Finite Group Databasebeta

← #10592447⋊C3 #13141676

Subgroups — #10592447⋊C3 #13141676

Order 1728.

410 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 80 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#64141⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3#64118⋊C3C211yesmaximal, characteristic, complemented
4#64116⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592447C311yesmaximal, characteristic, complemented
9C3×#4748—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14M16×H(3,𝔽3)C2211yescharacteristic
15#3753⋊C3—21no—
16#3754⋊C3—21no—
17#3754⋊C3—21no—
18#3754⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3744⋊C3—21no—
22#3743⋊C3—21no—
23C8×(C3⋊D3)⋊C3—21no—
24#3746⋊C3—21no—
25C2×#3754C2×C311yescharacteristic, complemented
26#64141C2×C311yescharacteristic
27#64118C2×C311yescharacteristic, complemented
28#64116C2×C311yescharacteristic, complemented
29C2×C3×#214—31no—
30C3×#932—31no—
31C3×#909—31no—
32C3×#907—31no—
33C32×#100—21no—
34C32×#100D311yescharacteristic, complemented
35C32×#100C2×C311yescharacteristic, complemented
36#63651C2×C311yescharacteristic, complemented
37#63649C2×C311yescharacteristic, complemented
38C3×#824—31no—
39C3×#822—31no—
40C23×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D411yescharacteristic
51C8×H(3,𝔽3)—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56C3×#471—91no—
57#4748—31no—
58#4748—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C3×#235—31no—
65C2×C3×Dic6—31no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16C2×D311yescharacteristic
69C32×M16C2×D311yescharacteristic
70C32×M16C22×C311yescharacteristic
71C32×M16C22×C311yescharacteristic
72#3754—21no—
73#3754—21no—
74#3754—21no—
75#3753—21no—
76#3748C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3746—21no—
79#3744—21no—
80#3743—21no—
81C8×C3⋊D3—21no—
82C3×#214—61no—
83C3×#214—61no—
84C3×#214—61no—
85C3×#213—61no—
86C3×#208—31no—
87C2×C3×C3⋊C8—31no—
88C3×Dic12—61no—
89C3×#204—61no—
90C3×ΓL(1,𝔽25)—61no—
91C3×D3×C8—61no—
92C22×H(3,𝔽3)—41no—
93C22×H(3,𝔽3)—81no—
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×(C3⋊D3)⋊C3—21no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100C3×#137—91no—
101C3×AGL(1,ℤ/8)—91no—
102C2×C3×SD16—91no—
103C3×#131—91no—
104C2×#214—31no—
105C2×#214—91no—
106#932—31no—
107#932—91no—
108#909—31no—
109#909—91no—
110#907—31no—
111#907—91no—
112C3×#108—91no—
113C3×#101—91no—
114C3×#100—31no—
115C3×#100—31no—
116C3×#100—61no—
117C3×#100C3×D311yescharacteristic
118#824—31no—
119#824—91no—
120#822—31no—
121#822—91no—
122C23×C32—22no—
123C23×C32—21no—
124C23×C32—21no—
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—41no—
130C32×D4—41no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134#638—21no—
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)C3×D411yescharacteristic
137C4×C3⋊D3—21no—
138#634—21no—
139#634—21no—
140#634C3×D411yescharacteristic
141C3×#67—61no—
142C2×C3×Dic3—61no—
143C3×D12—31no—
144C3×C4×D3—61no—
145C3×Dic6—31no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—41no—
153C32×C8—41no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)C3×D411yescharacteristic
156C3⋊(C3⋊C8)C3×D411yescharacteristic
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—61no—
160#471—91noSylow
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)—81no—
163C2×H(3,𝔽3)#12111yescharacteristic
164(C3⋊D3)⋊C3—41no—
165C3×#41—91no—
166C2×C3×Q8—91no—
167C2×C3×D4—31no—
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C2×C3×D4C2×C3×D311yescharacteristic
171#235—31no—
172#235—91no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×Q16—181no—
176C3×SD16—181no—
177C3×SD16—181no—
178C3×SD16—181no—
179C3×SD16—181no—
180C3×D8—181no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—31no—
184C3×M16—31no—
185C3×M16—61no—
186C3×M16—61no—
187C3×M16—91no—
188C3×M16—181no—
189C3×M16C2×C3×D311yescharacteristic
190C3×M16C2×C3×D311yescharacteristic
191C2×C3×C8—91no—
192C2×C3×C8—181no—
193#214—61no—
194#214—61no—
195#214—61no—
196#214—181no—
197#214—181no—
198#214—181no—
199#213—61no—
200#213—181no—
201#208—31no—
202#208—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205Dic12—61no—
206Dic12—181no—
207#204—61no—
208#204—181no—
209ΓL(1,𝔽25)—61no—
210ΓL(1,𝔽25)—181no—
211D3×C8—61no—
212D3×C8—181no—
213C22×C32—21no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32—82no—
220C22×C32#23511yescharacteristic
221C22×C32C3×#4111yescharacteristic
222C2×C3⋊D3—21no—
223C2×C3×D3—61no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4D3×D411yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4C2×C3×D411yescharacteristic
229C32×C4C2×C3×D411yescharacteristic
230C3⋊Dic3—21no—
231C3⋊Dic3—41no—
232C3×Dic3—61no—
233C3×Dic3—121no—
234#137—91no—
235AGL(1,ℤ/8)—91no—
236C2×SD16—91no—
237#131—91no—
238#108—91no—
239#101—91no—
240#100(C3⋊D3)⋊C311yescharacteristic, complemented
241H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
242C23×C3—21no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—62no—
246C3×Q8—91no—
247C3×Q8—181no—
248C3×Q8—181no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—61no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—61no—
255C3×D4—91no—
256C3×D4—121no—
257C3×D4—121no—
258C3×D4—181no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4C22×C3×D311yescharacteristic
265#67—61no—
266#67—181no—
267C2×Dic3—61no—
268C2×Dic3—181no—
269D12—31no—
270D12—91no—
271C4×D3—61no—
272C4×D3—181no—
273Dic6—31no—
274Dic6—61no—
275Dic6—61no—
276Dic6—91no—
277Dic6—181no—
278Dic6—181no—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—61no—
284C3×C8—61no—
285C3×C8—91no—
286C3×C8—91no—
287C3×C8—121no—
288C3×C8—121no—
289C3×C8—181no—
290C3⋊C8—31no—
291C3⋊C8—31no—
292C3⋊C8—61no—
293C3⋊C8—91no—
294C3⋊C8—91no—
295C3⋊C8—181no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—41no—
300C2×C32—82no—
301C2×C32—81no—
302C2×C32—81no—
303C2×C32#88411yescharacteristic
304C2×C32C3×#12111yescharacteristic
305C3⋊D3—41no—
306C3×D3—121no—
307#41—91no—
308C2×Q8—91no—
309C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
310Q16—181no—
311SD16—181no—
312SD16—181no—
313SD16—181no—
314SD16—181no—
315D8—181no—
316M16—91no—
317M16—181no—
318M16C2×(C3⋊D3)⋊C311yescharacteristic
319M16C2×(C3⋊D3)⋊C311yescharacteristic
320C2×C8—91no—
321C2×C8—181no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—41no—
325C22×C3—61no—
326C22×C3—81no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—122no—
330C22×C3—181no—
331C22×C3—241no—
332C22×C3—241no—
333C22×C3—242no—
334C22×C3C3×#23511yescharacteristic
335C2×D3—61no—
336C2×D3—181no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—181no—
344C3×C4—361no—
345C3×C4C3×D3×D411yescharacteristic
346C3×C4C3×D3×D411yescharacteristic
347Dic3—61no—
348Dic3—121no—
349Dic3—181no—
350Dic3—361no—
351C32—21no—
352C32#474811yescharacteristic, complemented
353C32C3×#47111yescharacteristic, complemented
354C23—21no—
355Q8—91no—
356Q8—181no—
357Q8—181no—
358D4—21no—
359D4—21no—
360D4—91no—
361D4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
365C8—21no—
366C8—21no—
367C8—91no—
368C8—91no—
369C8—181no—
370C2×C3—21no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—81no—
377C2×C3—121no—
378C2×C3—241no—
379C2×C3—241no—
380C2×C3—242no—
381C2×C3—361no—
382C2×C3C3×#88411yescharacteristic
383D3—121no—
384D3—361no—
385C22—41no—
386C22—81no—
387C22—181no—
388C22#3828⋊C311yescharacteristic
389C4—181no—
390C4—361no—
391C4D4×(C3⋊D3)⋊C311yescharacteristic
392C4D4×(C3⋊D3)⋊C311yescharacteristic
393C3—31no—
394C3—31no—
395C3—61no—
396C3C3×#474811yescharacteristic
397C2—21no—
398C2—81no—
399C2—361no—
400C2#64093⋊C311yescharacteristic
subgroupG/H
centerC2#64093⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×#884
prime radical#64141⋊C3C2
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. C2×#3754⋊C3C2semidirect
3. #64118⋊C3C2semidirect
4. #64116⋊C3C2semidirect
5. H(3,𝔽3)×#100C2semidirect
6. #63651⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592447C3semidirect
25. C2×#3754C2×C3semidirect
27. #64118C2×C3semidirect
28. #64116C2×C3semidirect
34. C32×#100D3semidirect
35. C32×#100C2×C3semidirect
36. #63651C2×C3semidirect
37. #63649C2×C3semidirect
240. #100(C3⋊D3)⋊C3semidirect
241. H(3,𝔽3)#471semidirect
352. C32#4748semidirect
353. C32C3×#471semidirect

Maximal subgroups

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

subgroupindex
#64141⋊C32
C2×#3754⋊C32
#64118⋊C32
#64116⋊C32
H(3,𝔽3)×#1002
#63651⋊C32
#63649⋊C32
#105924473
C3×#47483