Finite Group Databasebeta

← #10592521⋊C3 #13141982

Subgroups — #10592521⋊C3 #13141982

Order 1728.

632 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#11722638C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C211yesmaximal, characteristic, complemented
4#11721385C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63632⋊C3C211yesmaximal, characteristic, complemented
8#10592521C311yesmaximal, characteristic, complemented
9C3×#4822—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722—21no—
12D4×(C3⋊D3)⋊C3—22no—
13D4×(C3⋊D3)⋊C3—21no—
14#3828⋊C3—21no—
15#3828⋊C3C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)—21no—
17C2×C4⋊((C3⋊D3)⋊C3)—22no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
20M16×H(3,𝔽3)C2211yescharacteristic, complemented
21C42×H(3,𝔽3)C2211yescharacteristic, complemented
22#3755⋊C3—21no—
23#3753⋊C3—21no—
24#3748⋊C3C2211yescharacteristic, complemented
25#3745⋊C3—21no—
26#3744⋊C3—21no—
27#64366C2×C311yescharacteristic, complemented
28C3×#1009—31no—
29#64158C2×C311yescharacteristic, complemented
30#64117C2×C311yescharacteristic, complemented
31#64083C2×C311yescharacteristic, complemented
32C3×#949—31no—
33C3×#908—31no—
34C3×#874—31no—
35C32×#104—21no—
36C32×#104D311yescharacteristic, complemented
37C32×#104C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#823—31no—
41C3×#805—31no—
42C22×(C3⋊D3)⋊C3—22no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3D412yes—
45Q8×H(3,𝔽3)D411yescharacteristic, complemented
46D4×H(3,𝔽3)—21no—
47D4×H(3,𝔽3)D411yescharacteristic, complemented
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51#638⋊C3—22no—
52#638⋊C3—21no—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)—21no—
57C4⋊((C3⋊D3)⋊C3)—21no—
58C4⋊((C3⋊D3)⋊C3)—44no—
59C4⋊((C3⋊D3)⋊C3)D411yescharacteristic, complemented
60C4×(C3⋊D3)⋊C3—22no—
61C4×(C3⋊D3)⋊C3—21no—
62#634⋊C3D411yescharacteristic, complemented
63C8×H(3,𝔽3)—21no—
64(C3⋊(C3⋊C8))⋊C3—21no—
65C3×#453—91no—
66#4822—31no—
67#4822—91no—
68C32×#41—21no—
69C32×#41C2×D311yescharacteristic
70C32×#41C22×C311yescharacteristic
71#3832—21no—
72D4×C3⋊D3—22no—
73D4×C3⋊D3—21no—
74#3828—21no—
75#3828C22×C311yescharacteristic
76C2×C4⋊(C3⋊D3)—21no—
77C2×C4⋊(C3⋊D3)—22no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
80C3×#239—61no—
81C3×D3×D4—62no—
82C3×D3×D4—61no—
83C3×#235—31no—
84C3×#235—61no—
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×D12—62no—
89C32×M16—21no—
90C32×M16C2×D311yescharacteristic, complemented
91C32×M16C22×C311yescharacteristic, complemented
92C32×C42—21no—
93C32×C42C2×D311yescharacteristic, complemented
94C32×C42C22×C311yescharacteristic, complemented
95#3755—21no—
96#3753—21no—
97#3748C22×C311yescharacteristic, complemented
98#3745—21no—
99#3744—21no—
100C3×#215—61no—
101C3×#213—61no—
102C3×#208—31no—
103C3×D24—61no—
104C3×#204—61no—
105C22×H(3,𝔽3)—21no—
106C22×H(3,𝔽3)C2×D411yescharacteristic
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—21no—
109C2×(C3⋊D3)⋊C3—42no—
110C2×(C3⋊D3)⋊C3—42no—
111C2×(C3⋊D3)⋊C3—41no—
112C2×(C3⋊D3)⋊C3—82no—
113C4×H(3,𝔽3)—22no—
114C4×H(3,𝔽3)—21no—
115C4×H(3,𝔽3)C2×D411yescharacteristic
116C4×H(3,𝔽3)C2×D411yescharacteristic
117(C3⋊Dic3)⋊C3—21no—
118C3×H(5,𝔽2)—91no—
119#1009—31no—
120#1009—91no—
121C3×AGL(1,ℤ/8)—91no—
122C3×AGL(1,ℤ/8)—91no—
123C3×#127—91no—
124#949—31no—
125#949—91no—
126#908—31no—
127#908—91no—
128#874—31no—
129#874—91no—
130C3×#104—31no—
131C3×#104—31no—
132C3×#104—61no—
133C3×#104—91no—
134C3×#104C3×D311yescharacteristic
135C3×#100—91no—
136#823—31no—
137#823—91no—
138#805—31no—
139#805—91no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3—21no—
142C22×C3⋊D3C3×D412yes—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C32×Q8—21no—
147C32×Q8D1211yescharacteristic, complemented
148C32×Q8C3×D411yescharacteristic, complemented
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—21no—
152C32×D4—41no—
153C32×D4D1211yescharacteristic, complemented
154C32×D4C3×D411yescharacteristic, complemented
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C23×C311yescharacteristic
164#638—22no—
165#638—21no—
166C4⋊(C3⋊D3)—22no—
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—44no—
172C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
173C4×C3⋊D3—22no—
174C4×C3⋊D3—21no—
175#634C3×D411yescharacteristic, complemented
176C3×#67—62no—
177C3×#67—61no—
178C3×D12—31no—
179C3×D12—62no—
180C3×D12—61no—
181C3×D12—61no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—124no—
185C3×C4×D3—62no—
186C3×C4×D3—61no—
187C3×Dic6—31no—
188C32×C8—21no—
189C32×C8—21no—
190C32×C8—41no—
191C3⋊(C3⋊C8)—21no—
192C3×C3⋊C8—61no—
193#453—91noSylow
194C2×H(3,𝔽3)—21no—
195C2×H(3,𝔽3)—41no—
196C2×H(3,𝔽3)#12011yescharacteristic
197(C3⋊D3)⋊C3—42no—
198(C3⋊D3)⋊C3—41no—
199(C3⋊D3)⋊C3—81no—
200C3×#41—31no—
201C3×#41—31no—
202C3×#41—61no—
203C3×#41—91no—
204C3×#41—181no—
205C3×#41—181no—
206C3×#41C2×C3×D311yescharacteristic
207C2×C3×D4—91no—
208C2×C3×D4—91no—
209C2×C3×D4—181no—
210C2×C3×D4—182no—
211C2×C3×D4—181no—
212C2×C3×D4—182no—
213#239—61no—
214#239—181no—
215D3×D4—62no—
216D3×D4—61no—
217D3×D4—181no—
218D3×D4—182no—
219#235—31no—
220#235—61no—
221#235—91no—
222#235—181no—
223C2×D12—31no—
224C2×D12—31no—
225C2×D12—61no—
226C2×D12—62no—
227C2×D12—91no—
228C2×D12—91no—
229C2×D12—181no—
230C2×D12—182no—
231C3×SD16—181no—
232C3×SD16—181no—
233C3×D8—181no—
234C3×D8—181no—
235C3×M16—31no—
236C3×M16—31no—
237C3×M16—61no—
238C3×M16—91no—
239C3×M16C2×C3×D311yescharacteristic
240C3×C42—31no—
241C3×C42—31no—
242C3×C42—61no—
243C3×C42C2×C3×D311yescharacteristic
244#215—61no—
245#215—181no—
246#213—61no—
247#213—181no—
248#208—31no—
249#208—91no—
250D24—61no—
251D24—181no—
252#204—61no—
253#204—181no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32D3×D411yescharacteristic
259C22×C32C2×C3×D411yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—21no—
262C2×C3⋊D3—42no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3—82no—
266C2×C3×D3—61no—
267C2×C3×D3—62no—
268C2×C3×D3—122no—
269C2×C3×D3—122no—
270C2×C3×D3—121no—
271C2×C3×D3—242no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—41no—
279C32×C4—42no—
280C32×C4C2×D1211yescharacteristic
281C32×C4D3×D411yescharacteristic
282C32×C4C2×C3×D411yescharacteristic
283C32×C4C2×C3×D411yescharacteristic
284C3⋊Dic3—21no—
285C3×Dic3—61no—
286H(5,𝔽2)—91no—
287AGL(1,ℤ/8)—91no—
288AGL(1,ℤ/8)—91no—
289#127—91no—
290#104—91no—
291#104(C3⋊D3)⋊C311yescharacteristic, complemented
292#100—91no—
293H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
294C23×C3—92no—
295C23×C3—181no—
296C23×C3—182no—
297C22×D3—32no—
298C22×D3—62no—
299C22×D3—61no—
300C22×D3—92no—
301C22×D3—182no—
302C22×D3—181no—
303C3×Q8—31no—
304C3×Q8—31no—
305C3×Q8—61no—
306C3×Q8—91no—
307C3×Q8C3×D1211yescharacteristic
308C3×D4—21no—
309C3×D4—31no—
310C3×D4—31no—
311C3×D4—61no—
312C3×D4—61no—
313C3×D4—61no—
314C3×D4—91no—
315C3×D4—121no—
316C3×D4—181no—
317C3×D4—181no—
318C3×D4—181no—
319C3×D4—181no—
320C3×D4—182no—
321C3×D4—181no—
322C3×D4—182no—
323C3×D4—364no—
324C3×D4C3×D1211yescharacteristic
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—121no—
335C2×C3×C4—121no—
336C2×C3×C4—181no—
337C2×C3×C4—182no—
338C2×C3×C4C22×C3×D311yescharacteristic
339#67—62no—
340#67—61no—
341#67—181no—
342#67—182no—
343D12—31no—
344D12—62no—
345D12—61no—
346D12—61no—
347D12—61no—
348D12—61no—
349D12—91no—
350D12—124no—
351D12—181no—
352D12—181no—
353D12—181no—
354D12—181no—
355D12—182no—
356D12—364no—
357C4×D3—62no—
358C4×D3—61no—
359C4×D3—181no—
360C4×D3—182no—
361Dic6—31no—
362Dic6—91no—
363C3×C8—21no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—121no—
367C3×C8—181no—
368C3⋊C8—61no—
369C3⋊C8—181no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—81no—
377C2×C32#88211yescharacteristic
378C2×C32C3×#12011yescharacteristic
379C3⋊D3—42no—
380C3⋊D3—41no—
381C3⋊D3—81no—
382C3×D3—122no—
383C3×D3—121no—
384C3×D3—241no—
385#41—91no—
386#41—181no—
387#41—181no—
388#41C2×(C3⋊D3)⋊C311yescharacteristic
389C2×D4—91no—
390C2×D4—91no—
391C2×D4—181no—
392C2×D4—182no—
393C2×D4—182no—
394C2×D4—181no—
395SD16—181no—
396SD16—181no—
397D8—181no—
398D8—181no—
399M16—91no—
400M16C2×(C3⋊D3)⋊C311yescharacteristic, complemented
401C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
402C22×C3—21no—
403C22×C3—31no—
404C22×C3—31no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—121no—
409C22×C3—181no—
410C22×C3—182no—
411C22×C3—361no—
412C22×C3—362no—
413C22×C3—362no—
414C22×C3—722no—
415C22×C3C3×D3×D411yescharacteristic
416C2×D3—61no—
417C2×D3—62no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—121no—
421C2×D3—181no—
422C2×D3—182no—
423C2×D3—242no—
424C2×D3—362no—
425C2×D3—361no—
426C2×D3—362no—
427C2×D3—722no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—31no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—62no—
437C3×C4—62no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4—121no—
441C3×C4—122no—
442C3×C4—181no—
443C3×C4C2×C3×D1211yescharacteristic
444C3×C4C3×D3×D411yescharacteristic
445Dic3—61no—
446Dic3—181no—
447C32—21no—
448C32#482211yescharacteristic, complemented
449C32C3×#45311yescharacteristic, complemented
450C23—92no—
451C23—182no—
452C23—181no—
453Q8—91no—
454Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
455D4—21no—
456D4—91no—
457D4—182no—
458D4—182no—
459D4—181no—
460D4—181no—
461D4—181no—
462D4—181no—
463D4—181no—
464D4—364no—
465D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—182no—
469C2×C4—181no—
470C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
471C8—21no—
472C8—181no—
473C2×C3—21no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—41no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—121no—
481C2×C3—121no—
482C2×C3—121no—
483C2×C3—241no—
484C2×C3—362no—
485C2×C3—361no—
486C2×C3—721no—
487C2×C3C3×#88211yescharacteristic
488D3—122no—
489D3—121no—
490D3—241no—
491D3—362no—
492D3—361no—
493D3—721no—
494C22—21no—
495C22—181no—
496C22—182no—
497C22—362no—
498C22—362no—
499C22—361no—
500C22—722no—
501C22D4×(C3⋊D3)⋊C311yescharacteristic
502C4—22no—
503C4—21no—
504C4—181no—
505C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
506C4D4×(C3⋊D3)⋊C311yescharacteristic
507C3—31no—
508C3—31no—
509C3—61no—
510C3C3×#482211yescharacteristic
511C2—21no—
512C2—41no—
513C2—362no—
514C2—361no—
515C2—721no—
516C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C3×#882
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #11722638C2semidirect
2. #64158⋊C3C2semidirect
3. #64117⋊C3C2semidirect
4. #11721385C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #63650⋊C3C2semidirect
7. #63632⋊C3C2semidirect
8. #10592521C3semidirect
19. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. M16×H(3,𝔽3)C22semidirect
21. C42×H(3,𝔽3)C22semidirect
24. #3748⋊C3C22semidirect
27. #64366C2×C3semidirect
29. #64158C2×C3semidirect
30. #64117C2×C3semidirect
31. #64083C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63650C2×C3semidirect
39. #63632C2×C3semidirect
45. Q8×H(3,𝔽3)D4semidirect
47. D4×H(3,𝔽3)D4semidirect
59. C4⋊((C3⋊D3)⋊C3)D4semidirect
62. #634⋊C3D4semidirect
79. C2×C4⋊(C3⋊D3)C22×C3semidirect
90. C32×M16C2×D3semidirect
91. C32×M16C22×C3semidirect
93. C32×C42C2×D3semidirect
94. C32×C42C22×C3semidirect
97. #3748C22×C3semidirect
147. C32×Q8D12semidirect
148. C32×Q8C3×D4semidirect
153. C32×D4D12semidirect
154. C32×D4C3×D4semidirect
172. C4⋊(C3⋊D3)C3×D4semidirect
175. #634C3×D4semidirect
291. #104(C3⋊D3)⋊C3semidirect
293. H(3,𝔽3)#453semidirect
400. M16C2×(C3⋊D3)⋊C3semidirect
401. C42C2×(C3⋊D3)⋊C3semidirect
448. C32#4822semidirect
449. C32C3×#453semidirect
454. Q8C4⋊((C3⋊D3)⋊C3)semidirect
465. D4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
#117226382
#64158⋊C32
#64117⋊C32
#117213852
H(3,𝔽3)×#1042
#63650⋊C32
#63632⋊C32
#105925213
C3×#48223