Finite Group Databasebeta

← #13147871

Subgroups — #13147871

Order 1728.

797 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721718C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721152C211yesmaximal, characteristic
5#11721150C212yesmaximal
6#11721148C212yesmaximal
7#11721117C211yesmaximal, characteristic, complemented
8#11721095C211yesmaximal, characteristic, complemented
9#11721089C211yesmaximal, characteristic
10#11721086C212yesmaximal, complemented
11#11721079C212yesmaximal, complemented
12#10593959—31nomaximal
13#10593954—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×#6199C2212yes—
17C2×#6196C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#88542C2211yescharacteristic
20#88542C2211yescharacteristic
21#88540C2211yescharacteristic
22#88540C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#88451C2212yes—
32#88449C2212yes—
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64079D311yescharacteristic, complemented
38#64078D311yescharacteristic, complemented
39C3×#872—31no—
40C3×#872—31no—
41C3×#872—61no—
42C3×#870—31no—
43C3×#869—31no—
44#63918—31no—
45#63918—31no—
46#63899—32no—
47#63896—32no—
48#63893—32no—
49D3×C4⋊Dic3—31no—
50D3×C4⋊Dic3—31no—
51#63887—32no—
52#63869—31no—
53#63869—31no—
54#63846—32no—
55#63845—32no—
56#63841—31no—
57#63841—31no—
58#63840—32no—
59#63839—32no—
60C22×#272C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C4×#272Q814yescomplemented
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72#634⋊C3—42no—
73#6199—42no—
74#6196—42no—
75#5729—91no—
76#5689—91no—
77#5519—91no—
78C2×#634C2×D312yes—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—32no—
83C2×#627—32no—
84C2×#627—32no—
85C2×D3×Dic3—32no—
86C2×D3×Dic3—32no—
87C32×C42—21no—
88C32×C42C2×D311yescharacteristic
89C32×C42C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)C2×D312yes—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95#3750C2×D312yes—
96#3750C2×D312yes—
97C3×(C2×D3)⋊C4—31no—
98C3×(C2×D3)⋊C4—31no—
99C3×(C2×D3)⋊C4—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4—61no—
102C3×(C2×D3)⋊C4—61no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×Dic3⋊C4—31no—
109C3×Dic3⋊C4—31no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—31no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—31no—
114C3×Dic3⋊C4—61no—
115C3×Dic3⋊C4—61no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—61no—
119#3724—32no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129Dic32—32no—
130Dic32—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×#272—21no—
133C2×#272—21no—
134C2×#272C2×Q812yes—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)C2×Q812yes—
138#1084—21no—
139#1084—21no—
140#1084C2×Q812yes—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—22no—
145C3×C4⋊Q8—91no—
146C3×#125—91no—
147C3×#117—91no—
148#946—92no—
149#944—92no—
150#926—91no—
151C2×C4⋊Dic3—91no—
152#897—92no—
153#896—92no—
154#896—92no—
155#892—91no—
156#892—91no—
157D3×C4⋊C4—91no—
158D3×C4⋊C4—91no—
159#890—91no—
160#890—91no—
161#890—92no—
162#889—92no—
163#889—92no—
164#888—91no—
165#888—91no—
166#888—92no—
167#888—92no—
168#878—92no—
169#878—92no—
170#872—31no—
171#872—31no—
172#872—61no—
173#870—31no—
174#870—91no—
175#870—91no—
176#869—31no—
177#869—91no—
178#869—91no—
179C22×C3×D3—31no—
180C22×C3×D3—31no—
181C22×C3×D3—61no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195#634—42no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C3×C4×D3—34no—
210C3×C4×D3—34no—
211C3×C4×D3—64no—
212C3×Dic6—122no—
213#627—122no—
214#627—122no—
215D3×Dic3—122no—
216D3×Dic3—122no—
217#544—271noSylow
218C2×H(3,𝔽3)C22×Q811yescharacteristic
219C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
220C2×H(3,𝔽3)2−1+411yescharacteristic
221#272—22no—
222C2×C3×Q8—92no—
223C22×C3×C4—91no—
224C22×Dic3—92no—
225C2×C4×D3—31no—
226C2×C4×D3—31no—
227C2×C4×D3—61no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×Dic6—32no—
231C2×Dic6—92no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—91no—
246C3×C42—31no—
247C3×C42—31no—
248C3×C42—61no—
249C3×C42—91no—
250C3×C42D3211yescharacteristic
251#217—92no—
252(C2×D3)⋊C4—31no—
253(C2×D3)⋊C4—31no—
254(C2×D3)⋊C4—31no—
255(C2×D3)⋊C4—31no—
256(C2×D3)⋊C4—61no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—32no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—91no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274Dic3⋊C4—31no—
275Dic3⋊C4—31no—
276Dic3⋊C4—31no—
277Dic3⋊C4—31no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—61no—
281Dic3⋊C4—61no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—61no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C23×D311yescharacteristic
301C2×C3×D3—32no—
302C2×C3×D3—32no—
303C2×C3×D3—62no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—22no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4D3×Q812yes—
318C32×C4D3×Q812yes—
319C3⋊Dic3—22no—
320C3⋊Dic3—22no—
321C3⋊Dic3—22no—
322C3⋊Dic3—22no—
323C3×Dic3—32no—
324C3×Dic3—32no—
325C3×Dic3—62no—
326C3×Dic3—62no—
327C3×Dic3—61no—
328C3×Dic3—62no—
329C3×Dic3—61no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—121no—
335C3×Dic3—121no—
336C4⋊Q8—91no—
337C4⋊Q8—271no—
338C4⋊Q8—272no—
339#125—91no—
340#125—271no—
341#125—272no—
342C22⋊Q8—272no—
343C22⋊Q8—272no—
344#117—91no—
345#117—271no—
346C2×C4⋊C4—271no—
347H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
348C23×C3—91no—
349C22×D3—31no—
350C22×D3—31no—
351C22×D3—61no—
352C3×Q8—362no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—91no—
363C2×C3×C4—94no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4—92no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—92no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—92no—
390C2×Dic3—362no—
391C4×D3—34no—
392C4×D3—34no—
393C4×D3—64no—
394C4×D3—362no—
395C4×D3—362no—
396Dic6—122no—
397Dic6—362no—
398Dic6—362no—
399Dic6—362no—
400Dic6—362no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32#100411yescharacteristic
405C2×C32C2×D3×Q811yescharacteristic
406C2×C32C2×D3×Q811yescharacteristic
407C2×C32#100711yescharacteristic
408C2×C32#100911yescharacteristic
409C2×C32#101011yescharacteristic
410C3×D3—62no—
411C3×D3—62no—
412C3×D3—122no—
413C2×Q8—92no—
414C2×Q8—272no—
415C22×C4—91no—
416C22×C4—272no—
417C4⋊C4—91no—
418C4⋊C4—91no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—92no—
426C4⋊C4—272no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C22⋊C4—91no—
431C22⋊C4—91no—
432C22⋊C4—272no—
433C42—91no—
434C42—272no—
435C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—61no—
439C22×C3—92no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3C22×D3211yescharacteristic
443C2×D3—32no—
444C2×D3—32no—
445C2×D3—62no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—121no—
451C2×D3—121no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—61no—
459C3×C4—61no—
460C3×C4—62no—
461C3×C4—92no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—182no—
465C3×C4—182no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4#379712yes—
471Dic3—32no—
472Dic3—32no—
473Dic3—62no—
474Dic3—61no—
475Dic3—61no—
476Dic3—61no—
477Dic3—61no—
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—62no—
482Dic3—121no—
483Dic3—121no—
484Dic3—182no—
485Dic3—182no—
486Dic3—182no—
487Dic3—182no—
488C32—21no—
489C32#568911yescharacteristic, complemented
490C32#572911yescharacteristic, complemented
491C23—91no—
492Q8—362no—
493Q8—1082no—
494C2×C4—91no—
495C2×C4—94no—
496C2×C4—91no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—92no—
502C2×C4—1082no—
503C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
505C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—61no—
513C2×C3—61no—
514C2×C3—61no—
515C2×C3—182no—
516C2×C3#6429711yescharacteristic
517C2×C3C2×#379711yescharacteristic
518C2×C3#6430711yescharacteristic
519D3—62no—
520D3—62no—
521D3—122no—
522C22—92no—
523C22—181no—
524C22—181no—
525C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C4—21no—
527C4—21no—
528C4—92no—
529C4—181no—
530C4—182no—
531C4—182no—
532C4—182no—
533C4—182no—
534C4—181no—
535C4#8865112yes—
536C3—31no—
537C3—31no—
538C3—61no—
539C3#1059385511yescharacteristic
540C2—182no—
541C2#1172248211yescharacteristic
542C2C2×#8865111yescharacteristic
543C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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
2. #64079⋊C3C2semidirect
3. #64078⋊C3C2semidirect
7. #11721117C2semidirect
8. #11721095C2semidirect
10. #11721086C2semidirect
11. #11721079C2semidirect
37. #64079D3semidirect
38. #64078D3semidirect
67. C4×#272Q8semidirect
347. H(3,𝔽3)#544semidirect
489. C32#5689semidirect
490. C32#5729semidirect

Maximal subgroups

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

subgroupindex
#117217182
#64079⋊C32
#64078⋊C32
#117211522
#117211502
#117211482
#117211172
#117210952
#117210892
#117210862
#117210792
#105939593
#105939543