Finite Group Databasebeta

← #10592448⋊C3 #13141677

Subgroups — #10592448⋊C3 #13141677

Order 1728.

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

normal subgroups only

This group has 424 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, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C211yesmaximal, characteristic, complemented
4#64116⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592448C311yesmaximal, characteristic, complemented
9C3×#4749—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13M16×H(3,𝔽3)C2211yescharacteristic
14M16×H(3,𝔽3)C2211yescharacteristic
15#3753⋊C3—21no—
16#3753⋊C3—21no—
17#3753⋊C3—21no—
18#3754⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3745⋊C3—21no—
22#3744⋊C3—21no—
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25#64141C2×C311yescharacteristic, complemented
26C2×#3753C2×C311yescharacteristic, complemented
27#64117C2×C311yescharacteristic, complemented
28#64116C2×C311yescharacteristic, complemented
29C3×#932—31no—
30C2×C3×#213—31no—
31C3×#908—31no—
32C3×#907—31no—
33C32×#100—21no—
34C32×#100D311yescharacteristic, complemented
35C32×#100C2×C311yescharacteristic, complemented
36#63650C2×C311yescharacteristic, complemented
37#63649C2×C311yescharacteristic, complemented
38C3×#823—31no—
39C3×#822—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊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×#472—91no—
57#4749—31no—
58#4749—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C3×#235—31no—
65C2×C3×D12—31no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16C2×D311yescharacteristic
69C32×M16C2×D311yescharacteristic
70C32×M16C22×C311yescharacteristic
71C32×M16C22×C311yescharacteristic
72#3754—21no—
73#3753—21no—
74#3753—21no—
75#3753—21no—
76#3748C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3745—21no—
79#3744—21no—
80#3743—21no—
81C8×C3⋊D3—21no—
82C3×#214—61no—
83C3×#213—61no—
84C3×#213—61no—
85C3×#213—61no—
86C3×#208—31no—
87C2×C3×C3⋊C8—31no—
88C3×D24—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—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—81no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3—21no—
101C3×AGL(1,ℤ/8)—91no—
102C3×AGL(1,ℤ/8)—91no—
103C2×C3×D8—91no—
104C3×#131—91no—
105#932—31no—
106#932—91no—
107C2×#213—31no—
108C2×#213—91no—
109#908—31no—
110#908—91no—
111#907—31no—
112#907—91no—
113C3×#108—91no—
114C3×#100—31no—
115C3×#100—31no—
116C3×#100—61no—
117C3×#100—91no—
118C3×#100C3×D311yescharacteristic
119#823—31no—
120#823—91no—
121#822—31no—
122#822—91no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C23×C311yescharacteristic
137#638—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C3×D411yescharacteristic
141C4×C3⋊D3—21no—
142#634C3×D411yescharacteristic
143C3×#67—61no—
144C3×D12—31no—
145C3×D12—61no—
146C3×D12—61no—
147C3×C4×D3—61no—
148C3×Dic6—31no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—21no—
153C32×C8—41no—
154C32×C8—41no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)C3×D411yescharacteristic
157C3⋊(C3⋊C8)C3×D411yescharacteristic
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—61no—
161#472—91noSylow
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165(C3⋊D3)⋊C3—41no—
166(C3⋊D3)⋊C3—81no—
167C3×#41—91no—
168C2×C3×D4—31no—
169C2×C3×D4—31no—
170C2×C3×D4—61no—
171C2×C3×D4—91no—
172C2×C3×D4C2×C3×D311yescharacteristic
173#235—31no—
174#235—91no—
175C2×D12—31no—
176C2×D12—91no—
177C3×SD16—181no—
178C3×SD16—181no—
179C3×D8—181no—
180C3×D8—181no—
181C3×D8—181no—
182C3×D8—181no—
183C3×M16—31no—
184C3×M16—31no—
185C3×M16—31no—
186C3×M16—31no—
187C3×M16—61no—
188C3×M16—61no—
189C3×M16—91no—
190C3×M16—181no—
191C3×M16C2×C3×D311yescharacteristic
192C3×M16C2×C3×D311yescharacteristic
193C2×C3×C8—91no—
194C2×C3×C8—181no—
195#214—61no—
196#214—181no—
197#213—61no—
198#213—61no—
199#213—61no—
200#213—181no—
201#213—181no—
202#213—181no—
203#208—31no—
204#208—91no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—91no—
207D24—61no—
208D24—181no—
209#204—61no—
210#204—181no—
211ΓL(1,𝔽25)—61no—
212ΓL(1,𝔽25)—181no—
213D3×C8—61no—
214D3×C8—181no—
215C22×C32—21no—
216C22×C32—42no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—81no—
220C22×C32—81no—
221C22×C32—82no—
222C22×C32#23511yescharacteristic
223C22×C32C3×#4111yescharacteristic
224C2×C3⋊D3—21no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—81no—
227C2×C3×D3—61no—
228C2×C3×D3—121no—
229C2×C3×D3—241no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4D3×D411yescharacteristic
233C32×C4D3×D411yescharacteristic
234C32×C4C2×C3×D411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3⋊Dic3—21no—
237C3×Dic3—61no—
238AGL(1,ℤ/8)—91no—
239AGL(1,ℤ/8)—91no—
240C2×D8—91no—
241#131—91no—
242#108—91no—
243#100—91no—
244#100(C3⋊D3)⋊C311yescharacteristic, complemented
245H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
246C23×C3—21no—
247C23×C3—61no—
248C23×C3—61no—
249C23×C3—62no—
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—181no—
253C3×Q8—91no—
254C3×D4—21no—
255C3×D4—21no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—61no—
260C3×D4—91no—
261C3×D4—121no—
262C3×D4—121no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—181no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—181no—
270C2×C3×C4C22×C3×D311yescharacteristic
271#67—61no—
272#67—181no—
273D12—31no—
274D12—61no—
275D12—61no—
276D12—91no—
277D12—181no—
278D12—181no—
279C4×D3—61no—
280C4×D3—181no—
281Dic6—31no—
282Dic6—91no—
283C3×C8—21no—
284C3×C8—21no—
285C3×C8—61no—
286C3×C8—61no—
287C3×C8—61no—
288C3×C8—61no—
289C3×C8—91no—
290C3×C8—91no—
291C3×C8—121no—
292C3×C8—121no—
293C3×C8—181no—
294C3⋊C8—31no—
295C3⋊C8—31no—
296C3⋊C8—61no—
297C3⋊C8—91no—
298C3⋊C8—91no—
299C3⋊C8—181no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—41no—
304C2×C32—82no—
305C2×C32—81no—
306C2×C32—81no—
307C2×C32#88411yescharacteristic
308C2×C32C3×#12111yescharacteristic
309C3⋊D3—41no—
310C3⋊D3—81no—
311C3×D3—121no—
312C3×D3—241no—
313#41—91no—
314C2×D4—91no—
315C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
316SD16—181no—
317SD16—181no—
318D8—181no—
319D8—181no—
320D8—181no—
321D8—181no—
322M16—91no—
323M16—181no—
324M16C2×(C3⋊D3)⋊C311yescharacteristic
325M16C2×(C3⋊D3)⋊C311yescharacteristic
326C2×C8—91no—
327C2×C8—181no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—41no—
331C22×C3—61no—
332C22×C3—81no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—122no—
336C22×C3—181no—
337C22×C3—241no—
338C22×C3—241no—
339C22×C3—242no—
340C22×C3—361no—
341C22×C3—721no—
342C22×C3C3×#23511yescharacteristic
343C2×D3—61no—
344C2×D3—121no—
345C2×D3—181no—
346C2×D3—241no—
347C2×D3—361no—
348C2×D3—721no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—181no—
356C3×C4C3×D3×D411yescharacteristic
357C3×C4C3×D3×D411yescharacteristic
358Dic3—61no—
359Dic3—181no—
360C32—21no—
361C32#474911yescharacteristic, complemented
362C32C3×#47211yescharacteristic, complemented
363C23—21no—
364C23—181no—
365Q8—91no—
366D4—21no—
367D4—21no—
368D4—91no—
369D4—181no—
370D4—181no—
371D4—181no—
372C2×C4—181no—
373C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
374C8—21no—
375C8—21no—
376C8—91no—
377C8—91no—
378C8—181no—
379C2×C3—21no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—81no—
386C2×C3—121no—
387C2×C3—241no—
388C2×C3—241no—
389C2×C3—242no—
390C2×C3—361no—
391C2×C3—721no—
392C2×C3C3×#88411yescharacteristic
393D3—121no—
394D3—241no—
395D3—361no—
396D3—721no—
397C22—41no—
398C22—81no—
399C22—181no—
400C22—361no—
401C22—721no—
402C22#3828⋊C311yescharacteristic
403C4—181no—
404C4D4×(C3⋊D3)⋊C311yescharacteristic
405C4D4×(C3⋊D3)⋊C311yescharacteristic
406C3—31no—
407C3—31no—
408C3—61no—
409C3C3×#474911yescharacteristic
410C2—21no—
411C2—81no—
412C2—361no—
413C2—721no—
414C2#64093⋊C311yescharacteristic
subgroupG/H
centerC2#64093⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×#884
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×#472
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
1. #64141⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #64117⋊C3C2semidirect
4. #64116⋊C3C2semidirect
5. H(3,𝔽3)×#100C2semidirect
6. #63650⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592448C3semidirect
25. #64141C2×C3semidirect
26. C2×#3753C2×C3semidirect
27. #64117C2×C3semidirect
28. #64116C2×C3semidirect
34. C32×#100D3semidirect
35. C32×#100C2×C3semidirect
36. #63650C2×C3semidirect
37. #63649C2×C3semidirect
244. #100(C3⋊D3)⋊C3semidirect
245. H(3,𝔽3)#472semidirect
361. C32#4749semidirect
362. C32C3×#472semidirect

Maximal subgroups

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

subgroupindex
#64141⋊C32
C2×#3753⋊C32
#64117⋊C32
#64116⋊C32
H(3,𝔽3)×#1002
#63650⋊C32
#63649⋊C32
#105924483
C3×#47493