Finite Group Databasebeta

← #13147886

Subgroups — #13147886

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#272C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721156C211yesmaximal, characteristic
5#11721116C211yesmaximal, characteristic, complemented
6#11721114C211yesmaximal, characteristic, complemented
7#11721106C211yesmaximal, characteristic, complemented
8#11721104C212yesmaximal, complemented
9#11721103C212yesmaximal, complemented
10#11721096C212yesmaximal, complemented
11#11721092C212yesmaximal, complemented
12#10593983—31nomaximal
13#10593876—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6200C2212yes—
19C2×#6196C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22C4⋊#1084C2211yescharacteristic
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2211yescharacteristic
27#3750⋊C3C2211yescharacteristic
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88451C2212yes—
32#88451C2212yes—
33#88451C2212yes—
34#88450C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37#64081D311yescharacteristic, complemented
38C4×#634D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—61no—
42C3×#872—31no—
43C3×C4×Dic6—31no—
44#63919—31no—
45#63918—31no—
46#63916—31no—
47#63915—32no—
48#63908—31no—
49#63905—32no—
50#63871—31no—
51#63863—31no—
52#63862—32no—
53#63860—31no—
54#63859—32no—
55#63857—31no—
56#63852—32no—
57#63851—32no—
58#63851—32no—
59#63850—32no—
60C22×#272C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C4⋊#272—22no—
67C4×#272—42no—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—22no—
74#634⋊C3—22no—
75#6200—42no—
76#6196—42no—
77#5582—91no—
78#5561—91no—
79#5534—91no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×#634C2×D311yescharacteristic
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—62no—
89C2×C3×Dic6—31no—
90C2×#627—32no—
91C2×#627—32no—
92C2×#626—32no—
93C2×#625—32no—
94C32×C42—21no—
95C32×C42C2×D311yescharacteristic
96C32×C42C2×D311yescharacteristic
97#3752C2×D311yescharacteristic
98#3752C2×D311yescharacteristic
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100#3750C2×D312yes—
101#3750C2×D311yescharacteristic
102#3750C2×D311yescharacteristic
103C4×C3⋊Dic3C2×D312yes—
104C4×C3⋊Dic3C2×D311yescharacteristic
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—31no—
108C3×(C2×D3)⋊C4—31no—
109C3×(C2×D3)⋊C4—62no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—61no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—31no—
116C3×Dic3⋊C4—31no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—31no—
119#3724—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123#3722—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—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×#272—22no—
133C2×#272—22no—
134C2×#272—22no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141#1084—22no—
142(C3⋊Dic3)⋊C3—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—31no—
175C4×D12—61no—
176#872—31no—
177#872—91no—
178#869—91no—
179C4×Dic6—31no—
180C4×Dic6—91no—
181C22×C3⋊D3C22×D311yescharacteristic
182C22×C3×D3—31no—
183C22×C3×D3—32no—
184C22×C3×D3—32no—
185C22×C3×D3—62no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D312yes—
200C4×C3⋊D3—22no—
201#634—22no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—62no—
210C3×D12—64no—
211C3×D12—62no—
212C3×D12—62no—
213C3×C4×D3—62no—
214C3×C4×D3—122no—
215C3×C4×D3—122no—
216C3×C4×D3—124no—
217C3×Dic6—62no—
218#627—122no—
219#627—122no—
220#626—122no—
221#625—122no—
222#542—271noSylow
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
225C2×H(3,𝔽3)2−1+411yescharacteristic
226#272—42no—
227(C3⋊D3)⋊C3—41no—
228C2×C3×Q8—91no—
229C2×C3×D4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—92no—
232C2×#67—92no—
233C2×#67—92no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—61no—
237C2×D12—92no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C2×C4×D3—32no—
241C2×C4×D3—62no—
242C2×C4×D3—91no—
243C2×Dic6—31no—
244C2×Dic6—91no—
245C2×Dic6—92no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C42—31no—
257C3×C42—31no—
258C3×C42—61no—
259C3×C42—91no—
260C3×C42—92no—
261C3×C42D3211yescharacteristic
262#217—92no—
263#217—92no—
264#217—92no—
265#217—92no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—31no—
269(C2×D3)⋊C4—31no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—91no—
272(C2×D3)⋊C4—91no—
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—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—61no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288Dic3⋊C4—32no—
289Dic3⋊C4—31no—
290Dic3⋊C4—31no—
291Dic3⋊C4—92no—
292Dic3⋊C4—91no—
293Dic3⋊C4—91no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C4×Dic3—32no—
299C4×Dic3—31no—
300C4×Dic3—92no—
301C4×Dic3—91no—
302C22×C32—21no—
303C22×C32C23×D311yescharacteristic
304C22×C32C23×D311yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3⋊D3—21no—
307C2×C3⋊D3—21no—
308C2×C3×D3—64no—
309C2×C3×D3—61no—
310C2×C3×D3—62no—
311C2×C3×D3—62no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C2×C3×D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—41no—
327C32×C4#23512yes—
328C32×C4#23512yes—
329C3⋊Dic3—22no—
330C3⋊Dic3—21no—
331C3⋊Dic3—22no—
332C3⋊Dic3—21no—
333C3⋊Dic3—21no—
334C3×Dic3—62no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—62no—
338C3×Dic3—61no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—122no—
342C4⋊Q8—271no—
343#126—272no—
344#124—272no—
345#124—271no—
346#123—272no—
347C22⋊Q8—272no—
348C22⋊Q8—271no—
349#121—271no—
350C4×Q8—91no—
351C4×D4—91no—
352#117—91no—
353H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C23×C3—92no—
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—31no—
359C22×D3—62no—
360C22×D3—91no—
361C3×Q8—182no—
362C3×D4—182no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—91no—
373C2×C3×C4—92no—
374C2×C3×C4—92no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—182no—
379C2×C3×C4—362no—
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383#67—362no—
384#67—362no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—31no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—62no—
393C2×Dic3—91no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—91no—
397C2×Dic3—91no—
398D12—62no—
399D12—62no—
400D12—64no—
401D12—362no—
402C4×D3—62no—
403C4×D3—122no—
404C4×D3—122no—
405C4×D3—124no—
406C4×D3—182no—
407Dic6—62no—
408Dic6—182no—
409Dic6—362no—
410Dic6—362no—
411Dic6—362no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32C2×#23511yescharacteristic
416C2×C32C2×#23511yescharacteristic
417C2×C32#100711yescharacteristic
418C2×C32#100911yescharacteristic
419C2×C32#100911yescharacteristic
420C2×C32#101011yescharacteristic
421C3⋊D3—41no—
422C3×D3—124no—
423C3×D3—122no—
424C3×D3—121no—
425C3×D3—122no—
426C2×Q8—91no—
427C2×Q8—272no—
428C2×D4—91no—
429C2×D4—272no—
430C22×C4—91no—
431C22×C4—92no—
432C4⋊C4—91no—
433C4⋊C4—91no—
434C4⋊C4—91no—
435C4⋊C4—91no—
436C4⋊C4—92no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C22⋊C4—91no—
440C22⋊C4—91no—
441C22⋊C4—92no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22⋊C4—272no—
445C22⋊C4—272no—
446C42—91no—
447C42—92no—
448C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—61no—
452C22×C3—181no—
453C22×C3—181no—
454C22×C3—181no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3C22×D3211yescharacteristic
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—64no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—122no—
470C2×D3—122no—
471C2×D3—181no—
472C2×D3—181no—
473C2×D3—181no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—32no—
477C3×C4—32no—
478C3×C4—61no—
479C3×C4—61no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—62no—
483C3×C4—62no—
484C3×C4—121no—
485C3×C4—181no—
486C3×C4—182no—
487C3×C4—181no—
488C3×C4—182no—
489C3×C4—181no—
490C3×C4—182no—
491C3×C4#379812yes—
492Dic3—62no—
493Dic3—62no—
494Dic3—61no—
495Dic3—62no—
496Dic3—61no—
497Dic3—61no—
498Dic3—62no—
499Dic3—122no—
500Dic3—182no—
501Dic3—181no—
502Dic3—182no—
503Dic3—181no—
504Dic3—181no—
505C32—21no—
506C32#553411yescharacteristic, complemented
507C32#558211yescharacteristic, complemented
508C23—91no—
509C23—92no—
510Q8—182no—
511Q8—1082no—
512D4—182no—
513D4—1082no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4—91no—
520C2×C4—182no—
521C2×C4—362no—
522C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—361no—
535C2×C3—362no—
536C2×C3#6429711yescharacteristic
537C2×C3C2×#379811yescharacteristic
538C2×C3#6430811yescharacteristic
539D3—122no—
540D3—122no—
541D3—124no—
542D3—121no—
543D3—361no—
544C22—182no—
545C22—181no—
546C22—181no—
547C22—182no—
548C22—182no—
549C22—181no—
550C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
551C4—21no—
552C4—21no—
553C4—181no—
554C4—181no—
555C4—182no—
556C4—182no—
557C4—182no—
558C4—181no—
559C4#8865012yes—
560C3—31no—
561C3—31no—
562C3—61no—
563C3#1059387811yescharacteristic
564C2—362no—
565C2—361no—
566C2#1172248211yescharacteristic
567C2C2×#8865011yescharacteristic
568C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721156C2
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×C4⋊#272C2semidirect
2. #64081⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
5. #11721116C2semidirect
6. #11721114C2semidirect
7. #11721106C2semidirect
8. #11721104C2semidirect
9. #11721103C2semidirect
10. #11721096C2semidirect
11. #11721092C2semidirect
37. #64081D3semidirect
38. C4×#634D3semidirect
353. H(3,𝔽3)#542semidirect
506. C32#5534semidirect
507. C32#5582semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64081⋊C32
C4×#634⋊C32
#117211562
#117211162
#117211142
#117211062
#117211042
#117211032
#117210962
#117210922
#105939833
#105938763