Finite Group Databasebeta

← #10592840⋊C3 #13143258

Subgroups — #10592840⋊C3 #13143258

Order 1728.

426 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic
3#64160⋊C3C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C211yesmaximal, characteristic, complemented
5#63650⋊C3C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63646⋊C3C211yesmaximal, characteristic, complemented
8#10592840C311yesmaximal, characteristic, complemented
9C3×#5141—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13M16×H(3,𝔽3)—21no—
14M16×H(3,𝔽3)C2211yescharacteristic
15C2×C8×H(3,𝔽3)—21no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—21no—
18#3756⋊C3—21no—
19#3753⋊C3—21no—
20#3754⋊C3—21no—
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23#3744⋊C3—21no—
24#3744⋊C3—21no—
25C32×#131—21no—
26C32×#131D311yescharacteristic, complemented
27C32×#131C2×C311yescharacteristic, complemented
28#64160C2×C311yescharacteristic, complemented
29#64158C2×C311yescharacteristic
30C2×#3744C2×C311yescharacteristic, complemented
31C3×#951—31no—
32C3×#949—31no—
33C2×C3×#204—31no—
34#63651C2×C311yescharacteristic, complemented
35#63650C2×C311yescharacteristic, complemented
36#63646C2×C311yescharacteristic, complemented
37C3×#824—31no—
38C3×#823—31no—
39C3×#819—31no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)D411yescharacteristic
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51C8×H(3,𝔽3)—21no—
52C8×H(3,𝔽3)D411yescharacteristic
53C8×H(3,𝔽3)D411yescharacteristic
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#471—91no—
57#5141—31no—
58#5141—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C22×C311yescharacteristic
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×Dic6—31no—
66C32×M16—22no—
67C32×M16—21no—
68C32×M16—21no—
69C32×M16—21no—
70C32×M16C2×D311yescharacteristic
71C32×M16C22×C311yescharacteristic
72C2×C32×C8—22no—
73C2×C32×C8—21no—
74C2×C32×C8—21no—
75C2×C32×C8—21no—
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C311yescharacteristic
78#3756—21no—
79#3755—21no—
80#3754—21no—
81#3753—21no—
82#3748C22×C311yescharacteristic
83#3748C22×C311yescharacteristic
84#3744—21no—
85#3744—21no—
86C3×#216—61no—
87C3×#215—61no—
88C3×#214—61no—
89C3×#213—61no—
90C3×#208—31no—
91C3×#208—31no—
92C3×#204—61no—
93C3×#204—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—81no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—41no—
102C3×#137—91no—
103C3×AGL(1,ℤ/8)—91no—
104C2×C3×SD16—91no—
105C3×#131—31no—
106C3×#131—31no—
107C3×#131—61no—
108C3×#131C3×D311yescharacteristic
109#951—31no—
110#951—91no—
111#949—31no—
112#949—91no—
113C2×#204—31no—
114C2×#204—91no—
115C3×#108—91no—
116C3×#101—91no—
117C3×#100—91no—
118#824—31no—
119#824—91no—
120#823—31no—
121#823—91no—
122#819—31no—
123#819—91no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C32×Q8—21no—
127C32×Q8D1211yescharacteristic
128C32×Q8C3×D411yescharacteristic
129C32×D4—22no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4D1211yescharacteristic
134C32×D4C3×D411yescharacteristic
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C23×C311yescharacteristic
141C2×C3⋊Dic3—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144#634—21no—
145#634—21no—
146C2×C3×Dic3—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C32×C8—22no—
152C32×C8—21no—
153C32×C8—21no—
154C32×C8—21no—
155C32×C8—21no—
156C32×C8#6711yescharacteristic
157C32×C8#6711yescharacteristic
158C32×C8C3×D411yescharacteristic
159C32×C8C3×D411yescharacteristic
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#471—91noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)#12111yescharacteristic
168(C3⋊D3)⋊C3—81no—
169C3×#41—31no—
170C3×#41—31no—
171C3×#41—61no—
172C3×#41C2×C3×D311yescharacteristic
173C2×C3×Q8—91no—
174C2×C3×D4—91no—
175C2×D12—31no—
176C2×D12—91no—
177C2×Dic6—31no—
178C2×Dic6—91no—
179C3×Q16—181no—
180C3×SD16—181no—
181C3×SD16—181no—
182C3×SD16—181no—
183C3×SD16—181no—
184C3×D8—181no—
185C3×M16—21no—
186C3×M16—31no—
187C3×M16—31no—
188C3×M16—61no—
189C3×M16—61no—
190C3×M16—62no—
191C3×M16—61no—
192C3×M16—91no—
193C3×M16—91no—
194C3×M16C2×C3×D311yescharacteristic
195C2×C3×C8—21no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—61no—
200C2×C3×C8—62no—
201C2×C3×C8—61no—
202C2×C3×C8C2×C3×D311yescharacteristic
203#216—61no—
204#216—181no—
205#215—61no—
206#215—181no—
207#214—61no—
208#214—181no—
209#213—61no—
210#213—181no—
211#208—31no—
212#208—31no—
213#208—91no—
214#208—91no—
215#204—61no—
216#204—61no—
217#204—181no—
218#204—181no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32#23511yescharacteristic
224C22×C32C3×#4111yescharacteristic
225C2×C3⋊D3—41no—
226C2×C3⋊D3—81no—
227C2×C3×D3—121no—
228C2×C3×D3—241no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4C2×D1211yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C32×C4C2×C3×D411yescharacteristic
237C32×C4C2×C3×D411yescharacteristic
238C3⋊Dic3—41no—
239C3×Dic3—121no—
240#137—91no—
241AGL(1,ℤ/8)—91no—
242C2×SD16—91no—
243#131(C3⋊D3)⋊C311yescharacteristic, complemented
244#108—91no—
245#101—91no—
246#100—91no—
247H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—181no—
251C3×Q8—31no—
252C3×Q8—31no—
253C3×Q8—61no—
254C3×Q8—181no—
255C3×Q8—181no—
256C3×Q8C3×D1211yescharacteristic
257C3×D4—21no—
258C3×D4—31no—
259C3×D4—31no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—62no—
263C3×D4—61no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4C3×D1211yescharacteristic
267C2×C3×C4—21no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—181no—
275C2×C3×C4C22×C3×D311yescharacteristic
276C2×Dic3—61no—
277C2×Dic3—181no—
278D12—61no—
279D12—61no—
280D12—181no—
281D12—181no—
282Dic6—61no—
283Dic6—61no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—21no—
287C3×C8—31no—
288C3×C8—31no—
289C3×C8—31no—
290C3×C8—31no—
291C3×C8—61no—
292C3×C8—61no—
293C3×C8—62no—
294C3×C8—61no—
295C3×C8—61no—
296C3×C8—181no—
297C3×C8—181no—
298C3×C8C3×#6711yescharacteristic
299C3×C8C3×#6711yescharacteristic
300C3⋊C8—61no—
301C3⋊C8—61no—
302C3⋊C8—181no—
303C3⋊C8—181no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32#93011yescharacteristic
312C2×C32C3×#12111yescharacteristic
313C3⋊D3—81no—
314C3×D3—241no—
315#41C2×(C3⋊D3)⋊C311yescharacteristic
316C2×Q8—91no—
317C2×D4—91no—
318Q16—181no—
319SD16—181no—
320SD16—181no—
321SD16—181no—
322SD16—181no—
323D8—181no—
324M16—21no—
325M16—91no—
326M16—91no—
327M16C2×(C3⋊D3)⋊C311yescharacteristic
328C2×C8—21no—
329C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
330C22×C3—21no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—361no—
338C22×C3—721no—
339C22×C3C3×#23511yescharacteristic
340C2×D3—121no—
341C2×D3—241no—
342C2×D3—361no—
343C2×D3—721no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—62no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—361no—
355C3×C4C2×C3×D1211yescharacteristic
356C3×C4C2×C3×#6711yescharacteristic
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#514111yescharacteristic, complemented
361C32C3×#47111yescharacteristic, complemented
362C23—181no—
363Q8—181no—
364Q8—181no—
365Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
366D4—21no—
367D4—181no—
368D4—181no—
369D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
370C2×C4—21no—
371C2×C4—181no—
372C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
373C8—21no—
374C8—181no—
375C8—181no—
376C8#638⋊C311yescharacteristic
377C8#638⋊C311yescharacteristic
378C2×C3—21no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—41no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—121no—
386C2×C3—121no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—721no—
390C2×C3C3×#93011yescharacteristic
391D3—241no—
392D3—721no—
393C22—21no—
394C22—361no—
395C22—721no—
396C22#3828⋊C311yescharacteristic
397C4—21no—
398C4—361no—
399C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
400C4C2×#638⋊C311yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3C3×#514111yescharacteristic
405C2—21no—
406C2—41no—
407C2—721no—
408C2#64139⋊C311yescharacteristic
subgroupG/H
centerC2#64139⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C3×#930
prime radical#64158⋊C3C2
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#131C2semidirect
3. #64160⋊C3C2semidirect
4. C2×#3744⋊C3C2semidirect
5. #63650⋊C3C2semidirect
6. #63651⋊C3C2semidirect
7. #63646⋊C3C2semidirect
8. #10592840C3semidirect
26. C32×#131D3semidirect
27. C32×#131C2×C3semidirect
28. #64160C2×C3semidirect
30. C2×#3744C2×C3semidirect
34. #63651C2×C3semidirect
35. #63650C2×C3semidirect
36. #63646C2×C3semidirect
243. #131(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#471semidirect
360. C32#5141semidirect
361. C32C3×#471semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1312
#64158⋊C32
#64160⋊C32
C2×#3744⋊C32
#63650⋊C32
#63651⋊C32
#63646⋊C32
#105928403
C3×#51413