Finite Group Databasebeta

← #13149403

Subgroups — #13149403

Order 1728.

854 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721758C211yesmaximal, characteristic, complemented
2#64102⋊C3C211yesmaximal, characteristic
3#64099⋊C3C211yesmaximal, characteristic, complemented
4#11721136C211yesmaximal, characteristic
5#11721134C212yesmaximal, complemented
6#11721133C212yesmaximal, complemented
7#11721132C211yesmaximal, characteristic
8#11721129C212yesmaximal, complemented
9#11721116C211yesmaximal, characteristic, complemented
10C4×#6196C211yesmaximal, characteristic, complemented
11#11721101C212yesmaximal, complemented
12#10594484—31nomaximal
13#10594452—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×#6196C2211yescharacteristic
19C2×#6196C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22C4⋊#1084C2212yes—
23#88540C2212yes—
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88450C2212yes—
33#88449C2211yescharacteristic
34#88448C2212yes—
35#88448C2211yescharacteristic
36#88448C2211yescharacteristic
37#64102D311yescharacteristic
38#64099D311yescharacteristic, complemented
39C3×#893—31no—
40C3×#890—31no—
41C3×#890—31no—
42C3×#890—31no—
43C3×#890—61no—
44#63919—31no—
45#63918—31no—
46C4×#627—31no—
47C4×#627—31no—
48#63900—32no—
49#63899—32no—
50#63898—31no—
51#63895—32no—
52#63895—32no—
53#63894—31no—
54#63894—31no—
55#63893—32no—
56Dic3×D12—31no—
57D3×C4⋊Dic3—32no—
58#63865—32no—
59#63851—32no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
72C4×(C3⋊D3)⋊C3—24no—
73((C3⋊D3)⋊C3)⋊C4—42no—
74((C3⋊D3)⋊C3)⋊C4—42no—
75#6196—22no—
76#6196—42no—
77#5690—91no—
78#5690—91no—
79#5543—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D312yes—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C2×#627—31no—
85C2×#627—31no—
86C2×#627—32no—
87C2×#627—32no—
88C2×#624—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92C32×C4⋊C4—21no—
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C4⋊C4C2×D311yescharacteristic
95#3752C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D312yes—
99C4×C3⋊Dic3C2×D311yescharacteristic
100C4×C3⋊Dic3C2×D311yescharacteristic
101C3×(C2×D3)⋊C4—32no—
102C3×C4⋊Dic3—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—32no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—62no—
109C3×C4⋊Dic3—61no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—62no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—61no—
119#3724—32no—
120#3724—31no—
121#3724—31no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—31no—
124Dic3⋊Dic3—31no—
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—21no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×#125—91no—
146C3×#125—91no—
147C3×C4×D4—91no—
148D4×Dic3—91no—
149D4×Dic3—91no—
150C2×C4⋊Dic3—92no—
151#924—92no—
152#897—92no—
153#897—92no—
154#896—92no—
155#896—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—31no—
166#890—31no—
167#890—61no—
168#890—91no—
169#888—91no—
170#888—91no—
171#878—92no—
172#878—92no—
173C4×D12—91no—
174C4×D12—91no—
175#869—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—24no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—62no—
208C3×D12—34no—
209C3×C4×D3—64no—
210#627—62no—
211#627—62no—
212#627—122no—
213#627—122no—
214#624—122no—
215#624—122no—
216D3×Dic3—122no—
217D3×Dic3—122no—
218#554—271noSylow
219C2×H(3,𝔽3)C22×Q811yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
222(C3⋊D3)⋊C3—24no—
223C2×C3×D4—91no—
224C22×C3×C4—92no—
225C22×Dic3—92no—
226C22×Dic3—92no—
227C2×D12—31no—
228C2×D12—91no—
229C2×C4×D3—32no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×Dic6—91no—
236C2×Dic6—91no—
237C2×Dic6—91no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4D3211yescharacteristic
251C3×C22⋊C4—92no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—91no—
255#217—92no—
256#217—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—32no—
266C4⋊Dic3—32no—
267C4⋊Dic3—31no—
268C4⋊Dic3—32no—
269C4⋊Dic3—62no—
270C4⋊Dic3—61no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—91no—
276C4⋊Dic3—92no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—62no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—91no—
286Dic3⋊C4—91no—
287Dic3⋊C4—91no—
288Dic3⋊C4—91no—
289Dic3⋊C4—92no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—61no—
295C4×Dic3—91no—
296C4×Dic3—91no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3C2×Dic614yes—
303C2×C3×D3—34no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C32×C4—22no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—42no—
310C32×C4C2×Dic612yes—
311C32×C4D3×Q812yes—
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—22no—
315C3⋊Dic3—21no—
316C3⋊Dic3#23712yes—
317C3×Dic3—32no—
318C3×Dic3—61no—
319C3×Dic3—61no—
320C3×Dic3—61no—
321C3×Dic3—61no—
322C3×Dic3—62no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—121no—
329C3×Dic3—122no—
330C3×Dic3—121no—
331C4⋊Q8—271no—
332#125—91no—
333#125—91no—
334C22⋊Q8—272no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337C4×Q8—271no—
338C4×D4—91no—
339C4×D4—271no—
340C4×D4—271no—
341C2×C4⋊C4—272no—
342H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
343C23×C3—92no—
344C22×D3—32no—
345C22×D3—92no—
346C3×D4—94no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—32no—
350C2×C3×C4—32no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—92no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—91no—
361C2×C3×C4—184no—
362C2×C3×C4C2×D3212yes—
363C2×C3×C4C2×D3211yescharacteristic
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—62no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—92no—
383C2×Dic3—362no—
384C2×Dic3—362no—
385D12—34no—
386D12—94no—
387C4×D3—64no—
388C4×D3—184no—
389C4×D3—362no—
390C4×D3—362no—
391C4×D3—362no—
392C4×D3—362no—
393Dic6—182no—
394Dic6—182no—
395Dic6—182no—
396Dic6—362no—
397Dic6—362no—
398Dic6—362no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32C22×Dic611yescharacteristic
403C2×C32C2×#23711yescharacteristic
404C2×C32C2×D3×Q811yescharacteristic
405C2×C32D3×#4111yescharacteristic
406C2×C32#100911yescharacteristic
407C2×C32#100911yescharacteristic
408C3⋊D3—24no—
409C3×D3—64no—
410C2×Q8—271no—
411C2×Q8—272no—
412C2×D4—91no—
413C22×C4—92no—
414C22×C4—272no—
415C22×C4—272no—
416C4⋊C4—92no—
417C4⋊C4—92no—
418C4⋊C4—91no—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4—271no—
423C4⋊C4—272no—
424C4⋊C4—271no—
425C4⋊C4—271no—
426C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
427C22⋊C4—92no—
428C22⋊C4—272no—
429C22⋊C4—272no—
430C42—91no—
431C42—91no—
432C42—91no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—61no—
436C22×C3—94no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C22×D3211yescharacteristic
440C2×D3—34no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—94no—
444C2×D3—182no—
445C2×D3—182no—
446C3×C4—22no—
447C3×C4—32no—
448C3×C4—32no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—92no—
453C3×C4—122no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4—181no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4D3×Dic612yes—
462Dic3—32no—
463Dic3—61no—
464Dic3—61no—
465Dic3—62no—
466Dic3—62no—
467Dic3—61no—
468Dic3—61no—
469Dic3—61no—
470Dic3—61no—
471Dic3—62no—
472Dic3—61no—
473Dic3—92no—
474Dic3—121no—
475Dic3—121no—
476Dic3—122no—
477Dic3—181no—
478Dic3—181no—
479Dic3—182no—
480Dic3—181no—
481C32—21no—
482C32#554311yescharacteristic, complemented
483C32#569011yescharacteristic, complemented
484C23—92no—
485Q8—542no—
486Q8—1082no—
487D4—94no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—184no—
497C2×C4—1082no—
498C2×C4—1082no—
499C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
500C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—61no—
508C2×C3—61no—
509C2×C3—61no—
510C2×C3—184no—
511C2×C3C2×D3×Dic611yescharacteristic
512C2×C3D3×#23711yescharacteristic
513C2×C3#6432011yescharacteristic
514D3—64no—
515D3—184no—
516C22—94no—
517C22—182no—
518C22—182no—
519C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520C4—22no—
521C4—92no—
522C4—181no—
523C4—181no—
524C4—181no—
525C4—181no—
526C4—182no—
527C4—182no—
528C4—181no—
529C4((C3⋊D3)⋊C3)⋊Q812yes—
530C3—31no—
531C3—31no—
532C3—61no—
533C3#1059445211yescharacteristic
534C2—184no—
535C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
536C2#1172250411yescharacteristic
537C2#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 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. #11721758C2semidirect
3. #64099⋊C3C2semidirect
5. #11721134C2semidirect
6. #11721133C2semidirect
8. #11721129C2semidirect
9. #11721116C2semidirect
10. C4×#6196C2semidirect
11. #11721101C2semidirect
38. #64099D3semidirect
71. C4⋊((C3⋊D3)⋊C3)Q8semidirect
192. C4⋊(C3⋊D3)Dic6semidirect
342. H(3,𝔽3)#554semidirect
482. C32#5543semidirect
483. C32#5690semidirect

Maximal subgroups

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

subgroupindex
#117217582
#64102⋊C32
#64099⋊C32
#117211362
#117211342
#117211332
#117211322
#117211292
#117211162
C4×#61962
#117211012
#105944843
#105944523