Finite Group Databasebeta

← #13149469

Subgroups — #13149469

Order 1728.

777 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721758C211yesmaximal, characteristic, complemented
2#64101⋊C3C211yesmaximal, characteristic
3#64099⋊C3C211yesmaximal, characteristic, complemented
4#11721134C212yesmaximal
5#11721133C212yesmaximal
6#11721129C211yesmaximal, characteristic
7#11721117C211yesmaximal, characteristic, complemented
8#11721116C211yesmaximal, characteristic, complemented
9#11721116C212yesmaximal, complemented
10#11721105C211yesmaximal, characteristic
11#11721091C212yesmaximal, complemented
12#10594508—31nomaximal
13#10594505—31nomaximal
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
16C2×#6196C2212yes—
17C2×#6196C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4⋊#1084C2212yes—
21#88540C2212yes—
22C4×#1084C2211yescharacteristic
23#3752⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2211yescharacteristic
31#88450C2211yescharacteristic
32#88449C2212yes—
33#88448C2211yescharacteristic
34#88448C2212yes—
35#88448C2212yes—
36#88448C2211yescharacteristic
37#64101D311yescharacteristic
38#64099D311yescharacteristic, complemented
39C3×#892—31no—
40C3×#890—31no—
41C3×#890—31no—
42C3×#890—31no—
43C3×#890—61no—
44#63919—31no—
45#63919—32no—
46#63918—32no—
47#63918—31no—
48#63918—31no—
49#63918—31no—
50#63900—31no—
51#63899—32no—
52#63895—32no—
53#63895—32no—
54#63893—32no—
55D3×C4⋊Dic3—31no—
56#63874—31no—
57#63870—32no—
58#63867—32no—
59#63855—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2311yescharacteristic
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4×(C3⋊D3)⋊C3Q814yescomplemented
72((C3⋊D3)⋊C3)⋊C4—42no—
73#6196—42no—
74#6196—42no—
75#5689—91no—
76#5689—91no—
77#5519—91no—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×C3×C4×D3—31no—
80C2×#627—32no—
81C2×#627—32no—
82C2×#627—32no—
83C2×#627—32no—
84C2×#624—32no—
85C2×D3×Dic3—32no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D311yescharacteristic
89#3752C2×D312yes—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)C2×D312yes—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93#3750C2×D312yes—
94C4×C3⋊Dic3C2×D311yescharacteristic
95C4×C3⋊Dic3C2×D312yes—
96C3×(C2×D3)⋊C4—32no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—32no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—62no—
105C3×C4⋊Dic3—61no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—62no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—61no—
115#3724—32no—
116#3724—31no—
117#3724—32no—
118#3724—31no—
119Dic3⋊Dic3—31no—
120Dic3⋊Dic3—31no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—31no—
126#3722—31no—
127(C2×D3)⋊Dic3—31no—
128(C2×D3)⋊Dic3—31no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3C2×Q812yes—
133C4×H(3,𝔽3)—22no—
134C4×H(3,𝔽3)C2×Q812yes—
135#1084—21no—
136#1084—22no—
137#1084—21no—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143C3×#125—91no—
144C3×#125—91no—
145C3×#117—91no—
146#926—91no—
147C2×C4⋊Dic3—91no—
148#897—92no—
149#897—92no—
150#896—92no—
151#896—92no—
152#892—31no—
153#892—91no—
154D3×C4⋊C4—91no—
155D3×C4⋊C4—91no—
156#890—31no—
157#890—31no—
158#890—31no—
159#890—61no—
160#890—91no—
161#889—92no—
162#889—92no—
163#888—91no—
164#888—92no—
165#888—91no—
166#888—91no—
167#888—92no—
168#888—91no—
169#878—92no—
170#878—92no—
171#872—91no—
172#872—91no—
173#870—92no—
174#869—91no—
175#869—91no—
176#869—92no—
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—31no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C4×C3⋊D3Dic614yescomplemented
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—62no—
205C3×C4×D3—34no—
206#627—122no—
207#627—122no—
208#627—122no—
209#627—122no—
210#624—122no—
211D3×Dic3—122no—
212#544—271noSylow
213C2×H(3,𝔽3)C22×Q811yescharacteristic
214C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
215C2×H(3,𝔽3)2−1+411yescharacteristic
216(C3⋊D3)⋊C3—22no—
217C22×C3×C4—91no—
218C22×Dic3—92no—
219C2×C4×D3—31no—
220C2×C4×D3—91no—
221C2×C4×D3—92no—
222C2×C4×D3—92no—
223C2×Dic6—92no—
224C2×Dic6—92no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C22⋊C4—92no—
241C3×C42—92no—
242C3×C42—91no—
243C3×C42—91no—
244#217—91no—
245#217—91no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—91no—
252C4⋊Dic3—32no—
253C4⋊Dic3—31no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—32no—
259C4⋊Dic3—61no—
260C4⋊Dic3—62no—
261C4⋊Dic3—92no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—92no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—92no—
268C4⋊Dic3—91no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—62no—
273Dic3⋊C4—92no—
274Dic3⋊C4—91no—
275Dic3⋊C4—91no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—91no—
281Dic3⋊C4—91no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284C4×Dic3—31no—
285C4×Dic3—31no—
286C4×Dic3—31no—
287C4×Dic3—32no—
288C4×Dic3—61no—
289C4×Dic3—92no—
290C4×Dic3—91no—
291C22×C32—21no—
292C22×C32C23×D311yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—21no—
295C2×C3⋊D3—21no—
296C2×C3⋊D3C2×Dic612yes—
297C2×C3×D3—32no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—42no—
304C32×C4C2×Dic612yes—
305C32×C4D3×Q812yes—
306C3⋊Dic3—22no—
307C3⋊Dic3—21no—
308C3⋊Dic3—22no—
309C3⋊Dic3—21no—
310C3⋊Dic3C2×Dic612yes—
311C3×Dic3—32no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—61no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—61no—
320C3×Dic3—61no—
321C3×Dic3—61no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—122no—
325C4⋊Q8—272no—
326C4⋊Q8—271no—
327C4⋊Q8—271no—
328#125—91no—
329#125—91no—
330#125—272no—
331C22⋊Q8—272no—
332C22⋊Q8—272no—
333#117—91no—
334#117—271no—
335C2×C4⋊C4—271no—
336H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
337C23×C3—91no—
338C22×D3—31no—
339C22×D3—91no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—61no—
345C2×C3×C4—62no—
346C2×C3×C4—94no—
347C2×C3×C4—91no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—92no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4C2×D3212yes—
356C2×C3×C4C2×D3211yescharacteristic
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—62no—
371C2×Dic3—91no—
372C2×Dic3—91no—
373C2×Dic3—92no—
374C2×Dic3—91no—
375C2×Dic3—92no—
376C2×Dic3—362no—
377C4×D3—34no—
378C4×D3—94no—
379C4×D3—362no—
380C4×D3—362no—
381Dic6—362no—
382Dic6—362no—
383Dic6—362no—
384Dic6—362no—
385Dic6—362no—
386Dic6—362no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32C22×Dic611yescharacteristic
391C2×C32C2×D3×Q811yescharacteristic
392C2×C32#100911yescharacteristic
393C2×C32#100911yescharacteristic
394C2×C32#101011yescharacteristic
395C2×C32#101011yescharacteristic
396C3⋊D3—22no—
397C3×D3—62no—
398C2×Q8—272no—
399C2×Q8—272no—
400C22×C4—91no—
401C22×C4—272no—
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—272no—
410C4⋊C4—271no—
411C4⋊C4—271no—
412C4⋊C4—272no—
413C4⋊C4—272no—
414C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C22⋊C4—92no—
416C22⋊C4—271no—
417C22⋊C4—271no—
418C42—92no—
419C42—91no—
420C42—91no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—61no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3C22×D3211yescharacteristic
428C2×D3—32no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—92no—
432C2×D3—181no—
433C2×D3—181no—
434C3×C4—22no—
435C3×C4—32no—
436C3×C4—32no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—92no—
441C3×C4—122no—
442C3×C4—182no—
443C3×C4—181no—
444C3×C4—182no—
445C3×C4—181no—
446C3×C4—181no—
447C3×C4—181no—
448C3×C4—182no—
449C3×C4D3×Dic612yes—
450Dic3—32no—
451Dic3—61no—
452Dic3—61no—
453Dic3—62no—
454Dic3—62no—
455Dic3—61no—
456Dic3—61no—
457Dic3—61no—
458Dic3—61no—
459Dic3—62no—
460Dic3—62no—
461Dic3—92no—
462Dic3—121no—
463Dic3—122no—
464Dic3—121no—
465Dic3—181no—
466Dic3—182no—
467Dic3—182no—
468Dic3—181no—
469C32—21no—
470C32#551911yescharacteristic, complemented
471C32#568911yescharacteristic, complemented
472C23—91no—
473Q8—1082no—
474Q8—1082no—
475C2×C4—91no—
476C2×C4—94no—
477C2×C4—92no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—92no—
484C2×C4—1082no—
485C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
486C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—182no—
497C2×C3C2×D3×Dic611yescharacteristic
498C2×C3#6430911yescharacteristic
499C2×C3#6432011yescharacteristic
500D3—62no—
501D3—182no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
506C4—22no—
507C4—92no—
508C4—181no—
509C4—181no—
510C4—182no—
511C4—182no—
512C4—181no—
513C4—182no—
514C4—181no—
515C4((C3⋊D3)⋊C3)⋊Q812yes—
516C3—31no—
517C3—31no—
518C3—61no—
519C3#1059450511yescharacteristic
520C2—182no—
521C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
522C2#1172250211yescharacteristic
523C2#1172252111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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
1. #11721758C2semidirect
3. #64099⋊C3C2semidirect
7. #11721117C2semidirect
8. #11721116C2semidirect
9. #11721116C2semidirect
11. #11721091C2semidirect
38. #64099D3semidirect
71. C4×(C3⋊D3)⋊C3Q8semidirect
190. C4×C3⋊D3Dic6semidirect
336. H(3,𝔽3)#544semidirect
470. C32#5519semidirect
471. C32#5689semidirect

Maximal subgroups

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

subgroupindex
#117217582
#64101⋊C32
#64099⋊C32
#117211342
#117211332
#117211292
#117211172
#117211162
#117211052
#117210912
#105945083
#105945053