Finite Group Databasebeta

← #10592525⋊C3 #13141986

Subgroups — #10592525⋊C3 #13141986

Order 1728.

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

normal subgroups only

This group has 541 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
2#64160⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C211yesmaximal, characteristic, complemented
4#64078⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63632⋊C3C211yesmaximal, characteristic, complemented
8#10592525C311yesmaximal, characteristic, complemented
9C3×#4826—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×#634⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19M16×H(3,𝔽3)C2211yescharacteristic
20C42×H(3,𝔽3)C2211yescharacteristic
21#3756⋊C3—21no—
22#3754⋊C3—21no—
23C4⋊((C3⋊Dic3)⋊C3)—22no—
24#3748⋊C3C2211yescharacteristic
25#3744⋊C3—21no—
26#3746⋊C3—21no—
27#64367C2×C311yescharacteristic
28C3×#1010—31no—
29#64160C2×C311yescharacteristic, complemented
30#64118C2×C311yescharacteristic, complemented
31#64078C2×C311yescharacteristic, complemented
32C3×#951—31no—
33C3×#909—31no—
34C3×#869—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—
42Q8×H(3,𝔽3)D411yescharacteristic
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#638⋊C3—22no—
49#638⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3D412yes—
53C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
54C4×(C3⋊D3)⋊C3—22no—
55C4×(C3⋊D3)⋊C3—21no—
56#634⋊C3—22no—
57#634⋊C3—21no—
58#634⋊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×#456—91no—
65#4826—31no—
66#4826—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×#634—21no—
76C2×#634C22×C311yescharacteristic
77C2×#634C22×C311yescharacteristic
78C3×D3×Q8—61no—
79C3×#237—62no—
80C3×#237—61no—
81C3×#235—31no—
82C3×#235—61no—
83C2×C3×Dic6—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#3756—21no—
93#3754—21no—
94C4⋊(C3⋊Dic3)—22no—
95#3748C22×C311yescharacteristic
96#3746—21no—
97#3744—21no—
98C3×#216—61no—
99C3×#214—61no—
100C3×C4⋊Dic3—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—21no—
107C4×H(3,𝔽3)—22no—
108C4×H(3,𝔽3)—21no—
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—21no—
113(C3⋊Dic3)⋊C3—41no—
114C3×2−1+4—91no—
115#1010—31no—
116#1010—91no—
117C3×#137—91no—
118C3×#137—91no—
119C3×C4⋊Q8—91no—
120#951—31no—
121#951—91no—
122#909—31no—
123#909—91no—
124#869—31no—
125#869—91no—
126C3×#104—31no—
127C3×#104—31no—
128C3×#104—61no—
129C3×#104—91no—
130C3×#104C3×D311yescharacteristic
131C3×#101—91no—
132#824—31no—
133#824—91no—
134#805—31no—
135#805—91no—
136C32×Q8—21no—
137C32×Q8D1211yescharacteristic
138C32×Q8C3×D411yescharacteristic
139C32×D4—21no—
140C32×D4—21no—
141C32×D4—21no—
142C32×D4—41no—
143C32×D4D1211yescharacteristic
144C32×D4C3×D411yescharacteristic
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154#638—22no—
155#638—21no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3C3×D412yes—
159C4⋊(C3⋊D3)C3×D411yescharacteristic
160C4×C3⋊D3—22no—
161C4×C3⋊D3—21no—
162#634—22no—
163#634—21no—
164#634—21no—
165#634—21no—
166#634—21no—
167#634C3×D411yescharacteristic
168C3×#67—62no—
169C3×#67—61no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—61no—
173C3×D12—31no—
174C3×C4×D3—62no—
175C3×C4×D3—61no—
176C3×Dic6—31no—
177C3×Dic6—62no—
178C3×Dic6—61no—
179C3×Dic6—61no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182C32×C8—21no—
183C32×C8—21no—
184C32×C8—41no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—61no—
187#456—91noSylow
188C2×H(3,𝔽3)—21no—
189C2×H(3,𝔽3)—41no—
190C2×H(3,𝔽3)#12011yescharacteristic
191(C3⋊D3)⋊C3—41no—
192C3×#41—31no—
193C3×#41—31no—
194C3×#41—61no—
195C3×#41—91no—
196C3×#41—181no—
197C3×#41—182no—
198C3×#41—181no—
199C3×#41C2×C3×D311yescharacteristic
200C2×C3×Q8—91no—
201C2×C3×Q8—91no—
202C2×C3×Q8—181no—
203C2×C3×Q8—181no—
204D3×Q8—61no—
205D3×Q8—181no—
206#237—62no—
207#237—61no—
208#237—182no—
209#237—181no—
210#235—31no—
211#235—61no—
212#235—91no—
213#235—181no—
214C2×Dic6—31no—
215C2×Dic6—31no—
216C2×Dic6—61no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—181no—
220C3×Q16—181no—
221C3×Q16—181no—
222C3×SD16—181no—
223C3×SD16—181no—
224C3×M16—31no—
225C3×M16—31no—
226C3×M16—61no—
227C3×M16—91no—
228C3×M16C2×C3×D311yescharacteristic
229C3×C4⋊C4—182no—
230C3×C42—31no—
231C3×C42—31no—
232C3×C42—61no—
233C3×C42C2×C3×D311yescharacteristic
234#216—61no—
235#216—181no—
236#214—61no—
237#214—181no—
238C4⋊Dic3—62no—
239C4⋊Dic3—182no—
240#208—31no—
241#208—91no—
242Dic12—61no—
243Dic12—181no—
244#204—61no—
245#204—181no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32D3×D411yescharacteristic
251C22×C32C2×C3×D411yescharacteristic
252C2×C3⋊D3—21no—
253C2×C3×D3—61no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—41no—
261C32×C4—42no—
262C32×C4C2×D1211yescharacteristic
263C32×C4D3×D411yescharacteristic
264C32×C4C2×C3×D411yescharacteristic
265C32×C4C2×C3×D411yescharacteristic
266C3⋊Dic3—22no—
267C3⋊Dic3—21no—
268C3⋊Dic3—41no—
269C3×Dic3—61no—
270C3×Dic3—62no—
271C3×Dic3—121no—
2722−1+4—91no—
273#137—91no—
274#137—91no—
275C4⋊Q8—91no—
276#104—91no—
277#104(C3⋊D3)⋊C311yescharacteristic, complemented
278#101—91no—
279H(3,𝔽3)#45611yescharacteristic, complemented, Sylow
280C3×Q8—31no—
281C3×Q8—31no—
282C3×Q8—61no—
283C3×Q8—91no—
284C3×Q8—181no—
285C3×Q8—181no—
286C3×Q8—181no—
287C3×Q8—181no—
288C3×Q8—182no—
289C3×Q8C3×D1211yescharacteristic
290C3×D4—21no—
291C3×D4—31no—
292C3×D4—31no—
293C3×D4—61no—
294C3×D4—61no—
295C3×D4—61no—
296C3×D4—91no—
297C3×D4—121no—
298C3×D4—181no—
299C3×D4—182no—
300C3×D4C3×D1211yescharacteristic
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—92no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4—181no—
314C2×C3×C4—182no—
315C2×C3×C4—182no—
316C2×C3×C4—181no—
317C2×C3×C4C22×C3×D311yescharacteristic
318#67—62no—
319#67—61no—
320#67—181no—
321#67—182no—
322C2×Dic3—32no—
323C2×Dic3—62no—
324C2×Dic3—61no—
325C2×Dic3—92no—
326C2×Dic3—182no—
327C2×Dic3—181no—
328D12—31no—
329D12—91no—
330C4×D3—62no—
331C4×D3—61no—
332C4×D3—181no—
333C4×D3—182no—
334Dic6—31no—
335Dic6—62no—
336Dic6—61no—
337Dic6—61no—
338Dic6—61no—
339Dic6—61no—
340Dic6—91no—
341Dic6—181no—
342Dic6—181no—
343Dic6—182no—
344Dic6—181no—
345Dic6—181no—
346C3×C8—21no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—121no—
350C3×C8—181no—
351C3⋊C8—61no—
352C3⋊C8—181no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—41no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32—81no—
360C2×C32#88211yescharacteristic
361C2×C32C3×#12011yescharacteristic
362C3⋊D3—41no—
363C3×D3—121no—
364#41—91no—
365#41—181no—
366#41—182no—
367#41—181no—
368#41C2×(C3⋊D3)⋊C311yescharacteristic
369C2×Q8—91no—
370C2×Q8—91no—
371C2×Q8—181no—
372C2×Q8—181no—
373Q16—181no—
374Q16—181no—
375SD16—181no—
376SD16—181no—
377M16—91no—
378M16C2×(C3⋊D3)⋊C311yescharacteristic
379C4⋊C4—182no—
380C42C2×(C3⋊D3)⋊C311yescharacteristic
381C22×C3—21no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—121no—
388C22×C3—181no—
389C22×C3C3×D3×D411yescharacteristic
390C2×D3—61no—
391C2×D3—181no—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—31no—
395C3×C4—31no—
396C3×C4—31no—
397C3×C4—31no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—121no—
405C3×C4—122no—
406C3×C4—182no—
407C3×C4—181no—
408C3×C4—361no—
409C3×C4C2×C3×D1211yescharacteristic
410C3×C4C3×D3×D411yescharacteristic
411Dic3—61no—
412Dic3—62no—
413Dic3—121no—
414Dic3—181no—
415Dic3—182no—
416Dic3—361no—
417C32—21no—
418C32#482611yescharacteristic, complemented
419C32C3×#45611yescharacteristic, complemented
420Q8—91no—
421Q8—182no—
422Q8—181no—
423Q8—181no—
424Q8—181no—
425Q8—181no—
426Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
427D4—21no—
428D4—91no—
429D4—182no—
430D4—181no—
431D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—92no—
435C2×C4—182no—
436C2×C4—182no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
440C8—21no—
441C8—181no—
442C2×C3—21no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—41no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—61no—
449C2×C3—121no—
450C2×C3—121no—
451C2×C3—121no—
452C2×C3—241no—
453C2×C3—361no—
454C2×C3C3×#88211yescharacteristic
455D3—121no—
456D3—361no—
457C22—21no—
458C22—181no—
459C22D4×(C3⋊D3)⋊C311yescharacteristic
460C4—22no—
461C4—21no—
462C4—182no—
463C4—181no—
464C4—361no—
465C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
466C4D4×(C3⋊D3)⋊C311yescharacteristic
467C3—31no—
468C3—31no—
469C3—61no—
470C3C3×#482611yescharacteristic
471C2—21no—
472C2—41no—
473C2—361no—
474C2#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#11722639C2
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#456
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. #64078⋊C3C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #63651⋊C3C2semidirect
7. #63632⋊C3C2semidirect
8. #10592525C3semidirect
29. #64160C2×C3semidirect
30. #64118C2×C3semidirect
31. #64078C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63651C2×C3semidirect
39. #63632C2×C3semidirect
277. #104(C3⋊D3)⋊C3semidirect
279. H(3,𝔽3)#456semidirect
418. C32#4826semidirect
419. C32C3×#456semidirect

Maximal subgroups

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

subgroupindex
#117226392
#64160⋊C32
#64118⋊C32
#64078⋊C32
H(3,𝔽3)×#1042
#63651⋊C32
#63632⋊C32
#105925253
C3×#48263