Finite Group Databasebeta

← #10592524⋊C3 #13141985

Subgroups — #10592524⋊C3 #13141985

Order 1728.

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

normal subgroups only

This group has 576 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
2#64160⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63632⋊C3C211yesmaximal, characteristic, complemented
8#10592524C311yesmaximal, characteristic, complemented
9C3×#4825—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)C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic, complemented
19M16×H(3,𝔽3)C2211yescharacteristic, complemented
20C42×H(3,𝔽3)C2211yescharacteristic, complemented
21#3756⋊C3—21no—
22#3754⋊C3—21no—
23#3752⋊C3—22no—
24#3748⋊C3C2211yescharacteristic, complemented
25#3744⋊C3—21no—
26#3746⋊C3—21no—
27#64366C2×C311yescharacteristic
28C3×#1009—31no—
29#64160C2×C311yescharacteristic, complemented
30#64118C2×C311yescharacteristic, complemented
31#64084C2×C311yescharacteristic, complemented
32C3×#951—31no—
33C3×#909—31no—
34C3×#875—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)⋊C3D412yes—
44Q8×H(3,𝔽3)D411yescharacteristic
45D4×H(3,𝔽3)—21no—
46D4×H(3,𝔽3)D411yescharacteristic
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#638⋊C3—22no—
51#638⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
57C4×(C3⋊D3)⋊C3—22no—
58C4×(C3⋊D3)⋊C3—21no—
59#634⋊C3—21no—
60#634⋊C3—21no—
61#634⋊C3D411yescharacteristic
62C8×H(3,𝔽3)—21no—
63(C3⋊(C3⋊C8))⋊C3—21no—
64C3×#455—91no—
65#4825—31no—
66#4825—91no—
67C32×#41—21no—
68C32×#41C2×D311yescharacteristic
69C32×#41C22×C311yescharacteristic
70#3832—21no—
71D4×C3⋊D3—22no—
72D4×C3⋊D3—21no—
73#3828—21no—
74#3828C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)—21no—
76C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
77C2×#634C22×C311yescharacteristic, complemented
78C3×#239—61no—
79C3×D3×D4—62no—
80C3×D3×D4—61no—
81C3×#235—31no—
82C3×#235—61no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×Dic6—31no—
86C32×M16—21no—
87C32×M16C2×D311yescharacteristic, complemented
88C32×M16C22×C311yescharacteristic, complemented
89C32×C42—21no—
90C32×C42C2×D311yescharacteristic, complemented
91C32×C42C22×C311yescharacteristic, complemented
92#3756—21no—
93#3754—21no—
94#3752—22no—
95#3748C22×C311yescharacteristic, complemented
96#3746—21no—
97#3744—21no—
98C3×#216—61no—
99C3×#214—61no—
100C3×(C2×D3)⋊C4—62no—
101C3×#208—31no—
102C3×Dic12—61no—
103C3×#204—61no—
104C22×H(3,𝔽3)—21no—
105C22×H(3,𝔽3)C2×D411yescharacteristic
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—21no—
108C2×(C3⋊D3)⋊C3—42no—
109C2×(C3⋊D3)⋊C3—82no—
110C4×H(3,𝔽3)—21no—
111C4×H(3,𝔽3)—41no—
112C4×H(3,𝔽3)C2×D411yescharacteristic
113C4×H(3,𝔽3)C2×D411yescharacteristic
114(C3⋊Dic3)⋊C3—21no—
115(C3⋊Dic3)⋊C3—41no—
116C3×H(5,𝔽2)—91no—
117#1009—31no—
118#1009—91no—
119C3×#137—91no—
120C3×#137—91no—
121C3×#124—91no—
122#951—31no—
123#951—91no—
124#909—31no—
125#909—91no—
126#875—31no—
127#875—91no—
128C3×#104—31no—
129C3×#104—31no—
130C3×#104—61no—
131C3×#104—91no—
132C3×#104C3×D311yescharacteristic
133C3×#100—91no—
134#823—31no—
135#823—91no—
136#805—31no—
137#805—91no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3C3×D412yes—
140C22×C3×D3—32no—
141C22×C3×D3—62no—
142C32×Q8—21no—
143C32×Q8D1211yescharacteristic
144C32×Q8C3×D411yescharacteristic
145C32×D4—21no—
146C32×D4—21no—
147C32×D4—21no—
148C32×D4—41no—
149C32×D4D1211yescharacteristic
150C32×D4C3×D411yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160#638—22no—
161#638—21no—
162C2×C3⋊Dic3—21no—
163C4⋊(C3⋊D3)—22no—
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)C3×D411yescharacteristic
167C4×C3⋊D3—22no—
168C4×C3⋊D3—21no—
169#634—21no—
170#634—21no—
171#634C3×D411yescharacteristic
172C3×#67—62no—
173C3×#67—61no—
174C2×C3×Dic3—61no—
175C3×D12—31no—
176C3×D12—62no—
177C3×D12—61no—
178C3×D12—61no—
179C3×C4×D3—62no—
180C3×C4×D3—61no—
181C3×Dic6—31no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C32×C8—21no—
185C32×C8—21no—
186C32×C8—41no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—61no—
189#455—91noSylow
190C2×H(3,𝔽3)—21no—
191C2×H(3,𝔽3)—41no—
192C2×H(3,𝔽3)#12011yescharacteristic
193(C3⋊D3)⋊C3—42no—
194(C3⋊D3)⋊C3—41no—
195C3×#41—31no—
196C3×#41—31no—
197C3×#41—61no—
198C3×#41—91no—
199C3×#41—181no—
200C3×#41—181no—
201C3×#41C2×C3×D311yescharacteristic
202C2×C3×Q8—91no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4—181no—
206C2×C3×D4—182no—
207#239—61no—
208#239—181no—
209D3×D4—62no—
210D3×D4—61no—
211D3×D4—181no—
212D3×D4—182no—
213#235—31no—
214#235—61no—
215#235—91no—
216#235—181no—
217C2×D12—31no—
218C2×D12—61no—
219C2×D12—91no—
220C2×D12—181no—
221C2×Dic6—31no—
222C2×Dic6—91no—
223C3×Q16—181no—
224C3×Q16—181no—
225C3×SD16—181no—
226C3×SD16—181no—
227C3×M16—31no—
228C3×M16—31no—
229C3×M16—61no—
230C3×M16—91no—
231C3×M16C2×C3×D311yescharacteristic
232C3×C22⋊C4—182no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—61no—
236C3×C42C2×C3×D311yescharacteristic
237#216—61no—
238#216—181no—
239#214—61no—
240#214—181no—
241(C2×D3)⋊C4—62no—
242(C2×D3)⋊C4—182no—
243#208—31no—
244#208—91no—
245Dic12—61no—
246Dic12—181no—
247#204—61no—
248#204—181no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32D3×D411yescharacteristic
254C22×C32C2×C3×D411yescharacteristic
255C2×C3⋊D3—22no—
256C2×C3⋊D3—21no—
257C2×C3⋊D3—42no—
258C2×C3⋊D3—82no—
259C2×C3×D3—61no—
260C2×C3×D3—62no—
261C2×C3×D3—122no—
262C2×C3×D3—242no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—42no—
271C32×C4C2×D1211yescharacteristic
272C32×C4D3×D411yescharacteristic
273C32×C4C2×C3×D411yescharacteristic
274C32×C4C2×C3×D411yescharacteristic
275C3⋊Dic3—21no—
276C3⋊Dic3—41no—
277C3×Dic3—61no—
278C3×Dic3—121no—
279H(5,𝔽2)—91no—
280#137—91no—
281#137—91no—
282#124—91no—
283#104—91no—
284#104(C3⋊D3)⋊C311yescharacteristic, complemented
285#100—91no—
286H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
287C23×C3—92no—
288C23×C3—182no—
289C22×D3—32no—
290C22×D3—62no—
291C22×D3—92no—
292C22×D3—182no—
293C3×Q8—31no—
294C3×Q8—31no—
295C3×Q8—61no—
296C3×Q8—91no—
297C3×Q8—181no—
298C3×Q8—181no—
299C3×Q8C3×D1211yescharacteristic
300C3×D4—21no—
301C3×D4—31no—
302C3×D4—31no—
303C3×D4—61no—
304C3×D4—61no—
305C3×D4—61no—
306C3×D4—91no—
307C3×D4—121no—
308C3×D4—181no—
309C3×D4—181no—
310C3×D4—182no—
311C3×D4—181no—
312C3×D4—182no—
313C3×D4C3×D1211yescharacteristic
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—182no—
327C2×C3×C4—181no—
328C2×C3×C4C22×C3×D311yescharacteristic
329#67—62no—
330#67—61no—
331#67—181no—
332#67—182no—
333C2×Dic3—61no—
334C2×Dic3—181no—
335D12—31no—
336D12—62no—
337D12—61no—
338D12—61no—
339D12—91no—
340D12—181no—
341D12—181no—
342D12—182no—
343C4×D3—62no—
344C4×D3—61no—
345C4×D3—181no—
346C4×D3—182no—
347Dic6—31no—
348Dic6—61no—
349Dic6—61no—
350Dic6—91no—
351Dic6—181no—
352Dic6—181no—
353C3×C8—21no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—121no—
357C3×C8—181no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—81no—
367C2×C32#88211yescharacteristic
368C2×C32C3×#12011yescharacteristic
369C3⋊D3—42no—
370C3⋊D3—41no—
371C3×D3—122no—
372C3×D3—121no—
373#41—91no—
374#41—181no—
375#41—181no—
376#41C2×(C3⋊D3)⋊C311yescharacteristic
377C2×Q8—91no—
378C2×D4—91no—
379C2×D4—181no—
380C2×D4—182no—
381C2×D4—181no—
382Q16—181no—
383Q16—181no—
384SD16—181no—
385SD16—181no—
386M16—91no—
387M16C2×(C3⋊D3)⋊C311yescharacteristic, complemented
388C22⋊C4—182no—
389C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
390C22×C3—21no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—121no—
397C22×C3—181no—
398C22×C3—182no—
399C22×C3—362no—
400C22×C3—722no—
401C22×C3C3×D3×D411yescharacteristic
402C2×D3—61no—
403C2×D3—62no—
404C2×D3—122no—
405C2×D3—181no—
406C2×D3—182no—
407C2×D3—242no—
408C2×D3—362no—
409C2×D3—722no—
410C3×C4—21no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—31no—
415C3×C4—41no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—122no—
424C3×C4—181no—
425C3×C4—361no—
426C3×C4C2×C3×D1211yescharacteristic
427C3×C4C3×D3×D411yescharacteristic
428Dic3—61no—
429Dic3—121no—
430Dic3—181no—
431Dic3—361no—
432C32—21no—
433C32#482511yescharacteristic, complemented
434C32C3×#45511yescharacteristic, complemented
435C23—92no—
436C23—182no—
437Q8—91no—
438Q8—181no—
439Q8—181no—
440Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
441D4—21no—
442D4—91no—
443D4—182no—
444D4—182no—
445D4—181no—
446D4—181no—
447D4—181no—
448D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—182no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
455C8—21no—
456C8—181no—
457C2×C3—21no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—41no—
461C2×C3—61no—
462C2×C3—61no—
463C2×C3—61no—
464C2×C3—121no—
465C2×C3—121no—
466C2×C3—121no—
467C2×C3—241no—
468C2×C3—362no—
469C2×C3—361no—
470C2×C3C3×#88211yescharacteristic
471D3—122no—
472D3—121no—
473D3—362no—
474D3—361no—
475C22—21no—
476C22—181no—
477C22—182no—
478C22—362no—
479C22—722no—
480C22D4×(C3⋊D3)⋊C311yescharacteristic
481C4—21no—
482C4—41no—
483C4—181no—
484C4—361no—
485C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
486C4D4×(C3⋊D3)⋊C311yescharacteristic
487C3—31no—
488C3—31no—
489C3—61no—
490C3C3×#482511yescharacteristic
491C2—21no—
492C2—41no—
493C2—362no—
494C2—361no—
495C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C3×#882
prime radical#11722638C2
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#455
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
2. #64160⋊C3C2semidirect
3. #64118⋊C3C2semidirect
4. #64084⋊C3C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #63650⋊C3C2semidirect
7. #63632⋊C3C2semidirect
8. #10592524C3semidirect
18. C2×#634⋊C3C22semidirect
19. M16×H(3,𝔽3)C22semidirect
20. C42×H(3,𝔽3)C22semidirect
24. #3748⋊C3C22semidirect
29. #64160C2×C3semidirect
30. #64118C2×C3semidirect
31. #64084C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63650C2×C3semidirect
39. #63632C2×C3semidirect
77. C2×#634C22×C3semidirect
87. C32×M16C2×D3semidirect
88. C32×M16C22×C3semidirect
90. C32×C42C2×D3semidirect
91. C32×C42C22×C3semidirect
95. #3748C22×C3semidirect
284. #104(C3⋊D3)⋊C3semidirect
286. H(3,𝔽3)#455semidirect
387. M16C2×(C3⋊D3)⋊C3semidirect
389. C42C2×(C3⋊D3)⋊C3semidirect
433. C32#4825semidirect
434. C32C3×#455semidirect

Maximal subgroups

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

subgroupindex
#117226382
#64160⋊C32
#64118⋊C32
#64084⋊C32
H(3,𝔽3)×#1042
#63650⋊C32
#63632⋊C32
#105925243
C3×#48253