Finite Group Databasebeta

← #13148734

Subgroups — #13148734

Order 1728.

1315 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1315 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88677C211yesmaximal, characteristic
2C22⋊C4×#272C211yesmaximal, characteristic, complemented
3#64090⋊C3C212yesmaximal, complemented
4#11721314C212yesmaximal, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6#11721160C212yesmaximal, complemented
7C4×#6200C212yesmaximal, complemented
8#11721124C212yesmaximal, complemented
9#11721076C212yesmaximal, complemented
10#10594254—32nomaximal
11C23×#272C2211yescharacteristic
12C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
13C2×C4×#272C2212yescomplemented
14C2×#638⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2212yescomplemented
16C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
17#88677C418yescomplemented
18C2×#6200C2212yes—
19C2×#6199C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
21C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
22#88552C2211yescharacteristic, complemented
23#88542C2212yes—
24#3752⋊C3C2212yescomplemented
25#3750⋊C3C2212yescomplemented
26C4×(C3⋊Dic3)⋊C3C2212yescomplemented
27#88451C2212yes—
28#88451C2211yescharacteristic, complemented
29#88451C2211yescharacteristic, complemented
30#88450C2212yescomplemented
31#88448C2212yescomplemented
32#88447C2212yescomplemented
33C2×D3×#67—32no—
34#64090D312yescomplemented
35C3×#67⋊C4—32no—
36C3×D3×C22⋊C4—32no—
37C3×D3×C22⋊C4—61no—
38#67⋊Dic3—32no—
39C2×(C2×D3)⋊Dic3—32no—
40#63952—32no—
41#63921—32no—
42D3×(C2×D3)⋊C4—32no—
43#63904—32no—
44C4×#626—32no—
45C4×#625—32no—
46#63879—32no—
47#63878—32no—
48#63877—32no—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×#272—42no—
51C22×#272C2×C414yescomplemented
52C22×#272C2311yescharacteristic
53C22×#272C2311yescharacteristic
54C22×(C3⋊D3)⋊C3C2312yes—
55C2×((C3⋊D3)⋊C3)⋊C2—42no—
56C2×((C3⋊D3)⋊C3)⋊C2C2×C414yescomplemented
57C2×C4×H(3,𝔽3)C2312yes—
58C2×#1084C2312yes—
59C4×#272—42no—
60#638⋊C3C2×C418yescomplemented
61C2×(C3⋊Dic3)⋊C3—42no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C4×(C3⋊D3)⋊C3—42no—
66#6200C2×C418yescomplemented
67#6199C2×C414yescomplemented
68((C3⋊D3)⋊C3)⋊C4—42no—
69#5545—92no—
70#5486—91no—
71C23×C3×D3—32no—
72C23×C3×D3—61no—
73C22×D32—32no—
74C2×#638C2×D312yes—
75C22×C3⋊Dic3C2×D312yescomplemented
76C2×C4×C3⋊D3C2×D312yescomplemented
77C2×C3×#67—32no—
78C22×C3×Dic3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—62no—
83D3×#67—316no—
84C2×#626—32no—
85C2×#625—32no—
86C2×#624—32no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89C32×C22⋊C4—21no—
90C32×C22⋊C4C2×D312yescomplemented
91#3752C2×D312yescomplemented
92#3750C2×D312yescomplemented
93C4×C3⋊Dic3C2×D312yescomplemented
94C3×#217—32no—
95C3×#217—61no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4—62no—
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—32no—
102#3724—32no—
103Dic3⋊Dic3—32no—
104#3722—32no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111Dic32—32no—
112C22×H(3,𝔽3)—22no—
113C22×H(3,𝔽3)C22×C412yes—
114C22×H(3,𝔽3)C2411yescharacteristic
115C2×#272—22no—
116C2×#272—22no—
117C2×#272—44no—
118C2×#272C22×C412yes—
119C2×#272C22×C412yes—
120C2×(C3⋊D3)⋊C3—22no—
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3C22×C414yes—
123((C3⋊D3)⋊C3)⋊C2—44no—
124C4×H(3,𝔽3)—22no—
125#1084—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3C22×C414yes—
129C22×#67—91no—
130C2×D3×D4—92no—
131C3×C4×D4—92no—
132C2×C3×C22⋊C4—91no—
133C2×#217—91no—
134D4×Dic3—92no—
135C4×#67—92no—
136C4×#67—92no—
137C4×#67—92no—
138C4×#67—92no—
139C2×(C2×D3)⋊C4—92no—
140C2×(C2×D3)⋊C4—92no—
141#893—92no—
142#892—92no—
143#67⋊C4—32no—
144#67⋊C4—92no—
145#67⋊C4—92no—
146#67⋊C4—92no—
147D3×C22⋊C4—32no—
148D3×C22⋊C4—61no—
149D3×C22⋊C4—92no—
150D3×C22⋊C4—92no—
151#877—92no—
152C4×D12—92no—
153#872—92no—
154C23×C32—21no—
155C23×C32C22×D312yes—
156C22×C3⋊D3C22×D312yes—
157C22×C3×D3—32no—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—38no—
161C22×C3×D3—61no—
162C22×C3×D3—64no—
163C22×C3×D3—61no—
164C22×C3×D3—122no—
165C22×C3×D3—122no—
166C22×C3×D3—124no—
167C2×D32—38no—
168C2×D32—122no—
169C2×D32—122no—
170C2×C32×C4—22no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D312yes—
173#638C4×D318yescomplemented
174C2×C3⋊Dic3—42no—
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D312yes—
177C2×C3⋊Dic3C22×D312yes—
178C4×C3⋊D3—42no—
179C3×#67—38no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—122no—
187C3×C4×D3—122no—
188C3×C4×D3—122no—
189C3×C4×D3—122no—
190C3×C4×D3—124no—
191#626—38no—
192#625—38no—
193#624—122no—
194D3×Dic3—38no—
195D3×Dic3—122no—
196#518—271noSylow
197C2×H(3,𝔽3)—22no—
198C2×H(3,𝔽3)C23×C411yescharacteristic
199C2×H(3,𝔽3)H(5,𝔽2)12yes—
200#272—22no—
201#272—22no—
202(C3⋊D3)⋊C3—24no—
203C24×C3—91no—
204C23×D3—32no—
205C23×D3—61no—
206C23×D3—91no—
207C23×D3—92no—
208C2×C3×D4—92no—
209C22×C3×C4—92no—
210C22×C3×C4—92no—
211C22×C3×C4—92no—
212C2×#67—32no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—98no—
216C2×#67—92no—
217C22×Dic3—32no—
218C22×Dic3—92no—
219C22×Dic3—91no—
220C22×Dic3—92no—
221D3×D4—916no—
222C2×D12—92no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—62no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C3×C4⋊C4—92no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—61no—
234C3×C22⋊C4—92no—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4D3211yescharacteristic
238C3×C42—92no—
239#217—32no—
240#217—61no—
241#217—92no—
242#217—91no—
243#217—91no—
244#217—92no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256C4⋊Dic3—92no—
257Dic3⋊C4—32no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261C4×Dic3—32no—
262C4×Dic3—92no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—21no—
269C22×C32—22no—
270C22×C32—42no—
271C22×C32C2×C4×D314yes—
272C22×C32C23×D312yes—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3C2×C4×D314yes—
276C2×C3×D3—34no—
277C2×C3×D3—34no—
278C2×C3×D3—34no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—64no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—122no—
289C2×C3×D3—128no—
290C2×C3×D3—128no—
291D32—128no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—42no—
295C3⋊Dic3—22no—
296C3⋊Dic3—22no—
297C3⋊Dic3C2×C4×D314yes—
298C3×Dic3—34no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—122no—
304C22×D4—271no—
305C4×D4—92no—
306C4×D4—272no—
307C4×D4—272no—
308C4×D4—272no—
309#117—272no—
310C2×C22⋊C4—91no—
311C2×C22⋊C4—272no—
312C2×C22⋊C4—271no—
313H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
314C23×C3—32no—
315C23×C3—61no—
316C23×C3—92no—
317C23×C3—91no—
318C23×C3—91no—
319C23×C3—94no—
320C23×C3—362no—
321C23×C3C2×D3211yescharacteristic
322C22×D3—32no—
323C22×D3—32no—
324C22×D3—32no—
325C22×D3—38no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—64no—
329C22×D3—92no—
330C22×D3—94no—
331C22×D3—98no—
332C22×D3—122no—
333C22×D3—122no—
334C22×D3—124no—
335C22×D3—362no—
336C22×D3—362no—
337C22×D3—362no—
338C3×D4—98no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—62no—
342C2×C3×C4—92no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—362no—
347C2×C3×C4—362no—
348C2×C3×C4—362no—
349C2×C3×C4C2×D3212yes—
350#67—38no—
351#67—98no—
352#67—98no—
353#67—98no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—62no—
360C2×Dic3—92no—
361C2×Dic3—92no—
362C2×Dic3—92no—
363C2×Dic3—94no—
364C2×Dic3—122no—
365C2×Dic3—362no—
366C2×Dic3—362no—
367D12—98no—
368C4×D3—98no—
369C4×D3—122no—
370C4×D3—122no—
371C4×D3—122no—
372C4×D3—124no—
373C4×D3—362no—
374C4×D3—362no—
375C4×D3—362no—
376C2×C32—24no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—42no—
380C2×C32C22×C4×D312yes—
381C2×C32#100412yes—
382C2×C32#100912yes—
383C3⋊D3—24no—
384C3×D3—64no—
385C3×D3—64no—
386C3×D3—64no—
387C3×D3—64no—
388C3×D3—122no—
389C24—91no—
390C24—271no—
391C2×D4—92no—
392C2×D4—272no—
393C2×D4—278no—
394C22×C4—92no—
395C22×C4—92no—
396C22×C4—92no—
397C22×C4—271no—
398C22×C4—272no—
399C4⋊C4—92no—
400C4⋊C4—272no—
401C22⋊C4—91no—
402C22⋊C4—92no—
403C22⋊C4—92no—
404C22⋊C4—272no—
405C22⋊C4—271no—
406C22⋊C4—272no—
407C22⋊C4—271no—
408C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
409C42—92no—
410C42—272no—
411C22×C3—22no—
412C22×C3—32no—
413C22×C3—34no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—61no—
417C22×C3—62no—
418C22×C3—94no—
419C22×C3—92no—
420C22×C3—92no—
421C22×C3—122no—
422C22×C3—182no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—364no—
427C22×C3C4×D3212yes—
428C22×C3C22×D3211yescharacteristic
429C2×D3—34no—
430C2×D3—34no—
431C2×D3—34no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—64no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—94no—
442C2×D3—122no—
443C2×D3—128no—
444C2×D3—128no—
445C2×D3—182no—
446C2×D3—182no—
447C2×D3—364no—
448C2×D3—368no—
449C3×C4—22no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—94no—
453C3×C4—122no—
454C3×C4—182no—
455C3×C4—182no—
456C3×C4—182no—
457Dic3—34no—
458Dic3—62no—
459Dic3—62no—
460Dic3—62no—
461Dic3—62no—
462Dic3—94no—
463Dic3—122no—
464Dic3—182no—
465Dic3—182no—
466C32—21no—
467C32#554512yescomplemented
468C23—91no—
469C23—92no—
470C23—91no—
471C23—94no—
472C23—274no—
473C23—362no—
474C23—1082no—
475C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476D4—98no—
477D4—278no—
478C2×C4—92no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—274no—
483C2×C4—362no—
484C2×C4—362no—
485C2×C4—362no—
486C2×C4—1082no—
487C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
488C2×C3—22no—
489C2×C3—32no—
490C2×C3—32no—
491C2×C3—32no—
492C2×C3—64no—
493C2×C3—62no—
494C2×C3—61no—
495C2×C3—122no—
496C2×C3—184no—
497C2×C3—182no—
498C2×C3—182no—
499C2×C3C2×C4×D3211yescharacteristic
500C2×C3#6431412yes—
501D3—64no—
502D3—64no—
503D3—64no—
504D3—64no—
505D3—122no—
506D3—184no—
507C22—22no—
508C22—92no—
509C22—94no—
510C22—92no—
511C22—182no—
512C22—182no—
513C22—182no—
514C22—182no—
515C22—364no—
516C22—1084no—
517C22C4×((C3⋊D3)⋊C3)⋊C212yes—
518C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
519C4—22no—
520C4—94no—
521C4—182no—
522C4—182no—
523C4—182no—
524C3—32no—
525C3—61no—
526C3#1059425611yescharacteristic
527C2—22no—
528C2—182no—
529C2—182no—
530C2—184no—
531C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C2#1172250712yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#88677C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×#272C2semidirect
3. #64090⋊C3C2semidirect
4. #11721314C2semidirect
5. C2×#88451C2semidirect
6. #11721160C2semidirect
7. C4×#6200C2semidirect
8. #11721124C2semidirect
9. #11721076C2semidirect
13. C2×C4×#272C22semidirect
15. C22×(C3⋊Dic3)⋊C3C22semidirect
16. C2×C4×(C3⋊D3)⋊C3C22semidirect
17. #88677C4semidirect
20. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
21. C22⋊C4×H(3,𝔽3)C22semidirect
22. #88552C22semidirect
24. #3752⋊C3C22semidirect
25. #3750⋊C3C22semidirect
26. C4×(C3⋊Dic3)⋊C3C22semidirect
28. #88451C22semidirect
29. #88451C22semidirect
30. #88450C22semidirect
31. #88448C22semidirect
32. #88447C22semidirect
34. #64090D3semidirect
51. C22×#272C2×C4semidirect
56. C2×((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
60. #638⋊C3C2×C4semidirect
66. #6200C2×C4semidirect
67. #6199C2×C4semidirect
75. C22×C3⋊Dic3C2×D3semidirect
76. C2×C4×C3⋊D3C2×D3semidirect
90. C32×C22⋊C4C2×D3semidirect
91. #3752C2×D3semidirect
92. #3750C2×D3semidirect
93. C4×C3⋊Dic3C2×D3semidirect
173. #638C4×D3semidirect
313. H(3,𝔽3)#518semidirect
408. C22⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
467. C32#5545semidirect

Maximal subgroups

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

subgroupindex
C2×#886772
C22⋊C4×#2722
#64090⋊C32
#117213142
C2×#884512
#117211602
C4×#62002
#117211242
#117210762
#105942543