Finite Group Databasebeta

← #13140545

Subgroups — #13140545

Order 1728.

757 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 123 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591416—31nomaximal
9D3×#832—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3C414yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4—21no—
18C4⋊#1084C2211yescharacteristic, complemented
19C2×#6180C2211yescharacteristic, complemented
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
22#88448—21no—
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×D4×C3⋊D3D311yescharacteristic
25C2×C3×D3×D4—31no—
26#63900—31no—
27D3×C4⋊Dic3—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63659D311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35#63564—31no—
36#63564—31no—
37#63563—31no—
38#63562—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—42no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3D411yescharacteristic
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)C2×C412yescomplemented
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084—21no—
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
54C4×(C3⋊D3)⋊C3C2×C412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56((C3⋊D3)⋊C3)⋊C4—42no—
57#6180—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C2×#832—91no—
60D3×#102—91no—
61D3×#102—91no—
62C23×C3⋊D3—21no—
63C23×C3×D3—61no—
64C2×C32×D4—21no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D311yescharacteristic
67C2×#638—21no—
68D4×C3⋊D3—22no—
69D4×C3⋊D3Dic314yescomplemented
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×C3×#67—61no—
73C3×D3×D4—34no—
74C3×D3×D4—62no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—31no—
77C2×#624—61no—
78C2×D3×Dic3—61no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—61no—
89#3724—61no—
90Dic3⋊Dic3—61no—
91#3710—62no—
92D3×C3⋊C8—62no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)—41no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—48no—
101C2×(C3⋊D3)⋊C3C22⋊C412yes—
102C2×(C3⋊D3)⋊C3D812yes—
103C2×(C3⋊D3)⋊C3SD1612yes—
104C4×H(3,𝔽3)C22×C411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106#1084—41no—
107(C3⋊Dic3)⋊C3—41no—
108(C3⋊Dic3)⋊C3C22⋊C412yes—
109C22×C3×D4—91no—
110C2×D3×D4—31no—
111C2×D3×D4—91no—
112C2×C4⋊Dic3—91no—
113C22×C3⋊C8—91no—
114C2×D3×C8—91no—
115C2×D3×C8—91no—
116D3×C4⋊C4—91no—
117D3×C4⋊C4—91no—
118C3×#102—91no—
119C3×#102—91no—
120#832—31no—
121#832—31no—
122#832—31no—
123#832—61no—
124#832—91no—
125#832—91no—
126#832—91no—
127#821—91no—
128#821—91no—
129#809—91no—
130#809—91no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—41no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—24no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3#6711yescharacteristic
139C22×C3×D3—31no—
140C22×C3×D3—64no—
141C22×C3×D3—61no—
142C22×C3×D3—122no—
143C22×C3×D3—122no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—22no—
147C32×D4—41no—
148C32×D4C4×D312yescomplemented
149C32×D4C2×Dic312yescomplemented
150C2×C32×C4—21no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153#638—24no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3#6711yescharacteristic
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)C2×Dic312yescomplemented
158C4×C3⋊D3C2×Dic312yes—
159C4×C3⋊D3#6712yes—
160C3×#67—64no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—121no—
166C3×D12—32no—
167C3×D12—61no—
168C3×C4×D3—32no—
169C3×C4×D3—32no—
170#624—122no—
171D3×Dic3—122no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
174C3×C3⋊C8—61no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—121no—
177C2×#102—271noSylow
178C2×H(3,𝔽3)—42no—
179C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
180C2×H(3,𝔽3)C2×D811yescharacteristic
181C2×H(3,𝔽3)C2×SD1611yescharacteristic
182(C3⋊D3)⋊C3—42no—
183(C3⋊D3)⋊C3#10214yescomplemented
184C24×C3—181no—
185C23×D3—61no—
186C23×D3—181no—
187C2×C3×D4—31no—
188C2×C3×D4—31no—
189C2×C3×D4—61no—
190C2×C3×D4—91no—
191C2×C3×D4—94no—
192C2×C3×D4—182no—
193C2×C3×D4—181no—
194C2×C3×D4D3211yescharacteristic
195C22×C3×C4—91no—
196C2×#67—61no—
197C2×#67—181no—
198C22×Dic3—181no—
199D3×D4—34no—
200D3×D4—62no—
201D3×D4—94no—
202D3×D4—182no—
203C2×D12—31no—
204C2×D12—91no—
205C2×C4×D3—31no—
206C2×C4×D3—91no—
207C2×C4×D3—181no—
208C2×C4×D3—181no—
209C2×C3×C8—91no—
210C2×C3×C8—91no—
211C3×C4⋊C4—91no—
212C3×C4⋊C4—91no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—31no—
216C4⋊Dic3—61no—
217C4⋊Dic3—91no—
218C4⋊Dic3—181no—
219Dic3⋊C4—181no—
220Dic3⋊C4—181no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—61no—
225C2×C3⋊C8—91no—
226C2×C3⋊C8—182no—
227D3×C8—182no—
228D3×C8—182no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—21no—
232C22×C32—42no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—81no—
238C22×C32—81no—
239C22×C32D3×D411yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—48no—
245C2×C3⋊D3#21312yes—
246C2×C3⋊D3#21412yes—
247C2×C3⋊D3#21712yes—
248C2×C3×D3—32no—
249C2×C3×D3—32no—
250C2×C3×D3—32no—
251C2×C3×D3—62no—
252C2×C3×D3—128no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4C2×C4×D311yescharacteristic
258C32×C4D3×D411yescharacteristic
259C32×C4C22×Dic311yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C3⋊Dic3—41no—
262C3⋊Dic3#21712yes—
263C3×Dic3—32no—
264C3×Dic3—121no—
265C3×Dic3—121no—
266C3×Dic3—121no—
267C3×Dic3—241no—
268C22×D4—91no—
269C22×C8—271no—
270C2×C4⋊C4—271no—
271#102—91no—
272#102—91no—
273#102—271no—
274#102—271no—
275H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
276C23×C3—21no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—91no—
280C23×C3—121no—
281C23×C3—181no—
282C23×C3—184no—
283C23×C3—362no—
284C23×C3—362no—
285C22×D3—31no—
286C22×D3—64no—
287C22×D3—61no—
288C22×D3—91no—
289C22×D3—122no—
290C22×D3—122no—
291C22×D3—184no—
292C22×D3—181no—
293C22×D3—362no—
294C22×D3—362no—
295C3×D4—21no—
296C3×D4—32no—
297C3×D4—32no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—62no—
301C3×D4—92no—
302C3×D4—121no—
303C3×D4—181no—
304C3×D4—184no—
305C3×D4D3×Dic312yes—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—61no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4C2×D3211yescharacteristic
315#67—64no—
316#67—184no—
317C2×Dic3—31no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—91no—
322C2×Dic3—121no—
323C2×Dic3—181no—
324C2×Dic3—362no—
325D12—32no—
326D12—61no—
327D12—92no—
328D12—181no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—92no—
332C4×D3—92no—
333C4×D3—362no—
334C4×D3—362no—
335C3×C8—181no—
336C3×C8—181no—
337C3⋊C8—61no—
338C3⋊C8—61no—
339C3⋊C8—61no—
340C3⋊C8—121no—
341C3⋊C8—181no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—42no—
346C2×C32—42no—
347C2×C32—82no—
348C2×C32D3×C22⋊C411yescharacteristic
349C2×C32D3×D811yescharacteristic
350C2×C32D3×SD1611yescharacteristic
351C2×C32C2×#21311yescharacteristic
352C2×C32C2×#21411yescharacteristic
353C2×C32C2×#21711yescharacteristic
354C3⋊D3—42no—
355C3⋊D3#83214yescomplemented
356C3×D3—34no—
357C3×D3—122no—
358C24—181no—
359C2×D4—94no—
360C2×D4—91no—
361C2×D4—182no—
362C2×D4—181no—
363C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C22×C4—91no—
365C22×C4—541no—
366C2×C8—91no—
367C2×C8—91no—
368C2×C8—542no—
369C4⋊C4—91no—
370C4⋊C4—91no—
371C4⋊C4—541no—
372C22×C3—22no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—62no—
378C22×C3—62no—
379C22×C3—61no—
380C22×C3—92no—
381C22×C3—92no—
382C22×C3—92no—
383C22×C3—122no—
384C22×C3—121no—
385C22×C3—121no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—182no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—368no—
394C22×C3D3×#6711yescharacteristic
395C2×D3—32no—
396C2×D3—32no—
397C2×D3—32no—
398C2×D3—62no—
399C2×D3—92no—
400C2×D3—92no—
401C2×D3—92no—
402C2×D3—128no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—182no—
406C2×D3—368no—
407C2×D3—361no—
408C2×D3—361no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—92no—
416C3×C4—361no—
417C3×C4—361no—
418C3×C4C2×D3×Dic311yescharacteristic
419C3×C4D3×#6711yescharacteristic
420Dic3—32no—
421Dic3—92no—
422Dic3—121no—
423Dic3—121no—
424Dic3—121no—
425Dic3—241no—
426Dic3—361no—
427C32—21no—
428C32D3×#10211yescharacteristic, complemented
429C32C2×#83211yescharacteristic, complemented
430C23—21no—
431C23—91no—
432C23—184no—
433C23—181no—
434C23—362no—
435C23—362no—
436D4—21no—
437D4—92no—
438D4—184no—
439D4—181no—
440D4((C3⋊D3)⋊C3)⋊C412yescomplemented
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—1082no—
447C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
448C8—181no—
449C8—181no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—42no—
457C2×C3—61no—
458C2×C3—61no—
459C2×C3—61no—
460C2×C3—94no—
461C2×C3—122no—
462C2×C3—122no—
463C2×C3—242no—
464C2×C3—362no—
465C2×C3D3×#21311yescharacteristic
466C2×C3D3×#21411yescharacteristic
467C2×C3D3×#21711yescharacteristic
468D3—34no—
469D3—94no—
470D3—122no—
471D3—362no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—92no—
476C22—92no—
477C22—92no—
478C22—182no—
479C22—368no—
480C22—361no—
481C22—361no—
482C22#8867611yescharacteristic
483C4—92no—
484C4—361no—
485C4—361no—
486C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
487C4#8867611yescharacteristic
488C3—31no—
489C3—31no—
490C3—61no—
491C3D3×#83211yescharacteristic
492C2—42no—
493C2—94no—
494C2—362no—
495C2#1172119411yescharacteristic
496C2#1172119811yescharacteristic
497C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#102
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721129C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720578C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720308C2semidirect
14. D4×(C3⋊D3)⋊C3C4semidirect
18. C4⋊#1084C22semidirect
19. C2×#6180C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
30. #63659D3semidirect
46. D4×H(3,𝔽3)C2×C4semidirect
53. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
69. D4×C3⋊D3Dic3semidirect
79. C4⋊(C3⋊Dic3)C2×D3semidirect
80. C2×C3⋊(C3⋊C8)C2×D3semidirect
148. C32×D4C4×D3semidirect
149. C32×D4C2×Dic3semidirect
157. C4⋊(C3⋊D3)C2×Dic3semidirect
183. (C3⋊D3)⋊C3#102semidirect
275. H(3,𝔽3)C2×#102semidirect
355. C3⋊D3#832semidirect
428. C32D3×#102semidirect
429. C32C2×#832semidirect
440. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
#117211292
C2×((C3⋊D3)⋊C3)⋊C82
#117205782
#63659⋊C32
#117203182
#117203082
#105914163
D3×#8323