Finite Group Databasebeta

← #13140266

Subgroups — #13140266

Order 1728.

1363 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2C22×#6200C212yesmaximal, complemented
3C2×#3752⋊C3C211yesmaximal, characteristic, complemented
4C2×#88451C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591391—31nomaximal
7#10591389—31nomaximal
8C23×#272C2212yes—
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272—24no—
12C2×C4⋊#272—22no—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6200—24no—
16C2×#6200—24no—
17C2×#6200—24no—
18#3752⋊C3—22no—
19#88451—24no—
20C22×C3×D12—31no—
21C22×C3×D12—31no—
22C22×C3×D12—61no—
23C22×#626—32no—
24C22×#625—32no—
25C2×#3752D311yescharacteristic, complemented
26C2×C3×(C2×D3)⋊C4—31no—
27C2×(C2×D3)⋊Dic3—32no—
28C2×(C2×D3)⋊Dic3—32no—
29#63658D311yescharacteristic, complemented
30C3×C3⋊#95—31no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272—22no—
33C22×#272—22no—
34C22×#272—22no—
35C22×#272—22no—
36C22×#272—22no—
37C22×#272—22no—
38C22×#272D414yescomplemented
39C22×(C3⋊D3)⋊C3—22no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3D412yescomplemented
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)D412yescomplemented
46C4⋊#272—28no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D414yescomplemented
52#6200—216no—
53#5223—91no—
54#4960—91no—
55#4671—91no—
56C23×C3⋊D3C2×D311yescharacteristic
57C23×C3×D3—31no—
58C23×C3×D3—32no—
59C23×C3×D3—32no—
60C23×C3×D3—62no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D311yescharacteristic
65C22×C3⋊Dic3C2×D312yes—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—31no—
68C2×C3×D12—64no—
69C2×C3×D12—64no—
70C2×C3×D12—68no—
71C2×C3×D12—64no—
72C2×C3×D12—62no—
73C2×C3×D12—62no—
74C2×#626—64no—
75C2×#626—64no—
76C2×#626—64no—
77C2×#625—64no—
78C2×#625—64no—
79C2×#625—64no—
80#3752—22no—
81C3×(C2×D3)⋊C4—62no—
82(C2×D3)⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C22×H(3,𝔽3)C2×D412yes—
86C22×H(3,𝔽3)C2×D412yes—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)#4111yescharacteristic
89C2×#272—24no—
90C2×#272—24no—
91C2×#272—24no—
92C2×#272—42no—
93C2×#272—42no—
94C2×#272—42no—
95C2×#272—42no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—42no—
100C2×(C3⋊D3)⋊C3—41no—
101C4×H(3,𝔽3)—22no—
102(C3⋊Dic3)⋊C3—24no—
103(C3⋊Dic3)⋊C3—41no—
104C22×C3×D4—91no—
105C22×#67—92no—
106C22×#67—92no—
107C22×D12—31no—
108C22×D12—31no—
109C22×D12—61no—
110C22×D12—92no—
111C2×C3×C22⋊C4—91no—
112C2×#217—92no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117C3×#95—91no—
118C3⋊#95—31no—
119C3⋊#95—91no—
120C23×C32—21no—
121C23×C32C22×D311yescharacteristic
122C23×C32C22×D311yescharacteristic
123C22×C3⋊D3—21no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3D1212yescomplemented
128C22×C3×D3—32no—
129C22×C3×D3—34no—
130C22×C3×D3—34no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—64no—
134C22×C3×D3—61no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—61no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—122no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C22×C3×D3—122no—
154C2×C32×C4—24no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4D1212yescomplemented
159C2×C32×C4#6712yescomplemented
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3D1214yescomplemented
165C2×C3×Dic3—34no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170C3×D12—616no—
171C3×D12—68no—
172C3×D12—68no—
173#626—616no—
174#625—616no—
175#392—271noSylow
176C2×H(3,𝔽3)#12012yes—
177C2×H(3,𝔽3)#12011yescharacteristic
178C2×H(3,𝔽3)#12112yes—
179C2×H(3,𝔽3)#12111yescharacteristic
180C2×H(3,𝔽3)#12711yescharacteristic
181#272—44no—
182(C3⋊D3)⋊C3—42no—
183C24×C3—91no—
184C24×C3—92no—
185C23×D3—32no—
186C23×D3—32no—
187C23×D3—31no—
188C23×D3—62no—
189C23×D3—91no—
190C2×C3×D4—184no—
191C2×C3×D4—182no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—61no—
195C22×C3×C4—92no—
196C22×C3×C4—91no—
197C22×C3×C4D3211yescharacteristic
198C2×#67—184no—
199C2×#67—184no—
200C2×#67—184no—
201C2×#67—184no—
202C2×#67—184no—
203C2×#67—184no—
204C22×Dic3—32no—
205C22×Dic3—31no—
206C22×Dic3—92no—
207C22×Dic3—91no—
208C2×D12—62no—
209C2×D12—62no—
210C2×D12—64no—
211C2×D12—64no—
212C2×D12—68no—
213C2×D12—64no—
214C2×D12—184no—
215C2×D12—184no—
216C2×D12—184no—
217C3×C22⋊C4—182no—
218#217—184no—
219(C2×D3)⋊C4—62no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4—184no—
222(C2×D3)⋊C4—184no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32C2×D1212yes—
229C22×C32C2×D1211yescharacteristic
230C22×C32C2×D1211yescharacteristic
231C22×C32#23511yescharacteristic
232C22×C32D3×D411yescharacteristic
233C22×C32D3×D412yes—
234C22×C32D3×D412yes—
235C22×C32D3×D412yes—
236C22×C32#23911yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C2×C3⋊D3—24no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3×D3—68no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—62no—
250C2×C3×D3—64no—
251C2×C3×D3—64no—
252C2×C3×D3—124no—
253C2×C3×D3—124no—
254C2×C3×D3—124no—
255C2×C3×D3—124no—
256C2×C3×D3—124no—
257C2×C3×D3—124no—
258C2×C3×D3—121no—
259C2×C3×D3—122no—
260C2×C3×D3—121no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C32×C4—24no—
270C32×C4—22no—
271C32×C4—22no—
272C3⋊Dic3—24no—
273C3⋊Dic3—41no—
274C3×Dic3—64no—
275C3×Dic3—121no—
276C22×D4—91no—
277C22×D4—272no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—272no—
280#95—91no—
281H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
282C23×C3—31no—
283C23×C3—31no—
284C23×C3—61no—
285C23×C3—92no—
286C23×C3—94no—
287C23×C3—181no—
288C23×C3—182no—
289C23×C3—182no—
290C23×C3—181no—
291C23×C3—182no—
292C23×C3—182no—
293C23×C3—182no—
294C23×C3—182no—
295C23×C3—182no—
296C23×C3—182no—
297C23×C3C2×D3211yescharacteristic
298C22×D3—34no—
299C22×D3—34no—
300C22×D3—32no—
301C22×D3—64no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—62no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—64no—
315C22×D3—64no—
316C22×D3—61no—
317C22×D3—62no—
318C22×D3—61no—
319C22×D3—62no—
320C22×D3—92no—
321C22×D3—122no—
322C22×D3—122no—
323C22×D3—122no—
324C22×D3—122no—
325C22×D3—181no—
326C22×D3—182no—
327C22×D3—182no—
328C22×D3—181no—
329C3×D4—188no—
330C2×C3×C4—22no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—64no—
336C2×C3×C4—62no—
337C2×C3×C4—94no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340C2×C3×C4—182no—
341C2×C3×C4—181no—
342C2×C3×C4#62612yes—
343#67—1816no—
344#67—1816no—
345C2×Dic3—34no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—61no—
350C2×Dic3—94no—
351C2×Dic3—182no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354C2×Dic3—181no—
355D12—68no—
356D12—68no—
357D12—616no—
358D12—1816no—
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32#87411yescharacteristic
365C2×C32#88212yes—
366C2×C32#88212yes—
367C2×C32#88211yescharacteristic
368C2×C32#88412yes—
369C2×C32#89511yescharacteristic
370C2×C32#89512yes—
371C2×C32#93011yescharacteristic
372C2×C32#93711yescharacteristic
373C2×C32#94011yescharacteristic
374C3⋊D3—42no—
375C3×D3—128no—
376C3×D3—122no—
377C3×D3—124no—
378C3×D3—124no—
379C24—91no—
380C24—92no—
381C2×D4—184no—
382C2×D4—182no—
383C2×D4—544no—
384C2×D4—544no—
385C2×D4—544no—
386C22×C4—92no—
387C22×C4—91no—
388C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C22⋊C4—182no—
390C22⋊C4—544no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—31no—
395C22×C3—32no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—32no—
400C22×C3—32no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—62no—
404C22×C3—61no—
405C22×C3—62no—
406C22×C3—184no—
407C22×C3—182no—
408C22×C3—184no—
409C22×C3—184no—
410C22×C3—184no—
411C22×C3—361no—
412C22×C3—362no—
413C22×C3—361no—
414C22×C3—362no—
415C22×C3—362no—
416C22×C3—362no—
417C22×C3—362no—
418C22×C3#379911yescharacteristic
419C22×C3D3×D1211yescharacteristic
420C22×C3D3×D1212yes—
421C22×C3C2×#62611yescharacteristic
422C22×C3#381112yes—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—68no—
430C2×D3—62no—
431C2×D3—64no—
432C2×D3—124no—
433C2×D3—124no—
434C2×D3—122no—
435C2×D3—122no—
436C2×D3—122no—
437C2×D3—122no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—124no—
443C2×D3—124no—
444C2×D3—124no—
445C2×D3—124no—
446C2×D3—122no—
447C2×D3—121no—
448C2×D3—121no—
449C2×D3—184no—
450C2×D3—182no—
451C2×D3—361no—
452C2×D3—362no—
453C2×D3—361no—
454C3×C4—22no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—64no—
458C3×C4—184no—
459C3×C4—361no—
460Dic3—64no—
461Dic3—121no—
462Dic3—184no—
463Dic3—361no—
464C32—21no—
465C32#467111yescharacteristic, complemented
466C32#496011yescharacteristic, complemented
467C23—92no—
468C23—94no—
469C23—182no—
470C23—181no—
471C23—182no—
472C23—182no—
473C23—182no—
474C23—182no—
475C23—181no—
476C23—182no—
477C23—182no—
478C23—182no—
479C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480D4—188no—
481D4—5416no—
482C2×C4—22no—
483C2×C4—94no—
484C2×C4—182no—
485C2×C4—182no—
486C2×C4—182no—
487C2×C4—181no—
488C2×C4#620012yescomplemented
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—32no—
494C2×C3—32no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—32no—
498C2×C3—32no—
499C2×C3—61no—
500C2×C3—61no—
501C2×C3—62no—
502C2×C3—61no—
503C2×C3—62no—
504C2×C3—362no—
505C2×C3—364no—
506C2×C3#6390911yescharacteristic
507C2×C3#6391012yes—
508C2×C3#6391111yescharacteristic
509C2×C3#6392312yes—
510C2×C3#6398111yescharacteristic
511D3—124no—
512D3—124no—
513D3—128no—
514D3—122no—
515D3—362no—
516C22—184no—
517C22—182no—
518C22—184no—
519C22—184no—
520C22—184no—
521C22—361no—
522C22—362no—
523C22—362no—
524C22—362no—
525C22—362no—
526C22—361no—
527C22—362no—
528C22#8865211yescharacteristic
529C22#8865411yescharacteristic
530C22#8865412yes—
531C22C2×#620011yescharacteristic
532C22#8867712yes—
533C4—22no—
534C4—184no—
535C4—361no—
536C3—31no—
537C3—31no—
538C3—61no—
539C3#1059139511yescharacteristic
540C2—364no—
541C2—362no—
542C2#1172115611yescharacteristic
543C2#1172115712yes—
544C2#1172115811yescharacteristic
545C2#1172116312yes—
546C2#1172132311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. C22×#6200C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#88451C2semidirect
5. #63658⋊C3C2semidirect
25. C2×#3752D3semidirect
29. #63658D3semidirect
38. C22×#272D4semidirect
43. C22×(C3⋊D3)⋊C3D4semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
51. C2×(C3⋊Dic3)⋊C3D4semidirect
127. C22×C3⋊D3D12semidirect
158. C2×C32×C4D12semidirect
159. C2×C32×C4#67semidirect
164. C2×C3⋊Dic3D12semidirect
281. H(3,𝔽3)#392semidirect
465. C32#4671semidirect
466. C32#4960semidirect
488. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×#62002
C2×#3752⋊C32
C2×#884512
#63658⋊C32
#105913913
#105913893