Finite Group Databasebeta

← #13149454

Subgroups — #13149454

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721756C211yesmaximal, characteristic, complemented
2#64102⋊C3C211yesmaximal, characteristic
3#64099⋊C3C211yesmaximal, characteristic, complemented
4#11721136C211yesmaximal, characteristic
5#11721135C212yesmaximal, complemented
6#11721134C212yesmaximal, complemented
7#11721132C211yesmaximal, characteristic
8#11721119C212yesmaximal, complemented
9C4×#6196C211yesmaximal, characteristic, complemented
10#11721107C212yesmaximal, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10594470—31nomaximal
13#10594460—31nomaximal
14C2×#6229C2212yes—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2212yes—
23C4×#1084C2211yescharacteristic
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2212yes—
31#88450C2212yes—
32#88449C2212yes—
33#88449C2211yescharacteristic
34#88448C2212yes—
35#88448C2211yescharacteristic
36#88448C2211yescharacteristic
37#64102D311yescharacteristic
38#64099D311yescharacteristic, complemented
39C3×#893—31no—
40C3×#890—31no—
41C3×#888—31no—
42C3×#888—31no—
43C3×#888—61no—
44C4×#627—31no—
45C4×#627—31no—
46#63898—31no—
47#63895—32no—
48#63894—31no—
49#63894—31no—
50#63893—32no—
51Dic3×D12—31no—
52#63888—32no—
53#63887—32no—
54#63883—32no—
55D3×Dic3⋊C4—32no—
56#63872—31no—
57#63871—32no—
58#63869—31no—
59#63856—32no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66#6229—42no—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
73C4×(C3⋊D3)⋊C3—42no—
74((C3⋊D3)⋊C3)⋊C4—24no—
75((C3⋊D3)⋊C3)⋊C4—42no—
76#6196—22no—
77#5730—91no—
78#5690—91no—
79#5539—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D312yes—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—62no—
87C2×#627—31no—
88C2×#627—31no—
89C2×#624—32no—
90C2×#624—32no—
91C2×D3×Dic3—32no—
92C2×D3×Dic3—32no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C2×D311yescharacteristic
96#3752C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D312yes—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3750C2×D312yes—
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D311yescharacteristic
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—62no—
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—31no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—31no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—31no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—31no—
126Dic3⋊Dic3—31no—
127#3722—32no—
128#3722—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3C2×Q814yes—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)C2×Q812yes—
137#1084—21no—
138#1084—22no—
139#1084#4112yes—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
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—91no—
160#893—91no—
161#893—91no—
162D3×C4⋊C4—92no—
163D3×C4⋊C4—92no—
164#890—31no—
165#890—91no—
166#890—91no—
167#890—91no—
168#888—31no—
169#888—31no—
170#888—61no—
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⋊D3C22×D312yes—
180C22×C3×D3—32no—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4⋊(C3⋊D3)Dic614yescomplemented
193C4×C3⋊D3—42no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—62no—
208C3×D12—34no—
209C3×C4×D3—122no—
210C3×Dic6—122no—
211C3×Dic6—122no—
212C3×Dic6—124no—
213#627—62no—
214#627—62no—
215#624—64no—
216#624—122no—
217D3×Dic3—64no—
218D3×Dic3—122no—
219#554—271noSylow
220C2×H(3,𝔽3)C22×Q811yescharacteristic
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
223(C3⋊D3)⋊C3—24no—
224C2×C3×Q8—92no—
225C2×C3×D4—91no—
226C22×C3×C4—92no—
227C22×Dic3—92no—
228C22×Dic3—92no—
229C2×D12—31no—
230C2×D12—91no—
231C2×C4×D3—32no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×Dic6—32no—
238C2×Dic6—32no—
239C2×Dic6—62no—
240C2×Dic6—91no—
241C2×Dic6—91no—
242C2×Dic6—91no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4D3211yescharacteristic
252C3×C22⋊C4—92no—
253C3×C42—91no—
254C3×C42—91no—
255C3×C42—91no—
256#217—92no—
257#217—92no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—32no—
268C4⋊Dic3—61no—
269C4⋊Dic3—91no—
270C4⋊Dic3—92no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—92no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—62no—
280Dic3⋊C4—92no—
281Dic3⋊C4—91no—
282Dic3⋊C4—91no—
283Dic3⋊C4—92no—
284Dic3⋊C4—91no—
285Dic3⋊C4—91no—
286Dic3⋊C4—91no—
287Dic3⋊C4—91no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—31no—
295C4×Dic3—61no—
296C4×Dic3—91no—
297C4×Dic3—91no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C23×D311yescharacteristic
301C2×C3⋊D3—22no—
302C2×C3⋊D3—22no—
303C2×C3⋊D3C2×Dic614yes—
304C2×C3×D3—34no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—22no—
310C32×C4—42no—
311C32×C4C2×Dic612yes—
312C32×C4D3×Q812yes—
313C3⋊Dic3—21no—
314C3⋊Dic3—22no—
315C3⋊Dic3—21no—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3×Dic3—32no—
319C3×Dic3—32no—
320C3×Dic3—62no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—122no—
332C4⋊Q8—91no—
333#125—91no—
334#125—271no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337C22⋊Q8—272no—
338C4×Q8—271no—
339C4×D4—91no—
340C4×D4—271no—
341C4×D4—271no—
342C2×C4⋊C4—272no—
343H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
344C23×C3—92no—
345C22×D3—32no—
346C22×D3—92no—
347C3×Q8—362no—
348C3×D4—94no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—362no—
364C2×C3×C4C2×D3212yes—
365C2×C3×C4C2×D3211yescharacteristic
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—62no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—184no—
386C2×Dic3—362no—
387D12—34no—
388D12—94no—
389C4×D3—122no—
390C4×D3—184no—
391C4×D3—184no—
392C4×D3—362no—
393C4×D3—362no—
394C4×D3—362no—
395Dic6—122no—
396Dic6—122no—
397Dic6—124no—
398Dic6—182no—
399Dic6—182no—
400Dic6—182no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32C22×Dic611yescharacteristic
405C2×C32#100411yescharacteristic
406C2×C32C2×D3×Q811yescharacteristic
407C2×C32D3×#4111yescharacteristic
408C2×C32D3×#4111yescharacteristic
409C2×C32#100911yescharacteristic
410C3⋊D3—24no—
411C3×D3—64no—
412C2×Q8—92no—
413C2×Q8—271no—
414C2×D4—91no—
415C22×C4—92no—
416C22×C4—272no—
417C22×C4—272no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—92no—
423C4⋊C4—271no—
424C4⋊C4—272no—
425C4⋊C4—271no—
426C4⋊C4—271no—
427C4⋊C4—272no—
428C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
429C22⋊C4—92no—
430C22⋊C4—272no—
431C22⋊C4—272no—
432C42—91no—
433C42—91no—
434C42—91no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—61no—
438C22×C3—94no—
439C22×C3—182no—
440C22×C3—182no—
441C22×C3C22×D3211yescharacteristic
442C2×D3—34no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—94no—
446C2×D3—182no—
447C2×D3—182no—
448C3×C4—22no—
449C3×C4—32no—
450C3×C4—32no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—92no—
455C3×C4—122no—
456C3×C4—181no—
457C3×C4—181no—
458C3×C4—182no—
459C3×C4—181no—
460C3×C4—181no—
461C3×C4—182no—
462C3×C4—181no—
463C3×C4D3×Dic612yes—
464Dic3—32no—
465Dic3—32no—
466Dic3—62no—
467Dic3—61no—
468Dic3—61no—
469Dic3—62no—
470Dic3—62no—
471Dic3—62no—
472Dic3—61no—
473Dic3—61no—
474Dic3—61no—
475Dic3—61no—
476Dic3—62no—
477Dic3—122no—
478Dic3—181no—
479Dic3—181no—
480Dic3—181no—
481Dic3—182no—
482Dic3—181no—
483C32—21no—
484C32#553911yescharacteristic, complemented
485C32#569011yescharacteristic, complemented
486C23—92no—
487Q8—362no—
488Q8—542no—
489D4—94no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—91no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—362no—
499C2×C4—544no—
500C2×C4—1082no—
501C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
502C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—184no—
513C2×C3C2×D3×Dic611yescharacteristic
514C2×C3#6431411yescharacteristic
515C2×C3#6431911yescharacteristic
516D3—64no—
517D3—184no—
518C22—94no—
519C22—182no—
520C22—182no—
521C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C4—22no—
523C4—92no—
524C4—181no—
525C4—181no—
526C4—182no—
527C4—181no—
528C4—182no—
529C4—181no—
530C4—181no—
531C4((C3⋊D3)⋊C3)⋊Q812yes—
532C3—31no—
533C3—31no—
534C3—61no—
535C3#1059448511yescharacteristic
536C2—184no—
537C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
538C2#1172250711yescharacteristic
539C2#1172252011yescharacteristic
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⋊((C3⋊D3)⋊C3)C22
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
1. #11721756C2semidirect
3. #64099⋊C3C2semidirect
5. #11721135C2semidirect
6. #11721134C2semidirect
8. #11721119C2semidirect
9. C4×#6196C2semidirect
10. #11721107C2semidirect
11. #11721093C2semidirect
38. #64099D3semidirect
72. C4⋊((C3⋊D3)⋊C3)Q8semidirect
192. C4⋊(C3⋊D3)Dic6semidirect
343. H(3,𝔽3)#554semidirect
484. C32#5539semidirect
485. C32#5690semidirect

Maximal subgroups

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

subgroupindex
#117217562
#64102⋊C32
#64099⋊C32
#117211362
#117211352
#117211342
#117211322
#117211192
C4×#61962
#117211072
#117210932
#105944703
#105944603