Finite Group Databasebeta

← #13147801

Subgroups — #13147801

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#11721716C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721134C212yesmaximal, complemented
5#11721132C211yesmaximal, characteristic
6#11721132C211yesmaximal, characteristic
7#11721128C212yesmaximal, complemented
8#11721119C212yesmaximal, complemented
9#11721116C211yesmaximal, characteristic, complemented
10#11721101C212yesmaximal, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10593941—31nomaximal
13#10593938—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#634⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23#88540C2212yes—
24#88540C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic
38C4×#634D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×#870—31no—
41C3×#870—31no—
42C3×#870—61no—
43C3×C4×Dic6—31no—
44#63919—31no—
45#63918—31no—
46#63898—31no—
47#63898—31no—
48#63896—32no—
49#63895—32no—
50#63894—31no—
51#63894—31no—
52#63893—32no—
53#63888—32no—
54#63883—32no—
55D3×Dic3⋊C4—32no—
56#63872—31no—
57#63869—31no—
58#63865—32no—
59#63851—32no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2312yes—
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—42no—
73#634⋊C3—22no—
74((C3⋊D3)⋊C3)⋊C4—24no—
75((C3⋊D3)⋊C3)⋊C4—42no—
76#6196—42no—
77#5690—91no—
78#5579—91no—
79#5539—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D312yes—
82C2×#634C2×D311yescharacteristic
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6—31no—
86C2×#627—32no—
87C2×#627—32no—
88C2×#624—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92C32×C42—21no—
93C32×C42C2×D311yescharacteristic
94C32×C42C2×D311yescharacteristic
95#3752C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D311yescharacteristic
99#3750C2×D311yescharacteristic
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D311yescharacteristic
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3—61no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—31no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—62no—
118C3×Dic3⋊C4—62no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—31no—
121#3724—32no—
122#3724—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
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)—21no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)C2×Q812yes—
138#1084—22no—
139#1084—22no—
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×C4×Q8—91no—
147C3×C4×D4—91no—
148#946—92no—
149D4×Dic3—91no—
150D4×Dic3—91no—
151#924—92no—
152C2×Dic3⋊C4—92no—
153#897—92no—
154#896—92no—
155#896—92no—
156#896—92no—
157#896—92no—
158#893—91no—
159#893—91no—
160D3×C4⋊C4—92no—
161D3×C4⋊C4—92no—
162#890—91no—
163#890—91no—
164#888—91no—
165#888—91no—
166#878—92no—
167#878—92no—
168C4×D12—31no—
169C4×D12—91no—
170C4×D12—91no—
171C4×D12—91no—
172#870—31no—
173#870—31no—
174#870—61no—
175#870—91no—
176#869—91no—
177C4×Dic6—31no—
178C4×Dic6—91no—
179C22×C3⋊D3C22×D312yes—
180C22×C3×D3—32no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4⋊(C3⋊D3)Dic614yescomplemented
196C4×C3⋊D3—42no—
197#634—22no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—62no—
209C3×D12—34no—
210C3×C4×D3—122no—
211C3×Dic6—62no—
212#627—122no—
213#627—122no—
214#624—64no—
215#624—122no—
216D3×Dic3—64no—
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×Q8—91no—
224C2×C3×D4—91no—
225C22×C3×C4—92no—
226C22×Dic3—92no—
227C22×Dic3—92no—
228C2×D12—31no—
229C2×D12—91no—
230C2×C4×D3—32no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×Dic6—31no—
237C2×Dic6—91no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253C3×C42—91no—
254C3×C42—91no—
255C3×C42D3211yescharacteristic
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—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—61no—
271C4⋊Dic3—61no—
272C4⋊Dic3—91no—
273C4⋊Dic3—91no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—31no—
282Dic3⋊C4—31no—
283Dic3⋊C4—62no—
284Dic3⋊C4—62no—
285Dic3⋊C4—91no—
286Dic3⋊C4—91no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—31no—
294C4×Dic3—31no—
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—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—22no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4C2×Dic612yes—
314C32×C4D3×Q812yes—
315C3⋊Dic3—21no—
316C3⋊Dic3—22no—
317C3⋊Dic3—21no—
318C3⋊Dic3—21no—
319C3⋊Dic3#23512yes—
320C3×Dic3—32no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—61no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—122no—
330C3×Dic3—122no—
331C4⋊Q8—271no—
332#125—91no—
333#125—271no—
334C22⋊Q8—272no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337C4×Q8—91no—
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×Q8—182no—
347C3×D4—94no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—91no—
358C2×C3×C4—92no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—362no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C2×D3211yescharacteristic
368C2×Dic3—31no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—91no—
383C2×Dic3—92no—
384C2×Dic3—184no—
385C2×Dic3—362no—
386D12—34no—
387D12—94no—
388C4×D3—122no—
389C4×D3—184no—
390C4×D3—184no—
391C4×D3—362no—
392C4×D3—362no—
393C4×D3—362no—
394Dic6—62no—
395Dic6—182no—
396Dic6—362no—
397Dic6—362no—
398Dic6—362no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32C22×Dic611yescharacteristic
403C2×C32C2×#23511yescharacteristic
404C2×C32#100411yescharacteristic
405C2×C32C2×D3×Q811yescharacteristic
406C2×C32D3×#4111yescharacteristic
407C2×C32#100911yescharacteristic
408C3⋊D3—24no—
409C3×D3—64no—
410C2×Q8—91no—
411C2×Q8—272no—
412C2×D4—91no—
413C22×C4—92no—
414C22×C4—272no—
415C22×C4—272no—
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—91no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—272no—
425C4⋊C4—272no—
426C4⋊C4—272no—
427C22⋊C4—92no—
428C22⋊C4—272no—
429C22⋊C4—272no—
430C42—91no—
431C42—91no—
432C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
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—21no—
447C3×C4—21no—
448C3×C4—32no—
449C3×C4—32no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—62no—
455C3×C4—92no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—181no—
459C3×C4—182no—
460C3×C4—181no—
461C3×C4—181no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4D3×Dic612yes—
465Dic3—32no—
466Dic3—62no—
467Dic3—62no—
468Dic3—62no—
469Dic3—62no—
470Dic3—61no—
471Dic3—61no—
472Dic3—62no—
473Dic3—61no—
474Dic3—92no—
475Dic3—122no—
476Dic3—122no—
477Dic3—181no—
478Dic3—181no—
479Dic3—182no—
480Dic3—181no—
481C32—21no—
482C32#553911yescharacteristic, complemented
483C32#557911yescharacteristic, complemented
484C23—92no—
485Q8—182no—
486Q8—1082no—
487D4—94no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—362no—
496C2×C4—544no—
497C2×C4—1082no—
498C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
499C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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×#23511yescharacteristic
513C2×C3#6430711yescharacteristic
514D3—64no—
515D3—184no—
516C22—94no—
517C22—182no—
518C22—182no—
519C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520C4—21no—
521C4—21no—
522C4—92no—
523C4—182no—
524C4—181no—
525C4—182no—
526C4—181no—
527C4—182no—
528C4—181no—
529C4((C3⋊D3)⋊C3)⋊Q812yes—
530C3—31no—
531C3—31no—
532C3—61no—
533C3#1059392911yescharacteristic
534C2—184no—
535C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
536C2#1172249011yescharacteristic
537C2#1172249211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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
1. #11721716C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721134C2semidirect
7. #11721128C2semidirect
8. #11721119C2semidirect
9. #11721116C2semidirect
10. #11721101C2semidirect
11. #11721093C2semidirect
38. C4×#634D3semidirect
71. C4⋊((C3⋊D3)⋊C3)Q8semidirect
195. C4⋊(C3⋊D3)Dic6semidirect
342. H(3,𝔽3)#554semidirect
482. C32#5539semidirect
483. C32#5579semidirect

Maximal subgroups

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

subgroupindex
#117217162
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#117211342
#117211322
#117211282
#117211192
#117211162
#117211012
#117210932
#105939413
#105939383