic, complemented
8#11721112C212yesmaximal, complemented
9#11721110C211yesmaximal, characteristic, complemented
10#11721107C211yesmaximal, characteristic, complemented
11#11721101C212yesmaximal, complemented
12#10593878—31nomaximal
13#10593876—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#6198C2212yes—
19C2×#6196C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2212yes—
23C4×#1084C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2211yescharacteristic
26#3752⋊C3C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
38#64081D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×#872—31no—
41C3×C4×Dic6—31no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—61no—
44#63919—31no—
45#63918—31no—
46#63916—31no—
47#63909—31no—
48#63871—31no—
49#63868—32no—
50#63865—32no—
51#63864—31no—
52#63863—31no—
53#63859—32no—
54#63856—31no—
55#63853—32no—
56#63852—32no—
57#63852—32no—
58#63851—32no—
59#63850—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67#6229—22no—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—42no—
75#6198—42no—
76#6196—42no—
77#5582—91no—
78#5561—91no—
79#5534—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D312yes—
83C2×C3×D12—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#627—32no—
90C2×#627—32no—
91C2×#626—32no—
92C2×#626—32no—
93C32×C42—21no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96#3752C2×D311yescharacteristic
97#3752C2×D311yescharacteristic
98#3752C2×D312yes—
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100#3750C2×D311yescharacteristic
101#3750C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D311yescharacteristic
103C3×(C2×D3)⋊C4—31no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—31no—
114C3×Dic3⋊C4—62no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—62no—
119#3724—32no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123#3722—32no—
124#3722—32no—
125#3722—32no—
126#3722—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141#1084—22no—
142#1084—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—21no—
147C3×C4×Q8—91no—
148C3×C4×D4—91no—
149C3×#117—91no—
150#947—91no—
151#946—91no—
152#938—91no—
153#936—91no—
154#930—91no—
155#929—92no—
156#924—92no—
157#924—91no—
158#898—92no—
159#897—91no—
160#896—92no—
161#896—92no—
162#895—91no—
163#894—92no—
164#888—91no—
165#888—91no—
166#885—92no—
167#885—92no—
168#883—92no—
169#879—92no—
170#876—92no—
171#875—92no—
172#875—91no—
173C4×D12—31no—
174C4×D12—91no—
175#872—31no—
176#872—91no—
177#869—91no—
178C4×Dic6—31no—
179C4×Dic6—31no—
180C4×Dic6—61no—
181C22×C3⋊D3C22×D312yes—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3×D3—32no—
184C22×C3×D3—31no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C4⋊(C3⋊D3)—22no—
199C4×C3⋊D3—22no—
200C4×C3⋊D3—42no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C3×D12—62no—
212C3×C4×D3—62no—
213C3×C4×D3—122no—
214C3×Dic6—64no—
215C3×Dic6—62no—
216C3×Dic6—62no—
217#627—122no—
218#627—122no—
219#626—122no—
220#626—122no—
221#542—271noSylow
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
224C2×H(3,𝔽3)2−1+411yescharacteristic
225(C3⋊D3)⋊C3—42no—
226(C3⋊D3)⋊C3—41no—
227C2×C3×Q8—91no—
228C2×C3×D4—91no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C2×#67—92no—
232C2×#67—92no—
233C2×D12—31no—
234C2×D12—91no—
235C2×D12—92no—
236C2×C4×D3—31no—
237C2×C4×D3—32no—
238C2×C4×D3—91no—
239C2×C4×D3—92no—
240C2×Dic6—31no—
241C2×Dic6—31no—
242C2×Dic6—61no—
243C2×Dic6—92no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4—91no—
254C3×C42—31no—
255C3×C42—31no—
256C3×C42—61no—
257C3×C42—91no—
258C3×C42—92no—
259C3×C42D3211yescharacteristic
260#217—92no—
261#217—92no—
262#217—92no—
263#217—92no—
264(C2×D3)⋊C4—31no—
265(C2×D3)⋊C4—31no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—91no—
268(C2×D3)⋊C4—91no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—61no—
282C4⋊Dic3—91no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—31no—
288Dic3⋊C4—31no—
289Dic3⋊C4—62no—
290Dic3⋊C4—91no—
291Dic3⋊C4—91no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296C4×Dic3—32no—
297C4×Dic3—32no—
298C4×Dic3—31no—
299C4×Dic3—62no—
300C4×Dic3—91no—
301C22×C32—21no—
302C22×C32C23×D311yescharacteristic
303C22×C32C23×D311yescharacteristic
304C2×C3⋊D3—21no—
305C2×C3⋊D3—22no—
306C2×C3⋊D3—22no—
307C2×C3⋊D3—21no—
308C2×C3⋊D3—21no—
309C2×C3⋊D3—22no—
310C2×C3×D3—61no—
311C2×C3×D3—62no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C32×C4—22no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—41no—
323C32×C4#23512yes—
324C32×C4#23512yes—
325C3⋊Dic3—21no—
326C3⋊Dic3—22no—
327C3⋊Dic3—21no—
328C3⋊Dic3—21no—
329C3×Dic3—64no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—61no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C3×Dic3—122no—
339C4⋊Q8—271no—
340#126—272no—
341#124—272no—
342#124—271no—
343#123—272no—
344C22⋊Q8—272no—
345C22⋊Q8—271no—
346#121—271no—
347C4×Q8—91no—
348C4×D4—91no—
349#117—91no—
350H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
351C23×C3—92no—
352C23×C3—91no—
353C22×D3—31no—
354C22×D3—32no—
355C22×D3—92no—
356C22×D3—91no—
357C3×Q8—182no—
358C3×D4—182no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—91no—
369C2×C3×C4—92no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—182no—
375C2×C3×C4—362no—
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379#67—362no—
380#67—362no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—32no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—92no—
395D12—62no—
396D12—182no—
397D12—362no—
398C4×D3—62no—
399C4×D3—122no—
400C4×D3—182no—
401C4×D3—362no—
402Dic6—62no—
403Dic6—62no—
404Dic6—64no—
405Dic6—362no—
406Dic6—362no—
407Dic6—362no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32C2×#23511yescharacteristic
412C2×C32C2×#23511yescharacteristic
413C2×C32#100411yescharacteristic
414C2×C32#100911yescharacteristic
415C2×C32#101011yescharacteristic
416C2×C32#101011yescharacteristic
417C3⋊D3—41no—
418C3⋊D3—42no—
419C3×D3—122no—
420C3×D3—121no—
421C2×Q8—91no—
422C2×Q8—272no—
423C2×D4—91no—
424C2×D4—272no—
425C22×C4—91no—
426C22×C4—92no—
427C4⋊C4—91no—
428C4⋊C4—91no—
429C4⋊C4—91no—
430C4⋊C4—91no—
431C4⋊C4—92no—
432C4⋊C4—272no—
433C4⋊C4—272no—
434C22⋊C4—91no—
435C22⋊C4—91no—
436C22⋊C4—92no—
437C22⋊C4—272no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C42—91no—
442C42—92no—
443C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—61no—
447C22×C3—181no—
448C22×C3—182no—
449C22×C3—181no—
450C22×C3—182no—
451C22×C3—182no—
452C22×C3—181no—
453C22×C3C22×D3211yescharacteristic
454C2×D3—61no—
455C2×D3—62no—
456C2×D3—61no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—61no—
460C2×D3—181no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—182no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—32no—
469C3×C4—32no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—121no—
477C3×C4—181no—
478C3×C4—182no—
479C3×C4—181no—
480C3×C4—181no—
481C3×C4—182no—
482C3×C4—182no—
483C3×C4#379812yes—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—62no—
488Dic3—64no—
489Dic3—61no—
490Dic3—61no—
491Dic3—61no—
492Dic3—62no—
493Dic3—122no—
494Dic3—181no—
495Dic3—182no—
496Dic3—181no—
497Dic3—181no—
498C32—21no—
499C32#553411yescharacteristic, complemented
500C32#556111yescharacteristic, complemented
501C23—91no—
502C23—92no—
503Q8—182no—
504Q8—1082no—
505D4—182no—
506D4—1082no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—91no—
513C2×C4—182no—
514C2×C4—362no—
515C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
517C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—361no—
528C2×C3—362no—
529C2×C3#6429811yescharacteristic
530C2×C3C2×#379811yescharacteristic
531C2×C3#6430711yescharacteristic
532D3—121no—
533D3—122no—
534D3—362no—
535D3—361no—
536C22—181no—
537C22—182no—
538C22—181no—
539C22—182no—
540C22—182no—
541C22—181no—
542C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
543C4—21no—
544C4—21no—
545C4—181no—
546C4—181no—
547C4—182no—
548C4—182no—
549C4—182no—
550C4—181no—
551C4#8865012yes—
552C3—31no—
553C3—31no—
554C3—61no—
555C3#1059398311yescharacteristic
556C2—361no—
557C2—362no—
558C2#1172248311yescharacteristic
559C2C2×#8865011yescharacteristic
560C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721141C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#542
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6229C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64081⋊C3C2semidirect
4. #11721145C2semidirect
6. #11721139C2semidirect
7. #11721116C2semidirect
8. #11721112C2semidirect
9. #11721110C2semidirect
10. #11721107C2semidirect
11. #11721101C2semidirect
37. C4×C4⋊(C3⋊D3)D3semidirect
38. #64081D3semidirect
350. H(3,𝔽3)#542semidirect
499. C32#5534semidirect
500. C32#5561semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
#64081⋊C32
#117211452
#117211412
#117211392
#117211162
#117211122
#117211102
#117211072
#117211012
#105938783
#105938763