Finite Group Databasebeta

← #13155328

Subgroups — #13155328

Order 1728.

1126 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×D12C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic, complemented
3C3×#63818C211yesmaximal, characteristic, complemented
4#11722027C212yesmaximal, complemented
5C3×#63558C212yesmaximal, complemented
6#10591503—31nomaximal
7D3×#823—32nomaximal
8C22×D3×C3⋊D3C412yes—
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6291C412yes—
13C3⋊D3×D12—21no—
14C3⋊D3×D12—21no—
15C3⋊D3×D12—21no—
16C3⋊D3×D12—21no—
17C3×#3748C2212yescomplemented
18C32×#208C2212yescomplemented
19#88809—21no—
20#88808—21no—
21C3×#3712—21no—
22C3×#3710—21no—
23C2×D4×C3⋊D3—31no—
24C2×D3×D12—32no—
25C2×D3×D12—61no—
26C3×D3×M16—32no—
27#63818—31no—
28#63818D311yescharacteristic, complemented
29D3×#208—32no—
30#63650—32no—
31C3×#833—32no—
32C3×#823—32no—
33#63559—32no—
34#63559—32no—
35#63559—62no—
36#63558D312yescomplemented
37C22×C3⋊(C3⋊D3)C2×C412yes—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C22×C32×D3C2×C412yes—
40C2×D3×C3⋊D3—22no—
41C2×D3×C3⋊D3—22no—
42C2×D3×C3⋊D3—44no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3C2×C411yescharacteristic
47C3×C4×C3⋊D3D411yescharacteristic
48C3×C4×C3⋊D3D411yescharacteristic
49C3×C4×C3⋊D3D411yescharacteristic
50C3×C4×C3⋊D3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53#6291—22no—
54#6291—22no—
55C3×C3⋊(C3⋊C8)—22no—
56C32×C3⋊C8—22no—
57C2×#823—91no—
58D3×#100—92no—
59C23×C3⋊D3—32no—
60C22×D32—34no—
61C22×D32—62no—
62C2×C32×D4—31no—
63C2×#638—32no—
64D4×C3⋊D3—61no—
65D4×C3⋊D3—61no—
66D4×C3⋊D3—61no—
67D4×C3⋊D3—61no—
68C2×C4⋊(C3⋊D3)—32no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)—62no—
72C2×C4⋊(C3⋊D3)—121no—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C3×D12—21no—
75C2×C3×D12C2×D312yes—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—61no—
78C2×#626—34no—
79C2×#626—62no—
80D3×D12—62no—
81D3×D12—62no—
82D3×D12—62no—
83D3×D12—62no—
84D3×D12—62no—
85D3×D12—62no—
86D3×D12—62no—
87D3×D12—62no—
88C32×M16—32no—
89#3748C2×D312yescomplemented
90C3×#208—22no—
91C3×#208—32no—
92C3×#208—32no—
93C3×#208—62no—
94C3×#208C2×D312yescomplemented
95C3×#208C2×D312yescomplemented
96C3×ΓL(1,𝔽25)—62no—
97C3×D3×C8—62no—
98#3712—21no—
99#3712—61no—
100#3711—62no—
101#3710—21no—
102#3710—61no—
103D3×C3⋊C8—62no—
104C22×C33C22×C411yescharacteristic
105C2×C3⋊(C3⋊D3)—22no—
106C2×C3⋊(C3⋊D3)—42no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3C22⋊C411yescharacteristic
109C2×C3×C3⋊D3C22⋊C411yescharacteristic
110C2×C32×D3—22no—
111C2×C32×D3—42no—
112D3×C3⋊D3—44no—
113D3×C3⋊D3—44no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3C22⋊C411yescharacteristic
117C3×C3⋊Dic3C22⋊C411yescharacteristic
118C2×D3×D4—92no—
119C2×D3×D4—181no—
120C22×D12—91no—
121C2×C3×M16—91no—
122C2×#208—91no—
123D3×M16—32no—
124D3×M16—92no—
125C3×#100—92no—
126#833—32no—
127#823—32no—
128#823—92no—
129#823—92no—
130#823—92no—
131#823—182no—
132C23×C32—32no—
133C22×C3⋊D3—34no—
134C22×C3⋊D3—32no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—62no—
138C22×C3⋊D3—62no—
139C22×C3⋊D3—122no—
140C22×C3⋊D3—124no—
141C22×C3⋊D3C4×D311yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—32no—
144C22×C3×D3—61no—
145C22×C3×D3C4×D314yes—
146C2×D32—64no—
147C2×D32—64no—
148C2×D32—64no—
149C2×D32—64no—
150C2×D32—128no—
151C2×D32—128no—
152C32×D4—61no—
153C32×D4—61no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D312yes—
159#638—62no—
160#638—62no—
161C2×C3⋊Dic3C4×D311yescharacteristic
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—61no—
164C4⋊(C3⋊D3)—62no—
165C4⋊(C3⋊D3)—62no—
166C4⋊(C3⋊D3)—62no—
167C4⋊(C3⋊D3)—62no—
168C4⋊(C3⋊D3)—122no—
169C4⋊(C3⋊D3)—122no—
170C4⋊(C3⋊D3)—122no—
171C4⋊(C3⋊D3)—122no—
172C4×C3⋊D3D1211yescharacteristic
173C4×C3⋊D3D1211yescharacteristic
174C4×C3⋊D3#6711yescharacteristic
175C4×C3⋊D3#6711yescharacteristic
176C2×C3×Dic3—32no—
177C2×C3×Dic3—61no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12—22no—
181C3×D12—22no—
182C3×C4×D3—32no—
183C3×C4×D3—32no—
184C3×C4×D3—32no—
185C3×C4×D3—32no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190#626—64no—
191#626—64no—
192#626—64no—
193#626—64no—
194C32×C8—62no—
195C3⋊(C3⋊C8)—22no—
196C3×C3⋊C8—22no—
197C3×C3⋊C8—22no—
198C3×C3⋊C8—42no—
199C3×C3⋊C8—62no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8—122no—
202C2×#100—271noSylow
203C2×C33—21no—
204C2×C33C2×C22⋊C411yescharacteristic
205C3⋊(C3⋊D3)—42no—
206C3×C3⋊D3—21no—
207C3×C3⋊D3#10012yescomplemented
208C32×D3—42no—
209C23×D3—94no—
210C23×D3—92no—
211C23×D3—182no—
212C2×C3×D4—32no—
213C2×C3×D4—61no—
214C22×C3×C4—91no—
215C2×#67—94no—
216C2×#67—182no—
217D3×D4—182no—
218D3×D4—182no—
219D3×D4—182no—
220D3×D4—182no—
221D3×D4—182no—
222D3×D4—182no—
223D3×D4—182no—
224D3×D4—182no—
225C2×D12—92no—
226C2×D12—91no—
227C2×D12—92no—
228C2×D12—181no—
229C2×D12—182no—
230C2×D12—181no—
231C2×D12—181no—
232C2×D12—181no—
233C2×D12—181no—
234C2×D12—361no—
235C2×D12D3211yescharacteristic
236C2×C4×D3—32no—
237C2×C4×D3—61no—
238C3×M16—32no—
239C3×M16—32no—
240C3×M16—62no—
241C3×M16—92no—
242C3×M16—181no—
243C2×C3×C8—181no—
244#208—32no—
245#208—32no—
246#208—62no—
247#208—181no—
248#208D3212yescomplemented
249C2×C3⋊C8—181no—
250ΓL(1,𝔽25)—62no—
251ΓL(1,𝔽25)—182no—
252D3×C8—62no—
253D3×C8—182no—
254C22×C32—22no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—62no—
258C22×C32—122no—
259C22×C32C2×C4×D311yescharacteristic
260C22×C32C2×C4×D312yes—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—64no—
265C2×C3⋊D3—124no—
266C2×C3⋊D3—124no—
267C2×C3⋊D3—124no—
268C2×C3⋊D3—124no—
269C2×C3⋊D3—122no—
270C2×C3⋊D3—124no—
271C2×C3⋊D3—124no—
272C2×C3⋊D3—244no—
273C2×C3⋊D3—244no—
274C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
275C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—32no—
279C2×C3×D3—32no—
280C2×C3×D3—44no—
281C2×C3×D3—44no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286D32—128no—
287D32—128no—
288D32—128no—
289D32—128no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4C2×D1211yescharacteristic
297C32×C4D3×D412yes—
298C32×C4D3×D412yes—
299C32×C4C2×#6711yescharacteristic
300C3⋊Dic3(C2×D3)⋊C411yescharacteristic
301C3⋊Dic3(C2×D3)⋊C411yescharacteristic
302C3×Dic3—32no—
303C3×Dic3—32no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C22×D4—271no—
307C2×M16—91no—
308C2×M16—271no—
309#100—92no—
310#100—272no—
311C33C2×#10011yescharacteristic, complemented, Sylow
312C23×C3—34no—
313C23×C3—62no—
314C23×C3—91no—
315C22×D3—32no—
316C22×D3—61no—
317C22×D3—94no—
318C22×D3—92no—
319C22×D3—182no—
320C22×D3—184no—
321C22×D3—182no—
322C22×D3—182no—
323C22×D3—184no—
324C22×D3—184no—
325C22×D3—184no—
326C22×D3—184no—
327C22×D3—362no—
328C22×D3—364no—
329C22×D3—368no—
330C22×D3—368no—
331C22×D3#62412yes—
332C3×D4—62no—
333C3×D4—62no—
334C3×D4—62no—
335C3×D4—62no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4C2×D3212yes—
345C2×C3×C4C2×D3211yescharacteristic
346#67—184no—
347#67—184no—
348#67—184no—
349#67—184no—
350C2×Dic3—32no—
351C2×Dic3—61no—
352D12—21no—
353D12—21no—
354D12—182no—
355D12—182no—
356D12—182no—
357D12—182no—
358D12—182no—
359D12—182no—
360D12—181no—
361D12—181no—
362D12—362no—
363D12—362no—
364D12—362no—
365D12—362no—
366C4×D3—32no—
367C4×D3—32no—
368C4×D3—32no—
369C4×D3—32no—
370C4×D3—61no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C3×C8—62no—
375C3×C8—62no—
376C3×C8—122no—
377C3×C8—182no—
378C3⋊C8—22no—
379C3⋊C8—62no—
380C3⋊C8—62no—
381C3⋊C8—122no—
382C2×C32—21no—
383C2×C32—22no—
384C2×C32—22no—
385C2×C32—22no—
386C2×C32—21no—
387C2×C32—41no—
388C2×C32—42no—
389C2×C32—42no—
390C2×C32—122no—
391C2×C32D3×C22⋊C412yes—
392C2×C32C2×(C2×D3)⋊C411yescharacteristic
393C3⋊D3—21no—
394C3⋊D3—122no—
395C3⋊D3—124no—
396C3⋊D3—124no—
397C3⋊D3—244no—
398C3⋊D3—244no—
399C3⋊D3#82312yescomplemented
400C3×D3—34no—
401C3×D3—44no—
402C3×D3—44no—
403C3×D3—62no—
404C3×D3—62no—
405C3×D3—62no—
406C24—272no—
407C2×D4—31no—
408C2×D4—272no—
409C2×D4—271no—
410C2×D4—541no—
411C2×D4—541no—
412C2×D4—541no—
413C2×D4—541no—
414C22×C4—91no—
415M16—32no—
416M16—92no—
417M16—181no—
418M16—541no—
419C2×C8—181no—
420C2×C8—541no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—64no—
425C22×C3—64no—
426C22×C3—91no—
427C22×C3—91no—
428C22×C3—124no—
429C22×C3—124no—
430C22×C3—182no—
431C22×C3C4×D3212yes—
432C22×C3C2×#62411yescharacteristic
433C2×D3—22no—
434C2×D3—32no—
435C2×D3—32no—
436C2×D3—42no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—64no—
440C2×D3—64no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—182no—
444C2×D3—364no—
445C2×D3—364no—
446C2×D3—368no—
447C2×D3—364no—
448C2×D3—364no—
449C2×D3—368no—
450C2×D3—368no—
451C2×D3—368no—
452C2×D3—364no—
453C2×D3—362no—
454C2×D3—364no—
455C2×D3—724no—
456C2×D3—724no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—41no—
462C3×C4—41no—
463C3×C4—91no—
464C3×C4—91no—
465C3×C4D3×D1212yes—
466C3×C4D3×#6712yes—
467C3×C4#381111yescharacteristic
468C3×C4#381111yescharacteristic
469Dic3—32no—
470Dic3—32no—
471Dic3—61no—
472Dic3—61no—
473C32—21no—
474C32—22no—
475C32—41no—
476C32D3×#10012yescomplemented
477C32C2×#82311yescharacteristic, complemented
478C23—32no—
479C23—91no—
480C23—272no—
481C23—542no—
482C23—542no—
483C23—1084no—
484D4—61no—
485D4—61no—
486D4—542no—
487D4—542no—
488D4—541no—
489D4—541no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4D3311yescharacteristic
496C8—62no—
497C8—182no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—22no—
501C2×C3—21no—
502C2×C3—22no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—92no—
507C2×C3—124no—
508C2×C3—124no—
509C2×C3—181no—
510C2×C3D3×(C2×D3)⋊C412yes—
511C2×C3#6397411yescharacteristic
512D3—34no—
513D3—42no—
514D3—62no—
515D3—62no—
516D3—62no—
517D3—364no—
518D3—362no—
519D3—364no—
520D3—724no—
521D3—724no—
522C22—62no—
523C22—91no—
524C22—91no—
525C22—122no—
526C22—182no—
527C22—542no—
528C22—1082no—
529C22—1084no—
530C22—1084no—
531C22D3×#62411yescharacteristic
532C4—91no—
533C4—91no—
534C4#8895211yescharacteristic
535C4#8895511yescharacteristic
536C3—21no—
537C3—22no—
538C3—41no—
539C3D3×#82312yescomplemented
540C3#1059150311yescharacteristic, complemented
541C2—21no—
542C2—92no—
543C2—122no—
544C2—181no—
545C2—1082no—
546C2#1172290411yescharacteristic
subgroupG/H
centerC2#11722904
commutatorC22×C33C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C2×C22⋊C4
prime radicalC2×C3⋊D3×D12C2
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C3×#63818C2semidirect
4. #11722027C2semidirect
5. C3×#63558C2semidirect
17. C3×#3748C22semidirect
18. C32×#208C22semidirect
28. #63818D3semidirect
36. #63558D3semidirect
89. #3748C2×D3semidirect
94. C3×#208C2×D3semidirect
95. C3×#208C2×D3semidirect
207. C3×C3⋊D3#100semidirect
248. #208D32semidirect
311. C33C2×#100semidirect
399. C3⋊D3#823semidirect
476. C32D3×#100semidirect
477. C32C2×#823semidirect
539. C3D3×#823semidirect
540. C3#10591503semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
#117236562
C3×#638182
#117220272
C3×#635582
#105915033
D3×#8233