Finite Group Databasebeta

← #10592461⋊C3 #13141776

Subgroups — #10592461⋊C3 #13141776

Order 1728.

498 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 94 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#64149⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64096⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592461C311yesmaximal, characteristic, complemented
9C3×#4762—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#3754⋊C3—22no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23#64149C2×C311yescharacteristic
24C2×#3754C2×C311yescharacteristic, complemented
25C2×#3744C2×C311yescharacteristic, complemented
26#64096C2×C311yescharacteristic, complemented
27C3×#940—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30C3×#887—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic, complemented
33C32×#102C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×#809—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—42no—
45#638⋊C3—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#634⋊C3—21no—
52#634⋊C3D412yescomplemented
53C8×H(3,𝔽3)—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55C3×#460—91no—
56#4762—31no—
57#4762—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C22×C311yescharacteristic
61C2×#638—21no—
62C2×#638—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×#634C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×D12—31no—
68C2×C3×Dic6—31no—
69C2×C32×C8—21no—
70C2×C32×C8C2×D311yescharacteristic
71C2×C32×C8C22×C311yescharacteristic
72C32×C4⋊C4—21no—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C22×C311yescharacteristic
75#3754—22no—
76#3750—21no—
77C4×C3⋊Dic3—21no—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80#3744—22no—
81C3×#214—62no—
82C3×Dic3⋊C4—61no—
83C3×C4×Dic3—31no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—31no—
86C3×#204—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3SD1612yescomplemented
100C2×C3×SD16—91no—
101C2×C3×SD16—91no—
102C3×#127—91no—
103C3×C4×Q8—91no—
104#940—31no—
105#940—91no—
106C2×#214—31no—
107C2×#214—91no—
108C2×#204—31no—
109C2×#204—91no—
110#887—31no—
111#887—91no—
112C3×C4⋊C8—91no—
113C3×#102—31no—
114C3×#102—31no—
115C3×#102—61no—
116C3×#102—91no—
117C3×#102C3×D311yescharacteristic
118Dic3⋊C8—31no—
119Dic3⋊C8—91no—
120#809—31no—
121#809—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139#638—42no—
140#638—42no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3C3×D411yescharacteristic
143C2×C3⋊Dic3C3×D411yescharacteristic
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146#634—21no—
147#634C3×D412yescomplemented
148C3×#67—122no—
149C3×#67—122no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×Dic6—32no—
156C3×Dic6—61no—
157C32×C8—21no—
158C32×C8—21no—
159C32×C8—41no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#460—91noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)C2×SD1611yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167(C3⋊D3)⋊C3—81no—
168C2×C3×Q8—91no—
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C2×C3×D4—181no—
175C2×C3×D4C2×C3×D311yescharacteristic
176C2×#67—61no—
177C2×#67—61no—
178C2×#67—181no—
179C2×#67—181no—
180C2×D12—31no—
181C2×D12—91no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×SD16—182no—
185C3×SD16—182no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—61no—
189C2×C3×C8—91no—
190C2×C3×C8C2×C3×D311yescharacteristic
191C3×C4⋊C4—31no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—181no—
195C3×C4⋊C4C2×C3×D311yescharacteristic
196C3×C42—91no—
197C3×C42—181no—
198#214—62no—
199#214—182no—
200Dic3⋊C4—61no—
201Dic3⋊C4—181no—
202C4×Dic3—31no—
203C4×Dic3—61no—
204C4×Dic3—91no—
205C4×Dic3—181no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—91no—
208#204—62no—
209#204—182no—
210C22×C32—21no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—81no—
219C22×C32—81no—
220C22×C32D3×D411yescharacteristic
221C22×C32C2×C3×D411yescharacteristic
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4—81no—
233C32×C4#23511yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C32×C4C3×#4111yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—21no—
239C3⋊Dic3C3×SD1612yescomplemented
240C3×Dic3—32no—
241C3×Dic3—62no—
242C3×Dic3—61no—
243C2×SD16—91no—
244C2×SD16—91no—
245#127—91no—
246C4×Q8—91no—
247C4⋊C8—91no—
248#102—91no—
249#102(C3⋊D3)⋊C311yescharacteristic, complemented
250H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—121no—
255C23×C3—181no—
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—21no—
261C3×D4—21no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—62no—
267C3×D4—121no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—362no—
271C3×D4—362no—
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—91no—
279C2×C3×C4—91no—
280C2×C3×C4—121no—
281C2×C3×C4—181no—
282C2×C3×C4C22×C3×D311yescharacteristic
283#67—122no—
284#67—122no—
285#67—362no—
286#67—362no—
287C2×Dic3—31no—
288C2×Dic3—31no—
289C2×Dic3—61no—
290C2×Dic3—91no—
291C2×Dic3—91no—
292C2×Dic3—181no—
293D12—61no—
294D12—61no—
295D12—181no—
296D12—181no—
297Dic6—32no—
298Dic6—61no—
299Dic6—92no—
300Dic6—181no—
301C3×C8—21no—
302C3×C8—61no—
303C3×C8—61no—
304C3×C8—121no—
305C3×C8—181no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32#88411yescharacteristic
315C2×C32#91111yescharacteristic
316C2×C32D3×SD1611yescharacteristic
317C2×C32C3×#12111yescharacteristic
318C2×C32C2×C3×SD1611yescharacteristic
319C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
320C3⋊D3—81no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—181no—
325C2×D4—181no—
326C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
327SD16—182no—
328SD16—182no—
329C2×C8—91no—
330C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
331C4⋊C4—181no—
332C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
333C42—91no—
334C42—181no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—122no—
348C22×C3—241no—
349C22×C3—241no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3C3×D3×D411yescharacteristic
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—41no—
365C3×C4—61no—
366C3×C4—61no—
367C3×C4—92no—
368C3×C4—121no—
369C3×C4—121no—
370C3×C4—181no—
371C3×C4—182no—
372C3×C4—241no—
373C3×C4C3×#23511yescharacteristic
374C3×C4C3×D3×D411yescharacteristic
375Dic3—32no—
376Dic3—62no—
377Dic3—61no—
378Dic3—92no—
379Dic3—182no—
380Dic3—181no—
381C32—21no—
382C32#476211yescharacteristic, complemented
383C32C3×#46011yescharacteristic, complemented
384C23—21no—
385C23—181no—
386Q8—92no—
387Q8—181no—
388D4—21no—
389D4—21no—
390D4—181no—
391D4—181no—
392D4—362no—
393D4—362no—
394C2×C4—21no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—181no—
398C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
399C8—21no—
400C8—181no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—81no—
411C2×C3—241no—
412C2×C3—241no—
413C2×C3—242no—
414C2×C3—721no—
415C2×C3C3×#88411yescharacteristic
416C2×C3C3×#91111yescharacteristic
417C2×C3C3×D3×SD1611yescharacteristic
418D3—241no—
419D3—721no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22D4×(C3⋊D3)⋊C311yescharacteristic
427C4—41no—
428C4—92no—
429C4—181no—
430C4—182no—
431C4#3828⋊C311yescharacteristic
432C4D4×(C3⋊D3)⋊C311yescharacteristic
433C3—31no—
434C3—31no—
435C3—61no—
436C3C3×#476211yescharacteristic
437C2—81no—
438C2—721no—
439C2#64093⋊C311yescharacteristic
440C2#64120⋊C311yescharacteristic
441C2SD16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radical#64149⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#3754⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64096⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63641⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592461C3semidirect
24. C2×#3754C2×C3semidirect
25. C2×#3744C2×C3semidirect
26. #64096C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63641C2×C3semidirect
35. #63636C2×C3semidirect
52. #634⋊C3D4semidirect
99. (C3⋊Dic3)⋊C3SD16semidirect
147. #634C3×D4semidirect
239. C3⋊Dic3C3×SD16semidirect
249. #102(C3⋊D3)⋊C3semidirect
250. H(3,𝔽3)#460semidirect
382. C32#4762semidirect
383. C32C3×#460semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
C2×#3754⋊C32
C2×#3744⋊C32
#64096⋊C32
H(3,𝔽3)×#1022
#63641⋊C32
#63636⋊C32
#105924613
C3×#47623