Finite Group Databasebeta

← #13150894

Subgroups — #13150894

Order 1728.

1727 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1727 conjugacy classes of subgroups — too many to draw. Only its 171 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
2C2×#88677C212yesmaximal, complemented
3C2×#88655C211yesmaximal, characteristic, complemented
4#64139⋊C3C212yesmaximal, complemented
5#11721318C212yesmaximal, complemented
6#11721161C212yesmaximal, complemented
7#11721159C212yesmaximal, complemented
8#11721152C211yesmaximal, characteristic, complemented
9#11721142C212yesmaximal, complemented
10D3×#930—32nomaximal
11C23×#272C2211yescharacteristic
12C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
13C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C22×#1084C2211yescharacteristic, complemented
15C2×C4×#272—24no—
16C2×C4×#272C2211yescharacteristic
17C2×C4×#272C2211yescharacteristic
18C2×C4×#272C2211yescharacteristic, complemented
19C2×C4×#272C2211yescharacteristic, complemented
20C2×#638⋊C3C2214yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
22#88677—28no—
23C2×#6200C2214yescomplemented
24C2×#6199C2212yes—
25C2×#6198C2212yescomplemented
26#88655—24no—
27#3752⋊C3C2214yescomplemented
28C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
29#88450C2214yescomplemented
30#88449C2212yescomplemented
31C22×C3×C4×D3—32no—
32C22×C3×C4×D3—61no—
33C2×D3×#67—34no—
34C2×D3×D12—32no—
35#64139D312yescomplemented
36C3×#930—32no—
37#63978—34no—
38D3×(C2×D3)⋊C4—34no—
39#63909—32no—
40#63908—32no—
41#63906—34no—
42D3×C4⋊Dic3—32no—
43C23×H(3,𝔽3)C2311yescharacteristic
44C22×#272—22no—
45C22×#272—22no—
46C22×#272D414yescomplemented
47C22×#272C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C22×(C3⋊D3)⋊C3C2314yes—
50C2×((C3⋊D3)⋊C3)⋊C2—24no—
51C2×((C3⋊D3)⋊C3)⋊C2—24no—
52C2×((C3⋊D3)⋊C3)⋊C2—24no—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084—22no—
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C4×#272—22no—
60C4×#272—22no—
61C4×#272—22no—
62C4×#272D414yescomplemented
63#638⋊C3—28no—
64C2×(C3⋊Dic3)⋊C3C2314yes—
65C4⋊((C3⋊D3)⋊C3)—24no—
66#6200—28no—
67#6199—24no—
68#6198—24no—
69C2×#930—91no—
70D3×#121—92no—
71C23×C3×D3—32no—
72C23×C3×D3—61no—
73C22×D32—34no—
74C22×C32×C4—21no—
75C22×C32×C4C2×D312yescomplemented
76C2×#638C2×D314yescomplemented
77C2×C4⋊(C3⋊D3)C2×D312yescomplemented
78C2×C3×#67—34no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—61no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—68no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—61no—
89C2×C3×C4×D3—61no—
90C2×C3×C4×D3—61no—
91C2×C3×C4×D3—68no—
92D3×#67—616no—
93C2×#626—34no—
94C2×#626—34no—
95C2×#625—34no—
96C2×D3×Dic3—34no—
97D3×D12—68no—
98#3752C2×D314yescomplemented
99C4⋊(C3⋊Dic3)C2×D312yescomplemented
100C3×(C2×D3)⋊C4—34no—
101C3×C4⋊Dic3—32no—
102Dic3⋊Dic3—34no—
103#3722—34no—
104(C2×D3)⋊Dic3—34no—
105C22×H(3,𝔽3)—21no—
106C22×H(3,𝔽3)—21no—
107C22×H(3,𝔽3)C2×D412yescomplemented
108C22×H(3,𝔽3)C2411yescharacteristic
109C2×#272—21no—
110C2×#272—21no—
111C2×#272—28no—
112C2×#272C2×D412yes—
113C2×#272C2×D412yes—
114C2×#272C2×D412yescomplemented
115C2×#272#4112yes—
116C2×(C3⋊D3)⋊C3—24no—
117C2×(C3⋊D3)⋊C3—24no—
118C2×(C3⋊D3)⋊C3—24no—
119((C3⋊D3)⋊C3)⋊C2—48no—
120C4×H(3,𝔽3)—21no—
121C4×H(3,𝔽3)C2×D412yescomplemented
122#1084—21no—
123#1084C2×D412yescomplemented
124(C3⋊Dic3)⋊C3—24no—
125C23×C3×C4—91no—
126C22×#67—92no—
127C2×D3×D4—92no—
128C2×D3×D4—94no—
129C22×D12—91no—
130C22×C4×D3—32no—
131C22×C4×D3—61no—
132C3×#121—92no—
133#939—94no—
134#938—92no—
135#930—32no—
136#930—92no—
137#930—92no—
138#930—92no—
139#930—92no—
140C2×(C2×D3)⋊C4—92no—
141C2×C4⋊Dic3—91no—
142#895—92no—
143D3×C4⋊C4—92no—
144#884—94no—
145D3×C22⋊C4—94no—
146C23×C32—21no—
147C23×C32C22×D312yes—
148C22×C3⋊D3C22×D314yes—
149C22×C3×D3—32no—
150C22×C3×D3—38no—
151C22×C3×D3—32no—
152C22×C3×D3—34no—
153C22×C3×D3—64no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—64no—
157C22×C3×D3—64no—
158C22×C3×D3—64no—
159C22×C3×D3—64no—
160C2×D32—68no—
161C2×D32—68no—
162C2×D32—68no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D312yes—
169#638—28no—
170C2×C3⋊Dic3C22×D314yes—
171C4⋊(C3⋊D3)—24no—
172C3×#67—68no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—64no—
179C2×C3×Dic3—64no—
180C3×D12—64no—
181C3×C4×D3—38no—
182C3×C4×D3—64no—
183C3×C4×D3—64no—
184C3×C4×D3—64no—
185C3×C4×D3—64no—
186C3×C4×D3—64no—
187C3×C4×D3—64no—
188C3×C4×D3—64no—
189#626—68no—
190#626—68no—
191#625—68no—
192D3×Dic3—68no—
193C2×#121—271noSylow
194C2×H(3,𝔽3)—22no—
195C2×H(3,𝔽3)C22×D411yescharacteristic
196C2×H(3,𝔽3)C22×D411yescharacteristic
197C2×H(3,𝔽3)C2×#4111yescharacteristic
198#272—22no—
199#272#12114yescomplemented
200(C3⋊D3)⋊C3—44no—
201C24×C3—91no—
202C23×D3—32no—
203C23×D3—61no—
204C23×D3—92no—
205C23×D3—94no—
206C2×C3×D4—92no—
207C2×C3×D4—94no—
208C22×C3×C4—32no—
209C22×C3×C4—61no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—184no—
216C22×C3×C4D3211yescharacteristic
217C2×#67—34no—
218C2×#67—94no—
219C2×#67—94no—
220C2×#67—94no—
221C2×#67—94no—
222C2×#67—188no—
223C22×Dic3—32no—
224C22×Dic3—61no—
225C22×Dic3—92no—
226D3×D4—188no—
227D3×D4—1816no—
228C2×D12—32no—
229C2×D12—92no—
230C2×D12—92no—
231C2×D12—94no—
232C2×D12—184no—
233C2×C4×D3—32no—
234C2×C4×D3—32no—
235C2×C4×D3—32no—
236C2×C4×D3—32no—
237C2×C4×D3—61no—
238C2×C4×D3—61no—
239C2×C4×D3—61no—
240C2×C4×D3—61no—
241C2×C4×D3—68no—
242C2×C4×D3—68no—
243C2×C4×D3—94no—
244C3×C4⋊C4—92no—
245C3×C22⋊C4—94no—
246#217—94no—
247(C2×D3)⋊C4—34no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251C4⋊Dic3—32no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254Dic3⋊C4—94no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32D3×D414yescomplemented
262C22×C32C23×D312yes—
263C2×C3⋊D3—24no—
264C2×C3⋊D3—24no—
265C2×C3⋊D3—24no—
266C2×C3×D3—34no—
267C2×C3×D3—34no—
268C2×C3×D3—34no—
269C2×C3×D3—34no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—616no—
275C2×C3×D3—62no—
276C2×C3×D3—616no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—64no—
281C2×C3×D3—64no—
282C2×C3×D3—64no—
283D32—1216no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4D3×D414yescomplemented
288C3⋊Dic3—24no—
289C3×Dic3—34no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—64no—
293C3×Dic3—62no—
294C22×D4—271no—
295C22×D4—272no—
296C23×C4—91no—
297#121—92no—
298#121—272no—
299#121—272no—
300#121—272no—
301C2×C4⋊C4—271no—
302C2×C22⋊C4—272no—
303H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
304C23×C3—32no—
305C23×C3—61no—
306C23×C3—94no—
307C23×C3—91no—
308C23×C3—91no—
309C23×C3—94no—
310C23×C3—182no—
311C23×C3—182no—
312C23×C3C2×D3211yescharacteristic
313C22×D3—32no—
314C22×D3—32no—
315C22×D3—38no—
316C22×D3—34no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—64no—
320C22×D3—64no—
321C22×D3—64no—
322C22×D3—64no—
323C22×D3—64no—
324C22×D3—94no—
325C22×D3—188no—
326C22×D3—188no—
327C22×D3—188no—
328C22×D3—184no—
329C22×D3—184no—
330C22×D3—184no—
331C3×D4—184no—
332C3×D4—188no—
333C2×C3×C4—22no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—64no—
337C2×C3×C4—64no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—91no—
341C2×C3×C4—94no—
342C2×C3×C4—94no—
343C2×C3×C4—91no—
344C2×C3×C4—182no—
345C2×C3×C4—182no—
346C2×C3×C4—182no—
347C2×C3×C4—182no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350#67—68no—
351#67—188no—
352#67—188no—
353#67—188no—
354#67—188no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—34no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—64no—
361C2×Dic3—64no—
362C2×Dic3—94no—
363C2×Dic3—184no—
364D12—64no—
365D12—184no—
366D12—184no—
367D12—188no—
368C4×D3—38no—
369C4×D3—64no—
370C4×D3—64no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—64no—
374C4×D3—64no—
375C4×D3—64no—
376C4×D3—188no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32C2×D3×D412yes—
383C2×C32C2×D3×D412yes—
384C2×C32D3×#4112yes—
385C3⋊D3—44no—
386C3×D3—38no—
387C3×D3—64no—
388C3×D3—64no—
389C3×D3—64no—
390C3×D3—124no—
391C24—91no—
392C24—272no—
393C2×D4—94no—
394C2×D4—92no—
395C2×D4—272no—
396C2×D4—274no—
397C2×D4—544no—
398C2×D4—548no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—91no—
402C22×C4—91no—
403C22×C4—91no—
404C22×C4—184no—
405C22×C4—272no—
406C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
407C4⋊C4—92no—
408C4⋊C4—272no—
409C22⋊C4—94no—
410C22⋊C4—274no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—32no—
414C22×C3—34no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—61no—
420C22×C3—62no—
421C22×C3—92no—
422C22×C3—92no—
423C22×C3—92no—
424C22×C3—92no—
425C22×C3—188no—
426C22×C3—184no—
427C22×C3—184no—
428C22×C3—184no—
429C22×C3—181no—
430C22×C3—181no—
431C22×C3#380212yes—
432C22×C3C22×D3211yescharacteristic
433C2×D3—34no—
434C2×D3—34no—
435C2×D3—34no—
436C2×D3—34no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—616no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—616no—
446C2×D3—62no—
447C2×D3—64no—
448C2×D3—64no—
449C2×D3—64no—
450C2×D3—184no—
451C2×D3—184no—
452C2×D3—184no—
453C2×D3—3616no—
454C2×D3—368no—
455C3×C4—21no—
456C3×C4—34no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—92no—
461C3×C4—184no—
462C3×C4—181no—
463C3×C4#381112yes—
464Dic3—34no—
465Dic3—62no—
466Dic3—62no—
467Dic3—62no—
468Dic3—64no—
469Dic3—184no—
470C32—21no—
471C32D3×#12112yescomplemented
472C23—91no—
473C23—91no—
474C23—94no—
475C23—94no—
476C23—182no—
477C23—182no—
478C23—544no—
479C23—544no—
480C23—544no—
481C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
482D4—188no—
483D4—184no—
484D4—544no—
485D4—548no—
486C2×C4—22no—
487C2×C4—91no—
488C2×C4—94no—
489C2×C4—94no—
490C2×C4—91no—
491C2×C4—182no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—182no—
495C2×C4—544no—
496C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
497C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
498C2×C3—22no—
499C2×C3—32no—
500C2×C3—32no—
501C2×C3—32no—
502C2×C3—64no—
503C2×C3—64no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—61no—
507C2×C3—94no—
508C2×C3—182no—
509C2×C3—364no—
510C2×C3C2×#380211yescharacteristic
511C2×C3#6430611yescharacteristic
512C2×C3C2×#381111yescharacteristic
513D3—38no—
514D3—64no—
515D3—64no—
516D3—64no—
517D3—124no—
518D3—364no—
519C22—21no—
520C22—21no—
521C22—92no—
522C22—92no—
523C22—92no—
524C22—92no—
525C22—188no—
526C22—181no—
527C22—181no—
528C22—184no—
529C22—184no—
530C22—184no—
531C22—1088no—
532C22#8865512yescomplemented
533C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C4—21no—
535C4—92no—
536C4—181no—
537C4—184no—
538C4#8867712yescomplemented
539C3—32no—
540C3—61no—
541C3#1059508111yescharacteristic
542C2—22no—
543C2—94no—
544C2—182no—
545C2—364no—
546C2C2×#8865511yescharacteristic
547C2#1172249111yescharacteristic
548C2C2×#8867711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×#272C2semidirect
2. C2×#88677C2semidirect
3. C2×#88655C2semidirect
4. #64139⋊C3C2semidirect
5. #11721318C2semidirect
6. #11721161C2semidirect
7. #11721159C2semidirect
8. #11721152C2semidirect
9. #11721142C2semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
14. C22×#1084C22semidirect
18. C2×C4×#272C22semidirect
19. C2×C4×#272C22semidirect
20. C2×#638⋊C3C22semidirect
21. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
23. C2×#6200C22semidirect
25. C2×#6198C22semidirect
27. #3752⋊C3C22semidirect
28. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
29. #88450C22semidirect
30. #88449C22semidirect
35. #64139D3semidirect
46. C22×#272D4semidirect
62. C4×#272D4semidirect
75. C22×C32×C4C2×D3semidirect
76. C2×#638C2×D3semidirect
77. C2×C4⋊(C3⋊D3)C2×D3semidirect
98. #3752C2×D3semidirect
99. C4⋊(C3⋊Dic3)C2×D3semidirect
107. C22×H(3,𝔽3)C2×D4semidirect
114. C2×#272C2×D4semidirect
121. C4×H(3,𝔽3)C2×D4semidirect
123. #1084C2×D4semidirect
199. #272#121semidirect
261. C22×C32D3×D4semidirect
287. C32×C4D3×D4semidirect
303. H(3,𝔽3)C2×#121semidirect
406. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
471. C32D3×#121semidirect
532. C22#88655semidirect
538. C4#88677semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886772
C2×#886552
#64139⋊C32
#117213182
#117211612
#117211592
#117211522
#117211422
D3×#9303