Finite Group Databasebeta

← #13138053

Subgroups — #13138053

Order 1728.

757 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 111 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×#88654C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720531C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590593—31nomaximal
9D3×#809—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6200—21no—
16#88654—22no—
17#88654C414yescomplemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
19C2×#6180C2211yescharacteristic, complemented
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21#3750⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×#3802—31no—
25C2×D3×D12—31no—
26C4⋊C4×C3⋊D3D311yescharacteristic, complemented
27C3×D3×C4⋊C4—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63636D311yescharacteristic, complemented
31C3×#809—31no—
32C3×#809—31no—
33C3×#809—31no—
34C3×#809—61no—
35#63564—31no—
36#63563—31no—
37#63563—31no—
38#63562—31no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42C2×((C3⋊D3)⋊C3)⋊C2—24no—
43C2×((C3⋊D3)⋊C3)⋊C2—42no—
44C2×((C3⋊D3)⋊C3)⋊C2—42no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C4⋊#272—21no—
48C4⋊#272C2×C412yescomplemented
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
53C4×(C3⋊D3)⋊C3—42no—
54C4×(C3⋊D3)⋊C3C2×C412yes—
55C4×(C3⋊D3)⋊C3D412yes—
56#6200—24no—
57#6180—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C2×#809—91no—
60D3×#102—91no—
61D3×#102—91no—
62C22×D32—61no—
63C22×D32—61no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—21no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×#626—61no—
74C2×#625—61no—
75#3802—34no—
76#3802—62no—
77D3×D12—34no—
78D3×D12—62no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2×D311yescharacteristic, complemented
81C32×C4⋊C4C2×D311yescharacteristic, complemented
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
83#3750—21no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
85C3×C4⋊Dic3—31no—
86C3×Dic3⋊C4—61no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—61no—
91#3710—62no—
92D3×C3⋊C8—62no—
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C2×#272—22no—
95C2×#272—41no—
96C2×#272—41no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3C22⋊C412yes—
101C2×(C3⋊D3)⋊C3D812yes—
102C2×(C3⋊D3)⋊C3SD1612yes—
103((C3⋊D3)⋊C3)⋊C2—48no—
104C4×H(3,𝔽3)—41no—
105C4×H(3,𝔽3)C22×C411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107(C3⋊Dic3)⋊C3—41no—
108(C3⋊Dic3)⋊C3C22⋊C412yes—
109C2×D3×D4—91no—
110C2×D3×D4—91no—
111C22×D12—91no—
112C2×C3×C4⋊C4—91no—
113C22×C3⋊C8—91no—
114C2×D3×C8—91no—
115C2×D3×C8—91no—
116D3×C4⋊C4—31no—
117D3×C4⋊C4—91no—
118C3×#102—91no—
119C3×#102—91no—
120#832—91no—
121#832—91no—
122#821—91no—
123#821—91no—
124#809—31no—
125#809—31no—
126#809—31no—
127#809—61no—
128#809—91no—
129#809—91no—
130#809—91no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—121no—
138C2×D32—64no—
139C2×D32—64no—
140C2×D32—122no—
141C2×D32—122no—
142C2×D32—122no—
143C2×D32—122no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3#6711yescharacteristic
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)C4×D312yescomplemented
154C4×C3⋊D3—42no—
155C4×C3⋊D3C4×D312yes—
156C4×C3⋊D3D1212yes—
157C2×C3×Dic3—31no—
158C2×C3×Dic3—61no—
159C3×D12—32no—
160C3×D12—32no—
161C3×D12—32no—
162C3×D12—62no—
163C3×D12—61no—
164C3×D12—61no—
165C3×D12—61no—
166C3×D12—121no—
167C3×C4×D3—32no—
168C3×C4×D3—32no—
169C3×C4×D3—122no—
170#626—64no—
171#625—64no—
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)C2×C22⋊C411yescharacteristic
179C2×H(3,𝔽3)C2×D811yescharacteristic
180C2×H(3,𝔽3)C2×SD1611yescharacteristic
181#272—42no—
182(C3⋊D3)⋊C3—42no—
183(C3⋊D3)⋊C3#10214yescomplemented
184C23×D3—181no—
185C23×D3—181no—
186C23×D3—181no—
187C2×C3×D4—91no—
188C2×C3×D4—91no—
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191C2×#67—181no—
192C2×#67—181no—
193D3×D4—94no—
194D3×D4—94no—
195D3×D4—182no—
196D3×D4—182no—
197C2×D12—31no—
198C2×D12—31no—
199C2×D12—31no—
200C2×D12—61no—
201C2×D12—91no—
202C2×D12—94no—
203C2×D12—182no—
204C2×D12—181no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C4×D3—91no—
208C2×C4×D3—181no—
209C2×C3×C8—91no—
210C2×C3×C8—91no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—181no—
216C3×C4⋊C4D3211yescharacteristic
217C4⋊Dic3—31no—
218C4⋊Dic3—91no—
219Dic3⋊C4—61no—
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—21no—
230C22×C32D3×D411yescharacteristic
231C22×C32C2×#6711yescharacteristic
232C2×C3⋊D3—22no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3(C2×D3)⋊C412yes—
236C2×C3⋊D3#21312yes—
237C2×C3⋊D3#21512yes—
238C2×C3×D3—32no—
239C2×C3×D3—32no—
240C2×C3×D3—32no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—122no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—121no—
250C2×C3×D3—121no—
251C2×C3×D3—241no—
252C2×C3×D3—241no—
253D32—128no—
254D32—128no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—81no—
260C32×C4C2×C4×D311yescharacteristic
261C32×C4C2×C4×D311yescharacteristic
262C32×C4C2×D1211yescharacteristic
263C32×C4D3×D411yescharacteristic
264C3⋊Dic3—41no—
265C3⋊Dic3(C2×D3)⋊C412yes—
266C3×Dic3—32no—
267C3×Dic3—121no—
268C22×D4—271no—
269C22×C8—271no—
270C2×C4⋊C4—91no—
271#102—91no—
272#102—91no—
273#102—271no—
274#102—271no—
275H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
276C23×C3—91no—
277C23×C3—181no—
278C23×C3—181no—
279C22×D3—31no—
280C22×D3—61no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—91no—
284C22×D3—121no—
285C22×D3—181no—
286C22×D3—184no—
287C22×D3—184no—
288C22×D3—184no—
289C22×D3—362no—
290C22×D3—362no—
291C22×D3—362no—
292C22×D3—362no—
293C22×D3—362no—
294C22×D3—362no—
295C3×D4—92no—
296C3×D4—92no—
297C3×D4—181no—
298C3×D4—181no—
299C2×C3×C4—21no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—91no—
308C2×C3×C4—121no—
309C2×C3×C4—181no—
310C2×C3×C4—362no—
311C2×C3×C4C2×D3211yescharacteristic
312#67—184no—
313#67—184no—
314C2×Dic3—31no—
315C2×Dic3—61no—
316C2×Dic3—91no—
317C2×Dic3—181no—
318D12—32no—
319D12—32no—
320D12—32no—
321D12—62no—
322D12—61no—
323D12—61no—
324D12—61no—
325D12—92no—
326D12—121no—
327D12—181no—
328D12—184no—
329C4×D3—32no—
330C4×D3—32no—
331C4×D3—92no—
332C4×D3—92no—
333C4×D3—122no—
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×C32D3×C22⋊C411yescharacteristic
346C2×C32D3×D811yescharacteristic
347C2×C32D3×SD1611yescharacteristic
348C2×C32C2×(C2×D3)⋊C411yescharacteristic
349C2×C32C2×#21311yescharacteristic
350C2×C32C2×#21511yescharacteristic
351C3⋊D3—42no—
352C3⋊D3#80914yescomplemented
353C3×D3—34no—
354C3×D3—122no—
355C3×D3—122no—
356C3×D3—122no—
357C3×D3—242no—
358C24—541no—
359C2×D4—91no—
360C2×D4—91no—
361C2×D4—274no—
362C2×D4—542no—
363C2×D4—541no—
364C22×C4—91no—
365C22×C4—181no—
366C2×C8—91no—
367C2×C8—91no—
368C2×C8—542no—
369C4⋊C4—91no—
370C4⋊C4—181no—
371C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—61no—
375C22×C3—92no—
376C22×C3—92no—
377C22×C3—92no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3—361no—
383C22×C3—361no—
384C22×C3D3×#6711yescharacteristic
385C2×D3—32no—
386C2×D3—32no—
387C2×D3—32no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—92no—
392C2×D3—92no—
393C2×D3—92no—
394C2×D3—122no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—121no—
399C2×D3—121no—
400C2×D3—121no—
401C2×D3—182no—
402C2×D3—241no—
403C2×D3—241no—
404C2×D3—361no—
405C2×D3—361no—
406C2×D3—368no—
407C2×D3—368no—
408C2×D3—368no—
409C3×C4—31no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—41no—
414C3×C4—61no—
415C3×C4—61no—
416C3×C4—92no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4C4×D3211yescharacteristic
422C3×C4D3×D1211yescharacteristic
423Dic3—32no—
424Dic3—92no—
425Dic3—121no—
426Dic3—361no—
427C32—21no—
428C32D3×#10211yescharacteristic, complemented
429C32C2×#80911yescharacteristic, complemented
430C23—91no—
431C23—181no—
432C23—181no—
433C23—544no—
434C23—1082no—
435C23—1082no—
436D4—92no—
437D4—92no—
438D4—181no—
439D4—181no—
440D4—544no—
441C2×C4—21no—
442C2×C4—91no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—181no—
446C2×C4—362no—
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—61no—
457C2×C3—61no—
458C2×C3—61no—
459C2×C3—94no—
460C2×C3—362no—
461C2×C3—362no—
462C2×C3D3×(C2×D3)⋊C411yescharacteristic
463C2×C3D3×#21311yescharacteristic
464C2×C3D3×#21511yescharacteristic
465D3—34no—
466D3—94no—
467D3—122no—
468D3—122no—
469D3—122no—
470D3—242no—
471D3—362no—
472C22—92no—
473C22—92no—
474C22—92no—
475C22—182no—
476C22—182no—
477C22—361no—
478C22—361no—
479C22—361no—
480C22—361no—
481C22—1088no—
482C22#8867611yescharacteristic
483C4—41no—
484C4—92no—
485C4—361no—
486C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C4#8865411yescharacteristic
488C3—31no—
489C3—31no—
490C3—61no—
491C3D3×#80911yescharacteristic
492C2—94no—
493C2—362no—
494C2—362no—
495C2#1172116011yescharacteristic
496C2#1172119411yescharacteristic
497C2#1172123211yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88654C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720531C2semidirect
5. #63636⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720318C2semidirect
17. #88654C4semidirect
18. C4⋊C4×H(3,𝔽3)C22semidirect
19. C2×#6180C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. C4⋊C4×C3⋊D3D3semidirect
30. #63636D3semidirect
48. C4⋊#272C2×C4semidirect
52. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
80. C32×C4⋊C4C2×D3semidirect
81. C32×C4⋊C4C2×D3semidirect
82. C4⋊(C3⋊Dic3)C2×D3semidirect
84. C2×C3⋊(C3⋊C8)C2×D3semidirect
153. C4⋊(C3⋊D3)C4×D3semidirect
183. (C3⋊D3)⋊C3#102semidirect
275. H(3,𝔽3)C2×#102semidirect
352. C3⋊D3#809semidirect
371. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
428. C32D3×#102semidirect
429. C32C2×#809semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C4⋊C4×(C3⋊D3)⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
#117205312
#63636⋊C32
#117203242
#117203182
#105905933
D3×#8093