Finite Group Databasebeta

← #13147796

Subgroups — #13147796

Order 1728.

819 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6229C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721129C211yesmaximal, characteristic, complemented
5#11721125C212yesmaximal
6#11721121C212yesmaximal, complemented
7#11721112C212yesmaximal, complemented
8#11721107C211yesmaximal, characteristic, complemented
9#11721107C211yesmaximal, characteristic, complemented
10#11721087C212yesmaximal, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10593947—31nomaximal
13#10593917—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20C4⋊#1084C2211yescharacteristic
21#88540C2212yes—
22C4×#1084C2212yes—
23#3752⋊C3C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#88450C2212yes—
31#88450C2212yes—
32#88450C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic
38#64079D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×#870—31no—
41C3×C4×Dic6—31no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—61no—
44#63900—31no—
45#63897—32no—
46D3×C4⋊Dic3—31no—
47#63884—32no—
48#63877—32no—
49#63871—31no—
50#63871—31no—
51#63868—32no—
52#63859—32no—
53#63856—31no—
54#63856—31no—
55#63849—31no—
56#63847—32no—
57#63845—32no—
58#63841—31no—
59#63836—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67#6229—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75((C3⋊D3)⋊C3)⋊C4—24no—
76#5694—91no—
77#5581—91no—
78#5541—91no—
79C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C3×D12—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×#624—32no—
89C2×D3×Dic3—32no—
90C32×C42—21no—
91C32×C42C2×D311yescharacteristic
92C32×C42C2×D311yescharacteristic
93#3752C2×D311yescharacteristic
94#3752C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D312yes—
99#3750C2×D312yes—
100C3×(C2×D3)⋊C4—31no—
101C3×(C2×D3)⋊C4—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—61no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—62no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—62no—
116#3724—32no—
117#3724—32no—
118#3724—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123#3722—32no—
124#3722—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128Dic32—32no—
129Dic32—32no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3#4112yes—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)#4112yes—
140#1084—22no—
141#1084—22no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146C3×#125—91no—
147C3×C4×Q8—91no—
148C3×C4×D4—91no—
149#946—91no—
150#946—91no—
151#935—92no—
152#926—91no—
153C2×C4⋊Dic3—91no—
154#924—91no—
155#924—91no—
156#898—92no—
157#897—91no—
158#897—91no—
159#894—92no—
160#894—92no—
161#892—92no—
162#892—91no—
163#892—91no—
164D3×C4⋊C4—91no—
165D3×C4⋊C4—91no—
166#890—92no—
167#889—92no—
168#889—92no—
169#879—92no—
170#877—92no—
171#876—92no—
172C4×D12—31no—
173C4×D12—91no—
174#872—92no—
175#870—31no—
176#870—91no—
177C4×Dic6—31no—
178C4×Dic6—31no—
179C4×Dic6—61no—
180C22×C3⋊D3C22×D311yescharacteristic
181C22×C3⋊D3C22×D311yescharacteristic
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C4⋊(C3⋊D3)—22no—
198C4×C3⋊D3—22no—
199C4×C3⋊D3—22no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C3×D12—62no—
211C3×C4×D3—62no—
212C3×C4×D3—62no—
213C3×Dic6—64no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216#624—64no—
217D3×Dic3—64no—
218#552—271noSylow
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)2−1+411yescharacteristic
222(C3⋊D3)⋊C3—22no—
223(C3⋊D3)⋊C3—41no—
224C2×C3×Q8—91no—
225C2×C3×D4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×Dic3—92no—
229C2×D12—31no—
230C2×D12—91no—
231C2×C4×D3—31no—
232C2×C4×D3—31no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—61no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252C3×C42—92no—
253C3×C42D3211yescharacteristic
254#217—92no—
255#217—92no—
256#217—92no—
257(C2×D3)⋊C4—31no—
258(C2×D3)⋊C4—31no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272C4⋊Dic3—61no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—91no—
276C4⋊Dic3—92no—
277C4⋊Dic3—92no—
278C4⋊Dic3—92no—
279Dic3⋊C4—32no—
280Dic3⋊C4—32no—
281Dic3⋊C4—32no—
282Dic3⋊C4—32no—
283Dic3⋊C4—62no—
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—32no—
293C4×Dic3—32no—
294C4×Dic3—62no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C23×D311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3—21no—
304C2×C3⋊D3—21no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3#23512yes—
307C2×C3×D3—32no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C32×C4—22no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—41no—
320C32×C4#23712yes—
321C32×C4#23912yes—
322C3⋊Dic3—21no—
323C3⋊Dic3—21no—
324C3⋊Dic3—22no—
325C3⋊Dic3—22no—
326C3×Dic3—64no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—122no—
336#126—272no—
337#125—91no—
338#125—272no—
339#123—272no—
340C22⋊Q8—271no—
341C22⋊Q8—271no—
342C4×Q8—91no—
343C4×D4—91no—
344#117—272no—
345#117—271no—
346C2×C4⋊C4—271no—
347H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
348C23×C3—91no—
349C23×C3—91no—
350C22×D3—31no—
351C22×D3—31no—
352C22×D3—91no—
353C22×D3—91no—
354C3×Q8—182no—
355C3×D4—182no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—91no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4—182no—
372C2×C3×C4—182no—
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—31no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—31no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—91no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—91no—
390C2×Dic3—184no—
391D12—62no—
392D12—182no—
393C4×D3—62no—
394C4×D3—62no—
395C4×D3—182no—
396C4×D3—182no—
397C4×D3—184no—
398C4×D3—184no—
399Dic6—62no—
400Dic6—62no—
401Dic6—64no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32C2×#23511yescharacteristic
406C2×C32C2×#23711yescharacteristic
407C2×C32C2×#23911yescharacteristic
408C2×C32D3×#4111yescharacteristic
409C2×C32#101011yescharacteristic
410C2×C32#101011yescharacteristic
411C3⋊D3—22no—
412C3⋊D3—41no—
413C3×D3—62no—
414C3×D3—121no—
415C2×Q8—91no—
416C2×D4—91no—
417C22×C4—91no—
418C22×C4—91no—
419C22×C4—272no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—91no—
425C4⋊C4—92no—
426C4⋊C4—92no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C22⋊C4—91no—
431C22⋊C4—91no—
432C22⋊C4—272no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C42—92no—
436C42—272no—
437C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—61no—
441C22×C3—92no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3C22×D3211yescharacteristic
448C2×D3—32no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—92no—
455C2×D3—181no—
456C2×D3—181no—
457C2×D3—181no—
458C2×D3—181no—
459C2×D3—181no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—32no—
463C3×C4—32no—
464C3×C4—61no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—62no—
469C3×C4—62no—
470C3×C4—121no—
471C3×C4—182no—
472C3×C4—181no—
473C3×C4—181no—
474C3×C4—182no—
475C3×C4—182no—
476C3×C4—182no—
477C3×C4#379612yes—
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—62no—
482Dic3—64no—
483Dic3—61no—
484Dic3—61no—
485Dic3—62no—
486Dic3—62no—
487Dic3—122no—
488Dic3—181no—
489Dic3—181no—
490Dic3—182no—
491Dic3—182no—
492C32—21no—
493C32#554111yescharacteristic, complemented
494C32#569411yescharacteristic, complemented
495C23—91no—
496C23—91no—
497Q8—182no—
498D4—182no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—91no—
502C2×C4—92no—
503C2×C4—91no—
504C2×C4—92no—
505C2×C4—182no—
506C2×C4—182no—
507C2×C4—544no—
508C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
510C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—61no—
518C2×C3—61no—
519C2×C3—61no—
520C2×C3—182no—
521C2×C3—361no—
522C2×C3C2×#379611yescharacteristic
523C2×C3#6429811yescharacteristic
524C2×C3D3×#23511yescharacteristic
525D3—62no—
526D3—121no—
527D3—182no—
528D3—361no—
529C22—92no—
530C22—181no—
531C22—181no—
532C22—181no—
533C22—181no—
534C22—181no—
535C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C4—21no—
537C4—21no—
538C4—181no—
539C4—181no—
540C4—182no—
541C4—182no—
542C4—182no—
543C4—182no—
544C4#8864912yes—
545C3—31no—
546C3—31no—
547C3—61no—
548C3#1059392011yescharacteristic
549C2—182no—
550C2—361no—
551C2C2×#8864911yescharacteristic
552C2#1172248311yescharacteristic
553C2#1172249011yescharacteristic
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)#552
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. C4×#6229C2semidirect
3. #64079⋊C3C2semidirect
4. #11721129C2semidirect
6. #11721121C2semidirect
7. #11721112C2semidirect
8. #11721107C2semidirect
9. #11721107C2semidirect
10. #11721087C2semidirect
11. #11721081C2semidirect
38. #64079D3semidirect
347. H(3,𝔽3)#552semidirect
493. C32#5541semidirect
494. C32#5694semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#117211292
#117211252
#117211212
#117211122
#117211072
#117210872
#117210812
#105939473
#105939173