Finite Group Databasebeta

← #10592904⋊C3 #13143503

Subgroups — #10592904⋊C3 #13143503

Order 1728.

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

normal subgroups only

This group has 557 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
2H(3,𝔽3)×#137C211yesmaximal, characteristic, complemented
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64151⋊C3C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#63652⋊C3C211yesmaximal, characteristic, complemented
7#63651⋊C3C211yesmaximal, characteristic, complemented
8#10592904C311yesmaximal, characteristic, complemented
9C3×#5205—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12Q8×(C3⋊D3)⋊C3—21no—
13#3830⋊C3—21no—
14#3830⋊C3—22no—
15#3828⋊C3—21no—
16#3828⋊C3C2211yescharacteristic
17C2×#634⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19Q16×H(3,𝔽3)—21no—
20SD16×H(3,𝔽3)—21no—
21M16×H(3,𝔽3)C2211yescharacteristic
22#3755⋊C3—21no—
23#3756⋊C3—21no—
24#3750⋊C3—22no—
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
26#3748⋊C3C2211yescharacteristic
27#64367C2×C311yescharacteristic
28C3×#1010—31no—
29C32×#137—21no—
30C32×#137D311yescharacteristic, complemented
31C32×#137C2×C311yescharacteristic, complemented
32#64153C2×C311yescharacteristic, complemented
33#64151C2×C311yescharacteristic, complemented
34C3×#944—31no—
35C3×#942—31no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39C3×#837—31no—
40C3×#825—31no—
41C3×#824—31no—
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)—21no—
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—22no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3D412yes—
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×#456—91no—
65#5205—31no—
66#5205—91no—
67C32×#41—21no—
68C32×#41C2×D311yescharacteristic
69C32×#41C22×C311yescharacteristic
70C2×C32×Q8—21no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C22×C311yescharacteristic
73Q8×C3⋊D3—21no—
74#3830—21no—
75#3830—22no—
76#3828—21no—
77#3828C22×C311yescharacteristic
78C2×#634—21no—
79C2×#634C22×C311yescharacteristic
80C3×D3×Q8—61no—
81C3×#237—62no—
82C3×#237—61no—
83C3×#235—31no—
84C3×#235—61no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—61no—
87C32×Q16—22no—
88C32×Q16—21no—
89C32×Q16—21no—
90C32×SD16—22no—
91C32×SD16—21no—
92C32×SD16—21no—
93C32×M16—21no—
94C32×M16C2×D311yescharacteristic
95C32×M16C22×C311yescharacteristic
96#3756—21no—
97#3755—21no—
98#3750—22no—
99C4×C3⋊Dic3C22×C311yescharacteristic
100#3748C22×C311yescharacteristic
101C3×#216—61no—
102C3×#215—61no—
103C3×Dic3⋊C4—62no—
104C3×C4×Dic3—31no—
105C3×#208—31no—
106C22×H(3,𝔽3)—21no—
107C22×H(3,𝔽3)C2×D411yescharacteristic
108C2×(C3⋊D3)⋊C3—21no—
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—22no—
115(C3⋊Dic3)⋊C3—21no—
116C3×2−1+4—91no—
117#1010—31no—
118#1010—91no—
119C3×#137—31no—
120C3×#137—31no—
121C3×#137—61no—
122C3×#137—91no—
123C3×#137C3×D311yescharacteristic
124C3×C4⋊Q8—91no—
125#944—31no—
126#944—91no—
127#942—31no—
128#942—91no—
129C3×#104—91no—
130C3×#104—91no—
131C3×#101—91no—
132#837—31no—
133#837—91no—
134#825—31no—
135#825—91no—
136#824—31no—
137#824—91no—
138C32×Q8—22no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—21no—
144C32×Q8—41no—
145C32×Q8#6711yescharacteristic
146C32×Q8C3×D411yescharacteristic
147C32×D4—21no—
148C32×D4—21no—
149C32×D4—21no—
150C32×D4—41no—
151C32×D4#6711yescharacteristic
152C32×D4C3×D411yescharacteristic
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C23×C311yescharacteristic
162#638—22no—
163#638—21no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3C3×D412yes—
167C4⋊(C3⋊D3)C3×D411yescharacteristic
168C4×C3⋊D3—22no—
169C4×C3⋊D3—21no—
170#634—22no—
171#634—21no—
172#634—21no—
173#634C3×D411yescharacteristic
174C3×#67—62no—
175C3×#67—61no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—61no—
179C3×D12—31no—
180C3×C4×D3—62no—
181C3×C4×D3—61no—
182C3×Dic6—31no—
183C3×Dic6—62no—
184C3×Dic6—61no—
185C3×Dic6—61no—
186C32×C8—22no—
187C32×C8—21no—
188C32×C8—21no—
189C3⋊(C3⋊C8)—21no—
190C3×C3⋊C8—61no—
191#456—91noSylow
192C2×H(3,𝔽3)—21no—
193C2×H(3,𝔽3)—41no—
194C2×H(3,𝔽3)#12011yescharacteristic
195(C3⋊D3)⋊C3—41no—
196C3×#41—31no—
197C3×#41—31no—
198C3×#41—61no—
199C3×#41—91no—
200C3×#41—181no—
201C3×#41—182no—
202C3×#41—181no—
203C3×#41C2×C3×D311yescharacteristic
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×Q8—91no—
208C2×C3×Q8—181no—
209C2×C3×Q8—181no—
210C2×C3×Q8C2×C3×D311yescharacteristic
211D3×Q8—61no—
212D3×Q8—181no—
213#237—62no—
214#237—61no—
215#237—182no—
216#237—181no—
217#235—31no—
218#235—61no—
219#235—91no—
220#235—181no—
221C2×Dic6—31no—
222C2×Dic6—61no—
223C2×Dic6—91no—
224C2×Dic6—181no—
225C3×Q16—21no—
226C3×Q16—61no—
227C3×Q16—61no—
228C3×Q16—62no—
229C3×Q16—181no—
230C3×SD16—21no—
231C3×SD16—61no—
232C3×SD16—61no—
233C3×SD16—62no—
234C3×SD16—181no—
235C3×M16—31no—
236C3×M16—31no—
237C3×M16—61no—
238C3×M16—91no—
239C3×M16C2×C3×D311yescharacteristic
240C3×C4⋊C4—182no—
241C3×C42—91no—
242#216—61no—
243#216—181no—
244#215—61no—
245#215—181no—
246Dic3⋊C4—62no—
247Dic3⋊C4—182no—
248C4×Dic3—31no—
249C4×Dic3—91no—
250#208—31no—
251#208—91no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32D3×D411yescharacteristic
257C22×C32C2×C3×D411yescharacteristic
258C2×C3⋊D3—21no—
259C2×C3×D3—61no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—41no—
265C32×C4—42no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4D3×D411yescharacteristic
269C32×C4C2×#6711yescharacteristic
270C32×C4C2×C3×D411yescharacteristic
271C32×C4C2×C3×D411yescharacteristic
272C3⋊Dic3—22no—
273C3⋊Dic3—22no—
274C3⋊Dic3—21no—
275C3×Dic3—62no—
276C3×Dic3—61no—
277C3×Dic3—62no—
2782−1+4—91no—
279#137—91no—
280#137(C3⋊D3)⋊C311yescharacteristic, complemented
281C4⋊Q8—91no—
282#104—91no—
283#104—91no—
284#101—91no—
285H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
286C3×Q8—21no—
287C3×Q8—21no—
288C3×Q8—31no—
289C3×Q8—31no—
290C3×Q8—61no—
291C3×Q8—61no—
292C3×Q8—61no—
293C3×Q8—61no—
294C3×Q8—62no—
295C3×Q8—61no—
296C3×Q8—91no—
297C3×Q8—121no—
298C3×Q8—181no—
299C3×Q8—181no—
300C3×Q8—182no—
301C3×Q8C3×#6711yescharacteristic
302C3×D4—21no—
303C3×D4—31no—
304C3×D4—31no—
305C3×D4—61no—
306C3×D4—61no—
307C3×D4—61no—
308C3×D4—91no—
309C3×D4—121no—
310C3×D4—181no—
311C3×D4—182no—
312C3×D4C3×#6711yescharacteristic
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—92no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—182no—
327C2×C3×C4—181no—
328C2×C3×C4—182no—
329C2×C3×C4C22×C3×D311yescharacteristic
330#67—62no—
331#67—61no—
332#67—181no—
333#67—182no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—61no—
337C2×Dic3—92no—
338C2×Dic3—181no—
339C2×Dic3—182no—
340D12—31no—
341D12—91no—
342C4×D3—62no—
343C4×D3—61no—
344C4×D3—181no—
345C4×D3—182no—
346Dic6—31no—
347Dic6—62no—
348Dic6—61no—
349Dic6—61no—
350Dic6—91no—
351Dic6—181no—
352Dic6—181no—
353Dic6—182no—
354C3×C8—21no—
355C3×C8—61no—
356C3×C8—61no—
357C3×C8—62no—
358C3×C8—181no—
359C3⋊C8—61no—
360C3⋊C8—181no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—81no—
368C2×C32#93711yescharacteristic
369C2×C32C3×#12011yescharacteristic
370C3⋊D3—41no—
371C3×D3—121no—
372#41—91no—
373#41—181no—
374#41—182no—
375#41—181no—
376#41C2×(C3⋊D3)⋊C311yescharacteristic
377C2×Q8—91no—
378C2×Q8—181no—
379C2×Q8—181no—
380C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
381Q16—21no—
382Q16—181no—
383SD16—21no—
384SD16—181no—
385M16—91no—
386M16C2×(C3⋊D3)⋊C311yescharacteristic
387C4⋊C4—182no—
388C42—91no—
389C22×C3—21no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—121no—
396C22×C3—181no—
397C22×C3C3×D3×D411yescharacteristic
398C2×D3—61no—
399C2×D3—181no—
400C3×C4—21no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—31no—
404C3×C4—31no—
405C3×C4—41no—
406C3×C4—61no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—61no—
410C3×C4—121no—
411C3×C4—121no—
412C3×C4—121no—
413C3×C4—122no—
414C3×C4—182no—
415C3×C4—181no—
416C3×C4—182no—
417C3×C4C3×D3×D411yescharacteristic
418C3×C4C2×C3×#6711yescharacteristic
419Dic3—62no—
420Dic3—61no—
421Dic3—62no—
422Dic3—181no—
423Dic3—182no—
424Dic3—182no—
425C32—21no—
426C32#520511yescharacteristic, complemented
427C32C3×#45611yescharacteristic, complemented
428Q8—21no—
429Q8—21no—
430Q8—91no—
431Q8—182no—
432Q8—181no—
433Q8—181no—
434Q8#638⋊C311yescharacteristic
435D4—21no—
436D4—91no—
437D4—182no—
438D4—181no—
439D4#638⋊C311yescharacteristic
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—92no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
448C8—21no—
449C8—181no—
450C2×C3—21no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—41no—
454C2×C3—61no—
455C2×C3—61no—
456C2×C3—61no—
457C2×C3—121no—
458C2×C3—121no—
459C2×C3—121no—
460C2×C3—241no—
461C2×C3—361no—
462C2×C3C3×#93711yescharacteristic
463D3—121no—
464D3—361no—
465C22—21no—
466C22—181no—
467C22D4×(C3⋊D3)⋊C311yescharacteristic
468C4—21no—
469C4—41no—
470C4—182no—
471C4—182no—
472C4—181no—
473C4D4×(C3⋊D3)⋊C311yescharacteristic
474C4C2×#638⋊C311yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3C3×#520511yescharacteristic
479C2—21no—
480C2—41no—
481C2—361no—
482C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#137C2
socleC2×C3C3×#937
prime radical#11722639C2
hypercenter#137(C3⋊D3)⋊C3
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#137C2semidirect
3. #64153⋊C3C2semidirect
4. #64151⋊C3C2semidirect
5. #63664⋊C3C2semidirect
6. #63652⋊C3C2semidirect
7. #63651⋊C3C2semidirect
8. #10592904C3semidirect
30. C32×#137D3semidirect
31. C32×#137C2×C3semidirect
32. #64153C2×C3semidirect
33. #64151C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63651C2×C3semidirect
280. #137(C3⋊D3)⋊C3semidirect
285. H(3,𝔽3)#456semidirect
426. C32#5205semidirect
427. C32C3×#456semidirect

Maximal subgroups

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

subgroupindex
#117226392
H(3,𝔽3)×#1372
#64153⋊C32
#64151⋊C32
#63664⋊C32
#63652⋊C32
#63651⋊C32
#105929043
C3×#52053