Finite Group Databasebeta

← #10592844⋊C3 #13143261

Subgroups — #10592844⋊C3 #13143261

Order 1728.

426 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
2#64159⋊C3C211yesmaximal, characteristic
3#64110⋊C3C211yesmaximal, characteristic, complemented
4#63664⋊C3C211yesmaximal, characteristic, complemented
5#63652⋊C3C211yesmaximal, characteristic, complemented
6#63649⋊C3C211yesmaximal, characteristic, complemented
7#63642⋊C3C211yesmaximal, characteristic, complemented
8#10592844C311yesmaximal, characteristic, complemented
9C3×#5145—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12M16×H(3,𝔽3)—21no—
13M16×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3C411yescharacteristic, complemented
17#3756⋊C3C411yescharacteristic, complemented
18#3753⋊C3C411yescharacteristic, complemented
19#3754⋊C3C411yescharacteristic, complemented
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#3748⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25C32×#131—21no—
26C32×#131D311yescharacteristic, complemented
27C32×#131C2×C311yescharacteristic, complemented
28#64159C2×C311yescharacteristic
29#64110C2×C311yescharacteristic, complemented
30C3×#950—31no—
31C3×#901—31no—
32#63664C2×C311yescharacteristic, complemented
33#63652C2×C311yescharacteristic, complemented
34#63649C2×C311yescharacteristic, complemented
35#63642C2×C311yescharacteristic, complemented
36C3×#837—31no—
37C3×#825—31no—
38C3×#822—31no—
39C3×#815—31no—
40Q8×H(3,𝔽3)C2×C411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50C8×H(3,𝔽3)—21no—
51C8×H(3,𝔽3)D411yescharacteristic
52C8×H(3,𝔽3)D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
56C3×#444—91no—
57#5145—31no—
58#5145—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C22×C311yescharacteristic
62#3828C22×C311yescharacteristic
63C3×#235—31no—
64C32×M16—22no—
65C32×M16—21no—
66C32×M16—21no—
67C32×M16—21no—
68C32×M16C2×D311yescharacteristic
69C32×M16C22×C311yescharacteristic
70C2×C32×C8—22no—
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76#3756C3×C411yescharacteristic, complemented
77#3755C3×C411yescharacteristic, complemented
78#3754C3×C411yescharacteristic, complemented
79#3753C3×C411yescharacteristic, complemented
80C4×C3⋊Dic3C22×C311yescharacteristic
81#3748C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83#3743—21no—
84C8×C3⋊D3—21no—
85C3×#216—31no—
86C3×#215—31no—
87C3×#214—31no—
88C3×#213—31no—
89C3×C4×Dic3—31no—
90C3×#208—31no—
91C2×C3×C3⋊C8—31no—
92C3×ΓL(1,𝔽25)—61no—
93C3×D3×C8—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102C3×#135—91no—
103C3×#131—31no—
104C3×#131—31no—
105C3×#131—61no—
106C3×#131—91no—
107C3×#131C3×D311yescharacteristic
108#950—31no—
109#950—91no—
110#901—31no—
111#901—91no—
112C3×#108—91no—
113C3×#104—91no—
114C3×#104—91no—
115C3×#97—91no—
116#837—31no—
117#837—91no—
118#825—31no—
119#825—91no—
120#822—31no—
121#822—91no—
122#815—31no—
123#815—91no—
124C32×Q8—21no—
125C32×Q8C4×D311yescharacteristic
126C32×Q8C2×C3×C411yescharacteristic
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4C4×D311yescharacteristic
132C32×D4C2×C3×C411yescharacteristic
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139#638—21no—
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)C2×C3×C411yescharacteristic
142C4×C3⋊D3—21no—
143#634C2×C3×C411yescharacteristic
144C3×#67—61no—
145C2×C3×Dic3—61no—
146C3×D12—31no—
147C3×C4×D3—61no—
148C3×Dic6—31no—
149C32×C8—22no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—21no—
153C32×C8—21no—
154C32×C8#6711yescharacteristic
155C32×C8#6711yescharacteristic
156C32×C8C3×D411yescharacteristic
157C32×C8C3×D411yescharacteristic
158C3⋊(C3⋊C8)—21no—
159C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
160C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
161C3×C3⋊C8—31no—
162C3×C3⋊C8—31no—
163C3×C3⋊C8—61no—
164#444—91noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)C4×D411yescharacteristic
168(C3⋊D3)⋊C3—41no—
169C3×#41—31no—
170C3×#41—31no—
171C3×#41—61no—
172C3×#41—91no—
173C3×#41C2×C3×D311yescharacteristic
174#235—31no—
175#235—91no—
176C3×Q16—91no—
177C3×SD16—91no—
178C3×SD16—91no—
179C3×D8—91no—
180C3×M16—21no—
181C3×M16—31no—
182C3×M16—31no—
183C3×M16—61no—
184C3×M16—61no—
185C3×M16—62no—
186C3×M16—61no—
187C3×M16—91no—
188C3×M16—181no—
189C3×M16C2×C3×D311yescharacteristic
190C2×C3×C8—21no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—61no—
194C2×C3×C8—61no—
195C2×C3×C8—62no—
196C2×C3×C8—61no—
197C2×C3×C8—91no—
198C2×C3×C8—181no—
199C2×C3×C8C2×C3×D311yescharacteristic
200C3×C42—91no—
201#216—31no—
202#216—91no—
203#215—31no—
204#215—91no—
205#214—31no—
206#214—91no—
207#213—31no—
208#213—91no—
209C4×Dic3—31no—
210C4×Dic3—91no—
211#208—31no—
212#208—91no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—91no—
215ΓL(1,𝔽25)—61no—
216ΓL(1,𝔽25)—181no—
217D3×C8—61no—
218D3×C8—181no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32#23511yescharacteristic
224C22×C32C3×#4111yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3×D3—61no—
227C32×C4—22no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4C2×C4×D311yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C32×C4C22×C3×C411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3⋊Dic3—21no—
237C3⋊Dic3—41no—
238C3×Dic3—61no—
239C3×Dic3—121no—
240#135—91no—
241#131—91no—
242#131(C3⋊D3)⋊C311yescharacteristic, complemented
243#108—91no—
244#104—91no—
245#104—91no—
246#97—91no—
247H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
248C3×Q8—31no—
249C3×Q8—31no—
250C3×Q8—61no—
251C3×Q8—91no—
252C3×Q8C3×C4×D311yescharacteristic
253C3×D4—21no—
254C3×D4—31no—
255C3×D4—31no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—61no—
260C3×D4—91no—
261C3×D4—181no—
262C3×D4C3×C4×D311yescharacteristic
263C2×C3×C4—21no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C22×C3×D311yescharacteristic
273#67—61no—
274#67—181no—
275C2×Dic3—61no—
276C2×Dic3—181no—
277D12—31no—
278D12—91no—
279C4×D3—61no—
280C4×D3—181no—
281Dic6—31no—
282Dic6—91no—
283C3×C8—21no—
284C3×C8—31no—
285C3×C8—31no—
286C3×C8—31no—
287C3×C8—31no—
288C3×C8—61no—
289C3×C8—61no—
290C3×C8—62no—
291C3×C8—61no—
292C3×C8—61no—
293C3×C8—91no—
294C3×C8—91no—
295C3×C8—181no—
296C3×C8C3×#6711yescharacteristic
297C3×C8C3×#6711yescharacteristic
298C3⋊C8—31no—
299C3⋊C8—31no—
300C3⋊C8—61no—
301C3⋊C8—91no—
302C3⋊C8—91no—
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×C32C4×#6711yescharacteristic
312C2×C32C3×C4×D411yescharacteristic
313C3⋊D3—41no—
314C3×D3—121no—
315#41—91no—
316#41C2×(C3⋊D3)⋊C311yescharacteristic
317Q16—91no—
318SD16—91no—
319SD16—91no—
320D8—91no—
321M16—21no—
322M16—91no—
323M16—181no—
324M16C2×(C3⋊D3)⋊C311yescharacteristic
325C2×C8—21no—
326C2×C8—91no—
327C2×C8—181no—
328C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
329C42—91no—
330C22×C3—21no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—181no—
338C22×C3C3×#23511yescharacteristic
339C2×D3—61no—
340C2×D3—181no—
341C3×C4—21no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—62no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—181no—
352C3×C4—361no—
353C3×C4C2×C3×C4×D311yescharacteristic
354C3×C4C2×C3×#6711yescharacteristic
355Dic3—61no—
356Dic3—121no—
357Dic3—181no—
358Dic3—361no—
359C32—21no—
360C32#514511yescharacteristic, complemented
361C32C3×#44411yescharacteristic, complemented
362Q8—91no—
363Q8C4×(C3⋊D3)⋊C311yescharacteristic
364D4—21no—
365D4—91no—
366D4—181no—
367D4C4×(C3⋊D3)⋊C311yescharacteristic
368C2×C4—21no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
372C8—21no—
373C8—91no—
374C8—91no—
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—361no—
390C2×C3C3×C4×#6711yescharacteristic
391D3—121no—
392D3—361no—
393C22—21no—
394C22—181no—
395C22#3828⋊C311yescharacteristic
396C4—21no—
397C4—181no—
398C4—361no—
399C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
400C4C2×#638⋊C311yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3C3×#514511yescharacteristic
405C2—21no—
406C2—41no—
407C2—361no—
408C2C4×#638⋊C311yescharacteristic
subgroupG/H
centerC4C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C3×C4×#67
prime radical#64159⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ #131

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#131C2semidirect
3. #64110⋊C3C2semidirect
4. #63664⋊C3C2semidirect
5. #63652⋊C3C2semidirect
6. #63649⋊C3C2semidirect
7. #63642⋊C3C2semidirect
8. #10592844C3semidirect
16. #3755⋊C3C4semidirect
17. #3756⋊C3C4semidirect
18. #3753⋊C3C4semidirect
19. #3754⋊C3C4semidirect
26. C32×#131D3semidirect
27. C32×#131C2×C3semidirect
29. #64110C2×C3semidirect
32. #63664C2×C3semidirect
33. #63652C2×C3semidirect
34. #63649C2×C3semidirect
35. #63642C2×C3semidirect
76. #3756C3×C4semidirect
77. #3755C3×C4semidirect
78. #3754C3×C4semidirect
79. #3753C3×C4semidirect
242. #131(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#444semidirect
360. C32#5145semidirect
361. C32C3×#444semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1312
#64159⋊C32
#64110⋊C32
#63664⋊C32
#63652⋊C32
#63649⋊C32
#63642⋊C32
#105928443
C3×#51453