Finite Group Databasebeta

← #13149553

Subgroups — #13149553

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
1#11721755C211yesmaximal, characteristic, complemented
2#64101⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11721139C212yesmaximal
5C4×#6198C211yesmaximal, characteristic
6#11721112C212yesmaximal, complemented
7#11721107C211yesmaximal, characteristic, complemented
8#11721107C211yesmaximal, characteristic, complemented
9#11721105C212yesmaximal, complemented
10#11721095C211yesmaximal, characteristic, complemented
11#11721091C212yesmaximal, complemented
12#10594337—31nomaximal
13#10594333—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6198C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88540C2212yes—
21C4×#1084C2211yescharacteristic
22C4×#1084C2212yes—
23#3752⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88450C2211yescharacteristic
29#88450C2212yes—
30#88450C2212yes—
31#88450C2211yescharacteristic
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37#64101D311yescharacteristic, complemented
38C4⋊C4×C3⋊D3D311yescharacteristic, complemented
39C3×#892—31no—
40C3×D3×C4⋊C4—31no—
41C3×#887—31no—
42C3×#887—31no—
43C3×#887—61no—
44C4×#626—31no—
45C4×#626—31no—
46#63874—32no—
47#63871—31no—
48#63871—31no—
49#63870—32no—
50#63869—31no—
51#63869—31no—
52#63868—32no—
53#63867—32no—
54#63859—32no—
55#63856—31no—
56#63856—31no—
57#63855—32no—
58#63853—32no—
59#63850—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2311yescharacteristic
67#6229—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4×(C3⋊D3)⋊C3—24no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75#6198—22no—
76#5581—91no—
77#5529—91no—
78#5513—91no—
79C2×C4×C3⋊D3C2×D312yes—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×#626—31no—
89C2×#626—31no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C4⋊C4C2×D311yescharacteristic
93#3752C2×D312yes—
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96#3750C2×D312yes—
97C4×C3⋊Dic3C2×D312yes—
98C3×(C2×D3)⋊C4—32no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—62no—
105C3×C4×Dic3—32no—
106C3×C4×Dic3—32no—
107C3×C4×Dic3—31no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—62no—
111C3×C4×Dic3—61no—
112#3724—32no—
113#3724—32no—
114#3724—32no—
115#3724—32no—
116Dic3⋊Dic3—32no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—31no—
121Dic3⋊Dic3—31no—
122#3722—32no—
123#3722—32no—
124#3722—31no—
125#3722—31no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—31no—
129(C2×D3)⋊Dic3—31no—
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)—22no—
138C4×H(3,𝔽3)#4112yes—
139#1084—22no—
140#1084—21no—
141#1084—21no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146C3×C4×Q8—91no—
147C3×#117—91no—
148C2×C3×C4⋊C4—91no—
149#946—91no—
150#946—91no—
151#929—92no—
152C4×#67—91no—
153C4×#67—91no—
154#926—92no—
155#924—91no—
156#924—91no—
157#898—92no—
158#897—91no—
159#897—91no—
160#894—92no—
161#892—31no—
162#892—91no—
163D3×C4⋊C4—31no—
164D3×C4⋊C4—91no—
165#890—91no—
166#890—91no—
167#889—92no—
168#889—92no—
169#887—31no—
170#887—31no—
171#887—61no—
172#883—92no—
173#879—92no—
174#876—92no—
175C4×D12—91no—
176#872—92no—
177#872—92no—
178#870—92no—
179#870—91no—
180C22×C3⋊D3C22×D311yescharacteristic
181C22×C3⋊D3C22×D311yescharacteristic
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D312yes—
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C4×C3⋊D3—24no—
195C4×C3⋊D3—22no—
196C4×C3⋊D3—22no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—61no—
210C3×C4×D3—64no—
211C3×C4×D3—62no—
212C3×C4×D3—62no—
213C3×Dic6—64no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216#626—62no—
217#626—62no—
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—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—92no—
228C2×#67—91no—
229C2×#67—91no—
230C2×D12—91no—
231C2×C4×D3—32no—
232C2×C4×D3—31no—
233C2×C4×D3—31no—
234C2×C4×D3—92no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—61no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4D3211yescharacteristic
248C3×C22⋊C4—92no—
249C3×C42—92no—
250C3×C42—91no—
251C3×C42—92no—
252#217—91no—
253#217—92no—
254#217—92no—
255#217—91no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—91no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—91no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—91no—
269C4⋊Dic3—91no—
270C4⋊Dic3—91no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275Dic3⋊C4—32no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—62no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—91no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—91no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290C4×Dic3—32no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—32no—
294C4×Dic3—32no—
295C4×Dic3—62no—
296C4×Dic3—61no—
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#23712yes—
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—22no—
315C32×C4—22no—
316C32×C4—42no—
317C32×C4#23512yes—
318C32×C4#23512yes—
319C3⋊Dic3—21no—
320C3⋊Dic3—21no—
321C3⋊Dic3—22no—
322C3⋊Dic3—22no—
323C3×Dic3—64no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—61no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—121no—
336#126—272no—
337#125—272no—
338#125—271no—
339#123—272no—
340C22⋊Q8—271no—
341C22⋊Q8—271no—
342C4×Q8—91no—
343C4×D4—271no—
344#117—91no—
345#117—272no—
346C2×C4⋊C4—91no—
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—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—61no—
360C2×C3×C4—62no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4—182no—
369C2×C3×C4—182no—
370C2×C3×C4—184no—
371C2×C3×C4C2×D3212yes—
372C2×C3×C4C2×D3211yescharacteristic
373#67—182no—
374#67—182no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—31no—
382C2×Dic3—32no—
383C2×Dic3—31no—
384C2×Dic3—32no—
385C2×Dic3—61no—
386C2×Dic3—62no—
387C2×Dic3—61no—
388C2×Dic3—91no—
389C2×Dic3—92no—
390C2×Dic3—91no—
391C2×Dic3—92no—
392D12—182no—
393C4×D3—64no—
394C4×D3—62no—
395C4×D3—62no—
396C4×D3—182no—
397C4×D3—182no—
398C4×D3—184no—
399Dic6—62no—
400Dic6—62no—
401Dic6—64no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32C2×#23511yescharacteristic
406C2×C32C2×#23511yescharacteristic
407C2×C32C2×#23711yescharacteristic
408C2×C32#100711yescharacteristic
409C2×C32D3×#4111yescharacteristic
410C2×C32#101011yescharacteristic
411C3⋊D3—22no—
412C3⋊D3—41no—
413C3×D3—62no—
414C3×D3—121no—
415C2×Q8—91no—
416C2×D4—271no—
417C22×C4—92no—
418C22×C4—91no—
419C22×C4—91no—
420C4⋊C4—91no—
421C4⋊C4—92no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C4⋊C4—272no—
425C4⋊C4—272no—
426C4⋊C4—271no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C22⋊C4—92no—
431C22⋊C4—271no—
432C22⋊C4—272no—
433C22⋊C4—271no—
434C22⋊C4—272no—
435C42—92no—
436C42—92no—
437C42—91no—
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—22no—
461C3×C4—32no—
462C3×C4—32no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—62no—
466C3×C4—122no—
467C3×C4—182no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4—182no—
471C3×C4—182no—
472C3×C4—181no—
473C3×C4—181no—
474C3×C4#379812yes—
475Dic3—62no—
476Dic3—62no—
477Dic3—61no—
478Dic3—61no—
479Dic3—61no—
480Dic3—61no—
481Dic3—64no—
482Dic3—62no—
483Dic3—61no—
484Dic3—61no—
485Dic3—62no—
486Dic3—62no—
487Dic3—121no—
488Dic3—182no—
489Dic3—181no—
490Dic3—181no—
491Dic3—182no—
492C32—21no—
493C32#551311yescharacteristic, complemented
494C32#552911yescharacteristic, complemented
495C23—91no—
496C23—91no—
497Q8—182no—
498D4—542no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—91no—
504C2×C4—92no—
505C2×C4—91no—
506C2×C4—184no—
507C2×C4—182no—
508C2×C4—182no—
509C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
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×#379811yescharacteristic
523C2×C3D3×#23711yescharacteristic
524C2×C3#6431511yescharacteristic
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—22no—
537C4—182no—
538C4—181no—
539C4—181no—
540C4—182no—
541C4—181no—
542C4—181no—
543C4—182no—
544C4#8865012yes—
545C3—31no—
546C3—31no—
547C3—61no—
548C3#1059455011yescharacteristic
549C2—182no—
550C2—361no—
551C2C2×#8865011yescharacteristic
552C2#1172250411yescharacteristic
553C2#1172251611yescharacteristic
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×#6198C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
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. #11721755C2semidirect
2. #64101⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
6. #11721112C2semidirect
7. #11721107C2semidirect
8. #11721107C2semidirect
9. #11721105C2semidirect
10. #11721095C2semidirect
11. #11721091C2semidirect
37. #64101D3semidirect
38. C4⋊C4×C3⋊D3D3semidirect
347. H(3,𝔽3)#552semidirect
493. C32#5513semidirect
494. C32#5529semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#117211392
C4×#61982
#117211122
#117211072
#117211052
#117210952
#117210912
#105943373
#105943333