Finite Group Databasebeta

← #13149608

Subgroups — #13149608

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721763C211yesmaximal, characteristic
2#64097⋊C3C211yesmaximal, characteristic, complemented
3#64096⋊C3C211yesmaximal, characteristic, complemented
4#11721151C212yesmaximal, complemented
5#11721149C211yesmaximal, characteristic, complemented
6#11721148C212yesmaximal, complemented
7#11721114C211yesmaximal, characteristic, complemented
8#11721114C212yesmaximal, complemented
9#11721097C211yesmaximal, characteristic, complemented
10#11721094C211yesmaximal, characteristic, complemented
11#11721092C212yesmaximal, complemented
12#10594498—31nomaximal
13#10594494—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4⋊#272C2212yes—
16C2×C4×#272C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2212yes—
19C2×#6199C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28#88451C2212yes—
29#88451C2212yes—
30#88451C2212yes—
31#88451C2211yescharacteristic
32#88451C2211yescharacteristic
33#88451C2212yes—
34#88449C2211yescharacteristic
35#88449C2211yescharacteristic
36#88448C2212yes—
37#64097D311yescharacteristic, complemented
38#64096D311yescharacteristic, complemented
39C3×#895—31no—
40C3×#895—31no—
41C3×#895—61no—
42C3×#888—31no—
43C3×#887—31no—
44#63898—31no—
45#63896—32no—
46Dic3×D12—31no—
47#63887—32no—
48#63884—32no—
49#63884—32no—
50#63863—32no—
51#63863—31no—
52#63862—32no—
53#63860—31no—
54#63860—32no—
55#63859—32no—
56#63857—31no—
57#63856—31no—
58#63855—31no—
59#63855—31no—
60C22×#272C2312yes—
61C22×#272C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C4⋊#272—22no—
66C4⋊#272—42no—
67C4×#272—22no—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74#634⋊C3—22no—
75#634⋊C3—42no—
76#6199—42no—
77#5733—91no—
78#5603—91no—
79#5582—91no—
80C2×#634C2×D311yescharacteristic
81C2×#634C2×D312yes—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—32no—
85C2×C3×D12—32no—
86C2×C3×D12—61no—
87C2×C3×D12—62no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3—61no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—32no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C32×C4⋊C4—21no—
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C2×D311yescharacteristic
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3750C2×D312yes—
100#3750C2×D312yes—
101C4×C3⋊Dic3C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D312yes—
103C4×C3⋊Dic3C2×D311yescharacteristic
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—62no—
107C3×C4⋊Dic3—31no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—31no—
113#3724—32no—
114Dic3⋊Dic3—31no—
115Dic3⋊Dic3—31no—
116Dic3⋊Dic3—31no—
117Dic3⋊Dic3—31no—
118Dic3⋊Dic3—32no—
119(C2×D3)⋊Dic3—31no—
120(C2×D3)⋊Dic3—31no—
121(C2×D3)⋊Dic3—31no—
122(C2×D3)⋊Dic3—31no—
123(C2×D3)⋊Dic3—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—21no—
133C2×#272—22no—
134C2×#272—21no—
135C2×#272—22no—
136C2×#272—21no—
137C2×#272—22no—
138C4×H(3,𝔽3)—22no—
139C4×H(3,𝔽3)#4112yes—
140#1084—21no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147C3×C4⋊Q8—91no—
148C3×#121—91no—
149C3×C4×Q8—91no—
150#947—92no—
151#947—91no—
152#946—92no—
153#946—91no—
154#936—92no—
155#936—91no—
156#935—92no—
157D4×Dic3—91no—
158#926—91no—
159#924—91no—
160#898—92no—
161#897—91no—
162#896—92no—
163#895—31no—
164#895—31no—
165#895—61no—
166#894—92no—
167#894—92no—
168#893—91no—
169#888—31no—
170#888—91no—
171#887—31no—
172#887—91no—
173#879—92no—
174#878—92no—
175#876—92no—
176#875—91no—
177#875—92no—
178C4×D12—91no—
179#872—91no—
180#872—91no—
181C22×C3×D3—31no—
182C22×C3×D3—31no—
183C22×C3×D3—32no—
184C22×C3×D3—32no—
185C22×C3×D3—61no—
186C22×C3×D3—62no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199#634—22no—
200#634—42no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—61no—
210C3×D12—62no—
211C3×D12—62no—
212C3×D12—122no—
213C3×D12—122no—
214C3×D12—122no—
215C3×D12—124no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218C3×C4×D3—122no—
219C3×Dic6—62no—
220C3×Dic6—122no—
221D3×Dic3—122no—
222D3×Dic3—122no—
223#542—271noSylow
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
226C2×H(3,𝔽3)2−1+411yescharacteristic
227#272—41no—
228#272—42no—
229C2×C3×Q8—91no—
230C2×C3×Q8—92no—
231C2×C3×D4—91no—
232C2×C3×D4—92no—
233C22×C3×C4—91no—
234C22×Dic3—92no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—32no—
238C2×D12—32no—
239C2×D12—61no—
240C2×D12—62no—
241C2×C4×D3—31no—
242C2×C4×D3—31no—
243C2×C4×D3—61no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×Dic6—31no—
247C2×Dic6—32no—
248C2×Dic6—91no—
249C2×Dic6—92no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—61no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4D3211yescharacteristic
257C3×C22⋊C4—92no—
258C3×C42—91no—
259C3×C42—91no—
260C3×C42—92no—
261#217—92no—
262#217—92no—
263#217—92no—
264#217—92no—
265#217—91no—
266#217—91no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—62no—
270(C2×D3)⋊C4—91no—
271(C2×D3)⋊C4—91no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4—91no—
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—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—31no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—91no—
293Dic3⋊C4—91no—
294Dic3⋊C4—91no—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297C4×Dic3—32no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—91no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C22×C32—21no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C2×C3×D3—62no—
307C2×C3×D3—61no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—62no—
313C2×C3×D3—61no—
314C2×C3×D3—62no—
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—61no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—122no—
322C2×C3×D3—122no—
323C2×C3×D3—122no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4#23512yes—
329C32×C4#23912yes—
330C3⋊Dic3—22no—
331C3⋊Dic3—22no—
332C3⋊Dic3—21no—
333C3⋊Dic3—21no—
334C3⋊Dic3—21no—
335C3⋊Dic3—21no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—62no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—62no—
343C3×Dic3—61no—
344C3×Dic3—121no—
345C4⋊Q8—91no—
346#126—272no—
347#124—271no—
348#124—272no—
349#123—272no—
350C22⋊Q8—272no—
351C22⋊Q8—271no—
352#121—91no—
353C4×Q8—91no—
354C4×D4—271no—
355#117—271no—
356H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
357C23×C3—91no—
358C23×C3—92no—
359C22×D3—32no—
360C22×D3—32no—
361C22×D3—31no—
362C22×D3—31no—
363C22×D3—61no—
364C22×D3—62no—
365C3×Q8—182no—
366C3×Q8—362no—
367C3×D4—182no—
368C3×D4—362no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—32no—
372C2×C3×C4—32no—
373C2×C3×C4—61no—
374C2×C3×C4—62no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—91no—
382C2×C3×C4—182no—
383C2×C3×C4C2×D3212yes—
384C2×C3×C4C2×D3211yescharacteristic
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—31no—
392C2×Dic3—32no—
393C2×Dic3—61no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398C2×Dic3—91no—
399C2×Dic3—92no—
400C2×Dic3—362no—
401D12—62no—
402D12—62no—
403D12—122no—
404D12—122no—
405D12—122no—
406D12—124no—
407C4×D3—62no—
408C4×D3—62no—
409C4×D3—122no—
410C4×D3—362no—
411C4×D3—362no—
412Dic6—62no—
413Dic6—122no—
414Dic6—182no—
415Dic6—362no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32C2×#23511yescharacteristic
420C2×C32#100411yescharacteristic
421C2×C32C2×#23911yescharacteristic
422C2×C32#100711yescharacteristic
423C2×C32#100711yescharacteristic
424C2×C32#100911yescharacteristic
425C3×D3—122no—
426C3×D3—121no—
427C3×D3—122no—
428C3×D3—121no—
429C3×D3—122no—
430C3×D3—242no—
431C2×Q8—91no—
432C2×Q8—92no—
433C2×D4—91no—
434C2×D4—92no—
435C22×C4—91no—
436C22×C4—272no—
437C4⋊C4—91no—
438C4⋊C4—92no—
439C4⋊C4—92no—
440C4⋊C4—271no—
441C4⋊C4—271no—
442C4⋊C4—272no—
443C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C22⋊C4—92no—
445C22⋊C4—271no—
446C22⋊C4—271no—
447C22⋊C4—272no—
448C22⋊C4—272no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C42—91no—
452C42—91no—
453C42—92no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—61no—
457C22×C3—181no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—181no—
461C22×C3—181no—
462C22×C3—182no—
463C22×C3C22×D3211yescharacteristic
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—62no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—61no—
471C2×D3—61no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—61no—
475C2×D3—61no—
476C2×D3—62no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—122no—
482C3×C4—22no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—62no—
488C3×C4—122no—
489C3×C4—181no—
490C3×C4—182no—
491C3×C4—182no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—181no—
495C3×C4—181no—
496C3×C4#379912yes—
497Dic3—61no—
498Dic3—61no—
499Dic3—62no—
500Dic3—61no—
501Dic3—62no—
502Dic3—61no—
503Dic3—61no—
504Dic3—61no—
505Dic3—121no—
506Dic3—182no—
507Dic3—181no—
508Dic3—182no—
509Dic3—181no—
510Dic3—181no—
511Dic3—181no—
512C32—21no—
513C32#558211yescharacteristic, complemented
514C32#573311yescharacteristic, complemented
515C23—92no—
516C23—91no—
517Q8—182no—
518Q8—362no—
519D4—182no—
520D4—362no—
521C2×C4—91no—
522C2×C4—91no—
523C2×C4—92no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—91no—
527C2×C4—92no—
528C2×C4—182no—
529C2×C4—1082no—
530C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
531C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—361no—
542C2×C3—362no—
543C2×C3C2×#379911yescharacteristic
544C2×C3#6431411yescharacteristic
545C2×C3#6431611yescharacteristic
546D3—121no—
547D3—121no—
548D3—122no—
549D3—122no—
550D3—122no—
551D3—242no—
552C22—182no—
553C22—181no—
554C22—181no—
555C22—182no—
556C22—181no—
557C22—182no—
558C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
559C4—22no—
560C4—181no—
561C4—182no—
562C4—181no—
563C4—181no—
564C4—181no—
565C4—181no—
566C4—182no—
567C4#8865212yes—
568C3—31no—
569C3—31no—
570C3—61no—
571C3#1059435211yescharacteristic
572C2—361no—
573C2—362no—
574C2C2×#8865211yescharacteristic
575C2#1172250711yescharacteristic
576C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radical#11721763C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#542
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64097⋊C3C2semidirect
3. #64096⋊C3C2semidirect
4. #11721151C2semidirect
5. #11721149C2semidirect
6. #11721148C2semidirect
7. #11721114C2semidirect
8. #11721114C2semidirect
9. #11721097C2semidirect
10. #11721094C2semidirect
11. #11721092C2semidirect
37. #64097D3semidirect
38. #64096D3semidirect
356. H(3,𝔽3)#542semidirect
513. C32#5582semidirect
514. C32#5733semidirect

Maximal subgroups

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

subgroupindex
#117217632
#64097⋊C32
#64096⋊C32
#117211512
#117211492
#117211482
#117211142
#117210972
#117210942
#117210922
#105944983
#105944943