Finite Group Databasebeta

← #13149602

Subgroups — #13149602

Order 1728.

892 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721761C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3#64097⋊C3C211yesmaximal, characteristic, complemented
4#11721150C212yesmaximal, complemented
5#11721149C211yesmaximal, characteristic
6#11721149C211yesmaximal, characteristic
7#11721148C212yesmaximal, complemented
8#11721147C212yesmaximal, complemented
9C4×#6196C211yesmaximal, characteristic, complemented
10#11721095C211yesmaximal, characteristic, complemented
11#11721094C212yesmaximal, complemented
12#10594485—31nomaximal
13#10594460—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×#634⋊C3C2212yes—
17C2×#6199C2212yes—
18C2×#6199C2212yes—
19C2×#6196C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22C4×#1084C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88451C2212yes—
31#88451C2212yes—
32#88449C2211yescharacteristic
33#88448C2211yescharacteristic
34#88448C2212yes—
35#88448C2212yes—
36#88448C2211yescharacteristic
37#64099D311yescharacteristic, complemented
38#64097D311yescharacteristic, complemented
39C3×#893—31no—
40C3×#893—31no—
41C3×#893—61no—
42C3×#890—31no—
43C3×#888—31no—
44C4×#627—31no—
45C4×#627—31no—
46#63899—32no—
47#63898—31no—
48#63898—31no—
49#63896—32no—
50#63893—32no—
51Dic3×D12—31no—
52Dic3×D12—31no—
53#63887—32no—
54D3×Dic3⋊C4—32no—
55D3×Dic3⋊C4—32no—
56#63869—31no—
57#63869—31no—
58#63857—32no—
59#63856—32no—
60C22×#272C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C4⋊#272Q814yescomplemented
66C4×#272—42no—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—42no—
74#6199—24no—
75#6199—42no—
76#6196—22no—
77#5730—91no—
78#5690—91no—
79#5539—91no—
80C2×#634C2×D312yes—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—62no—
87C2×C3×Dic6—32no—
88C2×#627—31no—
89C2×#627—31no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98C4⋊(C3⋊Dic3)C2×D312yes—
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100#3750C2×D312yes—
101#3750C2×D312yes—
102C4×C3⋊Dic3C2×D311yescharacteristic
103C4×C3⋊Dic3C2×D311yescharacteristic
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—62no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—31no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—61no—
117#3724—31no—
118#3724—32no—
119#3724—32no—
120#3724—31no—
121Dic3⋊Dic3—31no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—31no—
125Dic3⋊Dic3—31no—
126Dic3⋊Dic3—31no—
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×#272C2×Q814yes—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)C2×Q812yes—
137#1084—21no—
138#1084—21no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4⋊Q8—91no—
146C3×#125—91no—
147C3×C4×D4—91no—
148#946—92no—
149#946—92no—
150D4×Dic3—91no—
151D4×Dic3—91no—
152#924—92no—
153C2×Dic3⋊C4—92no—
154#897—92no—
155#897—92no—
156#896—92no—
157#896—92no—
158#893—31no—
159#893—31no—
160#893—61no—
161#893—91no—
162#893—91no—
163D3×C4⋊C4—92no—
164D3×C4⋊C4—92no—
165#890—31no—
166#890—91no—
167#890—91no—
168#890—91no—
169#888—31no—
170#888—91no—
171#878—92no—
172#878—92no—
173C4×D12—91no—
174C4×D12—91no—
175#870—91no—
176C4×Dic6—91no—
177C4×Dic6—91no—
178C4×Dic6—91no—
179C22×C3×D3—32no—
180C22×C3×D3—32no—
181C22×C3×D3—62no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194#634—42no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C3×D12—34no—
208C3×D12—34no—
209C3×D12—64no—
210C3×C4×D3—122no—
211C3×C4×D3—122no—
212C3×C4×D3—124no—
213C3×Dic6—122no—
214#627—62no—
215#627—62no—
216D3×Dic3—64no—
217D3×Dic3—64no—
218D3×Dic3—122no—
219D3×Dic3—122no—
220#554—271noSylow
221C2×H(3,𝔽3)C22×Q811yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
224#272—24no—
225C2×C3×Q8—92no—
226C2×C3×D4—91no—
227C22×C3×C4—92no—
228C22×Dic3—92no—
229C22×Dic3—92no—
230C2×D12—31no—
231C2×D12—31no—
232C2×D12—61no—
233C2×C4×D3—32no—
234C2×C4×D3—32no—
235C2×C4×D3—62no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×Dic6—32no—
241C2×Dic6—92no—
242C2×Dic6—91no—
243C2×Dic6—91no—
244C2×Dic6—91no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4D3211yescharacteristic
254C3×C22⋊C4—92no—
255C3×C42—91no—
256C3×C42—91no—
257C3×C42—91no—
258#217—92no—
259#217—92no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—32no—
270C4⋊Dic3—91no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—91no—
276C4⋊Dic3—92no—
277C4⋊Dic3—92no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—91no—
287Dic3⋊C4—91no—
288Dic3⋊C4—91no—
289Dic3⋊C4—91no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—31no—
295C4×Dic3—31no—
296C4×Dic3—61no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C22×C32—21no—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D311yescharacteristic
302C2×C3×D3—34no—
303C2×C3×D3—34no—
304C2×C3×D3—64no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C32×C4—22no—
312C32×C4—22no—
313C32×C4—22no—
314C32×C4—42no—
315C32×C4D3×Q812yes—
316C32×C4D3×Q812yes—
317C3⋊Dic3—22no—
318C3⋊Dic3—21no—
319C3⋊Dic3—22no—
320C3⋊Dic3—21no—
321C3⋊Dic3—21no—
322C3⋊Dic3#23912yes—
323C3×Dic3—32no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—121no—
335C4⋊Q8—91no—
336#125—91no—
337#125—271no—
338C22⋊Q8—272no—
339C22⋊Q8—272no—
340C22⋊Q8—272no—
341C4×Q8—271no—
342C4×D4—91no—
343C4×D4—271no—
344C4×D4—271no—
345C2×C4⋊C4—272no—
346H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
347C23×C3—92no—
348C22×D3—32no—
349C22×D3—32no—
350C22×D3—62no—
351C3×Q8—362no—
352C3×D4—94no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—362no—
368C2×C3×C4C2×D3212yes—
369C2×C3×C4C2×D3211yescharacteristic
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387C2×Dic3—92no—
388C2×Dic3—184no—
389C2×Dic3—362no—
390D12—34no—
391D12—34no—
392D12—64no—
393C4×D3—122no—
394C4×D3—122no—
395C4×D3—124no—
396C4×D3—184no—
397C4×D3—184no—
398C4×D3—362no—
399C4×D3—362no—
400Dic6—122no—
401Dic6—182no—
402Dic6—182no—
403Dic6—182no—
404Dic6—362no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32#100411yescharacteristic
409C2×C32C2×D3×Q811yescharacteristic
410C2×C32C2×D3×Q811yescharacteristic
411C2×C32C2×#23911yescharacteristic
412C2×C32D3×#4111yescharacteristic
413C2×C32#100911yescharacteristic
414C3×D3—64no—
415C3×D3—64no—
416C3×D3—124no—
417C2×Q8—92no—
418C2×Q8—271no—
419C2×D4—91no—
420C22×C4—92no—
421C22×C4—272no—
422C22×C4—272no—
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—92no—
428C4⋊C4—271no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C4⋊C4—271no—
432C4⋊C4—272no—
433C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
434C22⋊C4—92no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C42—91no—
438C42—91no—
439C42—91no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—61no—
443C22×C3—94no—
444C22×C3—182no—
445C22×C3—182no—
446C22×C3C22×D3211yescharacteristic
447C2×D3—34no—
448C2×D3—34no—
449C2×D3—64no—
450C2×D3—62no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—122no—
455C2×D3—122no—
456C3×C4—22no—
457C3×C4—32no—
458C3×C4—32no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—92no—
463C3×C4—122no—
464C3×C4—181no—
465C3×C4—182no—
466C3×C4—181no—
467C3×C4—182no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4—181no—
471C3×C4#379712yes—
472Dic3—32no—
473Dic3—61no—
474Dic3—61no—
475Dic3—61no—
476Dic3—61no—
477Dic3—62no—
478Dic3—61no—
479Dic3—62no—
480Dic3—61no—
481Dic3—61no—
482Dic3—62no—
483Dic3—92no—
484Dic3—121no—
485Dic3—181no—
486Dic3—181no—
487Dic3—182no—
488Dic3—182no—
489Dic3—181no—
490C32—21no—
491C32#569011yescharacteristic, complemented
492C32#573011yescharacteristic, complemented
493C23—92no—
494Q8—362no—
495Q8—542no—
496D4—94no—
497C2×C4—91no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—362no—
506C2×C4—544no—
507C2×C4—1082no—
508C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
509C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—61no—
517C2×C3—61no—
518C2×C3—61no—
519C2×C3—184no—
520C2×C3C2×#379711yescharacteristic
521C2×C3#6431411yescharacteristic
522C2×C3D3×#23911yescharacteristic
523D3—64no—
524D3—64no—
525D3—124no—
526C22—94no—
527C22—182no—
528C22—182no—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—22no—
531C4—92no—
532C4—181no—
533C4—181no—
534C4—182no—
535C4—181no—
536C4—182no—
537C4—181no—
538C4—181no—
539C4#8865112yes—
540C3—31no—
541C3—31no—
542C3—61no—
543C3#1059447011yescharacteristic
544C2—184no—
545C2C2×#8865111yescharacteristic
546C2#1172250711yescharacteristic
547C2#1172251911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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. #64099⋊C3C2semidirect
3. #64097⋊C3C2semidirect
4. #11721150C2semidirect
7. #11721148C2semidirect
8. #11721147C2semidirect
9. C4×#6196C2semidirect
10. #11721095C2semidirect
11. #11721094C2semidirect
37. #64099D3semidirect
38. #64097D3semidirect
65. C4⋊#272Q8semidirect
346. H(3,𝔽3)#554semidirect
491. C32#5690semidirect
492. C32#5730semidirect

Maximal subgroups

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

subgroupindex
#117217612
#64099⋊C32
#64097⋊C32
#117211502
#117211492
#117211482
#117211472
C4×#61962
#117210952
#117210942
#105944853
#105944603