Finite Group Databasebeta

← #10592526⋊C3 #13141987

Subgroups — #10592526⋊C3 #13141987

Order 1728.

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

normal subgroups only

This group has 555 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#11722639C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63632⋊C3C211yesmaximal, characteristic, complemented
8#10592526C311yesmaximal, characteristic, complemented
9C3×#4827—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—21no—
12#3830⋊C3—21no—
13#3830⋊C3—22no—
14#3828⋊C3—21no—
15#3828⋊C3C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×#634⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19M16×H(3,𝔽3)C2211yescharacteristic
20C42×H(3,𝔽3)C2211yescharacteristic
21#3755⋊C3—21no—
22#3753⋊C3—21no—
23#3752⋊C3—22no—
24#3748⋊C3C2211yescharacteristic
25#3745⋊C3—21no—
26#3744⋊C3—21no—
27#64367C2×C311yescharacteristic, complemented
28C3×#1010—31no—
29#64158C2×C311yescharacteristic, complemented
30#64117C2×C311yescharacteristic, complemented
31#64084C2×C311yescharacteristic, complemented
32C3×#949—31no—
33C3×#908—31no—
34C3×#875—31no—
35C32×#104—21no—
36C32×#104D311yescharacteristic, complemented
37C32×#104C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#824—31no—
41C3×#805—31no—
42C22×(C3⋊D3)⋊C3—21no—
43Q8×H(3,𝔽3)D411yescharacteristic
44D4×H(3,𝔽3)—21no—
45D4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—22no—
50#638⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3D412yes—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
56C4×(C3⋊D3)⋊C3—22no—
57C4×(C3⋊D3)⋊C3—21no—
58#634⋊C3—22no—
59#634⋊C3—21no—
60#634⋊C3—21no—
61#634⋊C3D411yescharacteristic
62C8×H(3,𝔽3)—21no—
63(C3⋊(C3⋊C8))⋊C3—21no—
64C3×#454—91no—
65#4827—31no—
66#4827—91no—
67C32×#41—21no—
68C32×#41C2×D311yescharacteristic
69C32×#41C22×C311yescharacteristic
70Q8×C3⋊D3—21no—
71#3830—21no—
72#3830—22no—
73#3828—21no—
74#3828C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×#634—21no—
77C2×#634C22×C311yescharacteristic
78C3×D3×Q8—61no—
79C3×#237—62no—
80C3×#237—61no—
81C3×#235—31no—
82C3×#235—61no—
83C2×C3×D12—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C32×M16—21no—
87C32×M16C2×D311yescharacteristic
88C32×M16C22×C311yescharacteristic
89C32×C42—21no—
90C32×C42C2×D311yescharacteristic
91C32×C42C22×C311yescharacteristic
92#3755—21no—
93#3753—21no—
94#3752—22no—
95#3748C22×C311yescharacteristic
96#3745—21no—
97#3744—21no—
98C3×#215—61no—
99C3×#213—61no—
100C3×(C2×D3)⋊C4—62no—
101C3×#208—31no—
102C3×D24—61no—
103C3×#204—61no—
104C22×H(3,𝔽3)—21no—
105C22×H(3,𝔽3)C2×D411yescharacteristic
106C2×(C3⋊D3)⋊C3—21no—
107C2×(C3⋊D3)⋊C3—42no—
108C2×(C3⋊D3)⋊C3—41no—
109C4×H(3,𝔽3)—21no—
110C4×H(3,𝔽3)—41no—
111C4×H(3,𝔽3)C2×D411yescharacteristic
112C4×H(3,𝔽3)C2×D411yescharacteristic
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—21no—
115C3×2−1+4—91no—
116#1010—31no—
117#1010—91no—
118C3×AGL(1,ℤ/8)—91no—
119C3×AGL(1,ℤ/8)—91no—
120C3×#124—91no—
121#949—31no—
122#949—91no—
123#908—31no—
124#908—91no—
125#875—31no—
126#875—91no—
127C3×#104—31no—
128C3×#104—31no—
129C3×#104—61no—
130C3×#104—91no—
131C3×#104C3×D311yescharacteristic
132C3×#101—91no—
133#824—31no—
134#824—91no—
135#805—31no—
136#805—91no—
137C22×C3⋊D3—21no—
138C22×C3×D3—61no—
139C32×Q8—21no—
140C32×Q8D1211yescharacteristic
141C32×Q8C3×D411yescharacteristic
142C32×D4—21no—
143C32×D4—21no—
144C32×D4—21no—
145C32×D4—41no—
146C32×D4D1211yescharacteristic
147C32×D4C3×D411yescharacteristic
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C311yescharacteristic
157#638—22no—
158#638—21no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3C3×D412yes—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)C3×D411yescharacteristic
164C4×C3⋊D3—22no—
165C4×C3⋊D3—21no—
166#634—22no—
167#634—21no—
168#634—21no—
169#634C3×D411yescharacteristic
170C3×#67—62no—
171C3×#67—61no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—62no—
174C3×D12—31no—
175C3×D12—61no—
176C3×D12—61no—
177C3×C4×D3—62no—
178C3×C4×D3—61no—
179C3×Dic6—31no—
180C3×Dic6—62no—
181C3×Dic6—61no—
182C3×Dic6—61no—
183C32×C8—21no—
184C32×C8—21no—
185C32×C8—41no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—61no—
188#454—91noSylow
189C2×H(3,𝔽3)—21no—
190C2×H(3,𝔽3)—41no—
191C2×H(3,𝔽3)#12011yescharacteristic
192(C3⋊D3)⋊C3—41no—
193(C3⋊D3)⋊C3—81no—
194C3×#41—31no—
195C3×#41—31no—
196C3×#41—61no—
197C3×#41—91no—
198C3×#41—181no—
199C3×#41—182no—
200C3×#41—181no—
201C3×#41C2×C3×D311yescharacteristic
202C2×C3×Q8—91no—
203C2×C3×Q8—181no—
204C2×C3×Q8—181no—
205C2×C3×D4—91no—
206D3×Q8—61no—
207D3×Q8—181no—
208#237—62no—
209#237—61no—
210#237—182no—
211#237—181no—
212#235—31no—
213#235—61no—
214#235—91no—
215#235—181no—
216C2×D12—31no—
217C2×D12—91no—
218C2×Dic6—31no—
219C2×Dic6—61no—
220C2×Dic6—91no—
221C2×Dic6—181no—
222C3×SD16—181no—
223C3×SD16—181no—
224C3×D8—181no—
225C3×D8—181no—
226C3×M16—31no—
227C3×M16—31no—
228C3×M16—61no—
229C3×M16—91no—
230C3×M16C2×C3×D311yescharacteristic
231C3×C22⋊C4—182no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42—61no—
235C3×C42C2×C3×D311yescharacteristic
236#215—61no—
237#215—181no—
238#213—61no—
239#213—181no—
240(C2×D3)⋊C4—62no—
241(C2×D3)⋊C4—182no—
242#208—31no—
243#208—91no—
244D24—61no—
245D24—181no—
246#204—61no—
247#204—181no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—41no—
252C22×C32D3×D411yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C2×C3⋊D3—21no—
255C2×C3⋊D3—42no—
256C2×C3⋊D3—41no—
257C2×C3×D3—61no—
258C2×C3×D3—122no—
259C2×C3×D3—121no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—41no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—42no—
268C32×C4C2×D1211yescharacteristic
269C32×C4D3×D411yescharacteristic
270C32×C4C2×C3×D411yescharacteristic
271C32×C4C2×C3×D411yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3—21no—
274C3×Dic3—61no—
275C3×Dic3—62no—
2762−1+4—91no—
277AGL(1,ℤ/8)—91no—
278AGL(1,ℤ/8)—91no—
279#124—91no—
280#104—91no—
281#104(C3⋊D3)⋊C311yescharacteristic, complemented
282#101—91no—
283H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
284C23×C3—181no—
285C22×D3—61no—
286C22×D3—181no—
287C3×Q8—31no—
288C3×Q8—31no—
289C3×Q8—61no—
290C3×Q8—91no—
291C3×Q8—181no—
292C3×Q8—181no—
293C3×Q8—182no—
294C3×Q8C3×D1211yescharacteristic
295C3×D4—21no—
296C3×D4—31no—
297C3×D4—31no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—61no—
301C3×D4—91no—
302C3×D4—121no—
303C3×D4—181no—
304C3×D4—181no—
305C3×D4—181no—
306C3×D4—182no—
307C3×D4C3×D1211yescharacteristic
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—181no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4C22×C3×D311yescharacteristic
324#67—62no—
325#67—61no—
326#67—181no—
327#67—182no—
328C2×Dic3—32no—
329C2×Dic3—62no—
330C2×Dic3—92no—
331C2×Dic3—182no—
332D12—31no—
333D12—61no—
334D12—61no—
335D12—91no—
336D12—181no—
337D12—181no—
338C4×D3—62no—
339C4×D3—61no—
340C4×D3—181no—
341C4×D3—182no—
342Dic6—31no—
343Dic6—62no—
344Dic6—61no—
345Dic6—61no—
346Dic6—91no—
347Dic6—181no—
348Dic6—181no—
349Dic6—182no—
350C3×C8—21no—
351C3×C8—61no—
352C3×C8—61no—
353C3×C8—121no—
354C3×C8—181no—
355C3⋊C8—61no—
356C3⋊C8—181no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—81no—
364C2×C32#88211yescharacteristic
365C2×C32C3×#12011yescharacteristic
366C3⋊D3—41no—
367C3⋊D3—81no—
368C3×D3—121no—
369C3×D3—241no—
370#41—91no—
371#41—181no—
372#41—182no—
373#41—181no—
374#41C2×(C3⋊D3)⋊C311yescharacteristic
375C2×Q8—91no—
376C2×Q8—181no—
377C2×Q8—181no—
378C2×D4—91no—
379SD16—181no—
380SD16—181no—
381D8—181no—
382D8—181no—
383M16—91no—
384M16C2×(C3⋊D3)⋊C311yescharacteristic
385C22⋊C4—182no—
386C42C2×(C3⋊D3)⋊C311yescharacteristic
387C22×C3—21no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—121no—
394C22×C3—181no—
395C22×C3—361no—
396C22×C3—362no—
397C22×C3C3×D3×D411yescharacteristic
398C2×D3—61no—
399C2×D3—122no—
400C2×D3—121no—
401C2×D3—181no—
402C2×D3—361no—
403C2×D3—362no—
404C3×C4—21no—
405C3×C4—31no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—41no—
410C3×C4—61no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—121no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—122no—
418C3×C4—182no—
419C3×C4—181no—
420C3×C4C2×C3×D1211yescharacteristic
421C3×C4C3×D3×D411yescharacteristic
422Dic3—61no—
423Dic3—62no—
424Dic3—181no—
425Dic3—182no—
426C32—21no—
427C32#482711yescharacteristic, complemented
428C32C3×#45411yescharacteristic, complemented
429C23—181no—
430Q8—91no—
431Q8—182no—
432Q8—181no—
433Q8—181no—
434Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
435D4—21no—
436D4—91no—
437D4—182no—
438D4—181no—
439D4—181no—
440D4—181no—
441D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—92no—
445C2×C4—182no—
446C2×C4—182no—
447C2×C4—181no—
448C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
449C8—21no—
450C8—181no—
451C2×C3—21no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—41no—
455C2×C3—61no—
456C2×C3—61no—
457C2×C3—61no—
458C2×C3—121no—
459C2×C3—121no—
460C2×C3—121no—
461C2×C3—241no—
462C2×C3—361no—
463C2×C3—721no—
464C2×C3C3×#88211yescharacteristic
465D3—121no—
466D3—241no—
467D3—361no—
468D3—721no—
469C22—21no—
470C22—181no—
471C22—362no—
472C22—361no—
473C22D4×(C3⋊D3)⋊C311yescharacteristic
474C4—21no—
475C4—41no—
476C4—182no—
477C4—181no—
478C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
479C4D4×(C3⋊D3)⋊C311yescharacteristic
480C3—31no—
481C3—31no—
482C3—61no—
483C3C3×#482711yescharacteristic
484C2—21no—
485C2—41no—
486C2—361no—
487C2—721no—
488C2#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×#454
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. #11722639C2semidirect
2. #64158⋊C3C2semidirect
3. #64117⋊C3C2semidirect
4. #64084⋊C3C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #63651⋊C3C2semidirect
7. #63632⋊C3C2semidirect
8. #10592526C3semidirect
27. #64367C2×C3semidirect
29. #64158C2×C3semidirect
30. #64117C2×C3semidirect
31. #64084C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63651C2×C3semidirect
39. #63632C2×C3semidirect
281. #104(C3⋊D3)⋊C3semidirect
283. H(3,𝔽3)#454semidirect
427. C32#4827semidirect
428. C32C3×#454semidirect

Maximal subgroups

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

subgroupindex
#117226392
#64158⋊C32
#64117⋊C32
#64084⋊C32
H(3,𝔽3)×#1042
#63651⋊C32
#63632⋊C32
#105925263
C3×#48273