Finite Group Databasebeta

← #10592584⋊C3 #13142226

Subgroups — #10592584⋊C3 #13142226

Order 1728.

407 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 86 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#64104⋊C3C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#63635⋊C3C211yesmaximal, characteristic, complemented
8#10592584C311yesmaximal, characteristic, complemented
9C3×#4885—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64104C2×C311yescharacteristic
22#64101C2×C311yescharacteristic, complemented
23C3×#895—31no—
24C3×#892—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#809—31no—
35C3×#808—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)—42no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#482—91no—
51#4885—31no—
52#4885—91no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
55C2×C4×C3⋊D3C22×C311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—31no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67C32×C4⋊C4C22×C311yescharacteristic
68#3752—21no—
69#3752—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4×C3⋊Dic3—21no—
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×(C2×D3)⋊C4—61no—
75C3×C4⋊Dic3—31no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
91C3×#121—91no—
92C3×#117—91no—
93#895—31no—
94#895—91no—
95#892—31no—
96#892—91no—
97C3×#107—31no—
98C3×#107—31no—
99C3×#107—61no—
100C3×#107C3×D311yescharacteristic
101C3×#106—91no—
102C3×#102—91no—
103C3×#102—91no—
104C3×C22⋊C8—91no—
105#821—31no—
106#821—91no—
107(C2×D3)⋊C8—31no—
108(C2×D3)⋊C8—91no—
109#809—31no—
110#809—91no—
111#808—31no—
112#808—91no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3C3×D411yescharacteristic
115C22×C3×D3—31no—
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3C3×D411yescharacteristic
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—42no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—122no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C32×C8—22no—
141C32×C8—21no—
142C32×C8—21no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145#482—91noSylow
146C2×H(3,𝔽3)#12311yescharacteristic
147C2×H(3,𝔽3)#13511yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149(C3⋊D3)⋊C3—41no—
150(C3⋊D3)⋊C3—81no—
151C2×C3×D4—91no—
152C2×C3×D4—181no—
153C22×C3×C4—91no—
154C2×D12—31no—
155C2×D12—61no—
156C2×D12—91no—
157C2×D12—181no—
158C2×C4×D3—31no—
159C2×C4×D3—91no—
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—61no—
163C2×C3×C8—91no—
164C2×C3×C8C2×C3×D311yescharacteristic
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—61no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4C2×C3×D311yescharacteristic
173C3×C4⋊C4C2×C3×D311yescharacteristic
174C3×C22⋊C4—181no—
175C3×C22⋊C4—181no—
176C3×C42—181no—
177(C2×D3)⋊C4—61no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—91no—
183C4×Dic3—61no—
184C4×Dic3—181no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—91no—
187C22×C32—21no—
188C22×C32D3×D411yescharacteristic
189C22×C32C2×C3×D411yescharacteristic
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—41no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—81no—
209C32×C4—81no—
210C32×C4#23511yescharacteristic
211C32×C4#23911yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C32×C4C3×#4111yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3×Dic3—61no—
217C3×Dic3—121no—
218#121—91no—
219#117—91no—
220#107(C3⋊D3)⋊C311yescharacteristic, complemented
221#106—91no—
222#102—91no—
223#102—91no—
224C22⋊C8—91no—
225H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C23×C3—181no—
228C22×D3—31no—
229C22×D3—61no—
230C22×D3—91no—
231C22×D3—181no—
232C3×D4—181no—
233C3×D4—181no—
234C3×D4—362no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—91no—
245C2×C3×C4—121no—
246C2×C3×C4—121no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4C22×C3×D311yescharacteristic
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—91no—
254C2×Dic3—181no—
255D12—61no—
256D12—61no—
257D12—122no—
258D12—181no—
259D12—181no—
260D12—362no—
261C4×D3—61no—
262C4×D3—61no—
263C4×D3—181no—
264C4×D3—181no—
265C3×C8—21no—
266C3×C8—61no—
267C3×C8—61no—
268C3×C8—62no—
269C3×C8—181no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32#89411yescharacteristic
276C2×C32#91111yescharacteristic
277C2×C32#91911yescharacteristic
278C2×C32C3×#12311yescharacteristic
279C2×C32C3×#13511yescharacteristic
280C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
281C3⋊D3—41no—
282C3⋊D3—81no—
283C3×D3—121no—
284C3×D3—241no—
285C2×D4—91no—
286C2×D4—181no—
287C22×C4—91no—
288C2×C8—91no—
289C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
290C4⋊C4—91no—
291C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
292C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
293C22⋊C4—181no—
294C22⋊C4—181no—
295C42—181no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—181no—
300C22×C3—181no—
301C22×C3—181no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3—361no—
305C22×C3C3×D3×D411yescharacteristic
306C2×D3—61no—
307C2×D3—61no—
308C2×D3—61no—
309C2×D3—121no—
310C2×D3—121no—
311C2×D3—121no—
312C2×D3—181no—
313C2×D3—181no—
314C2×D3—181no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—361no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—61no—
325C3×C4—61no—
326C3×C4—121no—
327C3×C4—121no—
328C3×C4—121no—
329C3×C4—121no—
330C3×C4—181no—
331C3×C4—241no—
332C3×C4—241no—
333C3×C4—361no—
334C3×C4C3×#23511yescharacteristic
335C3×C4C3×#23911yescharacteristic
336Dic3—61no—
337Dic3—121no—
338Dic3—181no—
339Dic3—361no—
340C32—21no—
341C32#488511yescharacteristic, complemented
342C32C3×#48211yescharacteristic, complemented
343C23—91no—
344C23—181no—
345D4—181no—
346D4—181no—
347D4—362no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
355C8—21no—
356C8—181no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—361no—
367C2×C3—721no—
368C2×C3C3×#89411yescharacteristic
369C2×C3C3×#91111yescharacteristic
370C2×C3C3×#91911yescharacteristic
371D3—121no—
372D3—241no—
373D3—361no—
374D3—721no—
375C22—181no—
376C22—181no—
377C22—181no—
378C22—361no—
379C22—361no—
380C22—361no—
381C22D4×(C3⋊D3)⋊C311yescharacteristic
382C4—41no—
383C4—41no—
384C4—181no—
385C4—361no—
386C4#3828⋊C311yescharacteristic
387C4#8872211yescharacteristic
388C3—31no—
389C3—31no—
390C3—61no—
391C3C3×#488511yescharacteristic
392C2—361no—
393C2—721no—
394C2#64103⋊C311yescharacteristic
395C2#64120⋊C311yescharacteristic
396C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radical#64104⋊C3C2
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #64101⋊C3C2semidirect
3. H(3,𝔽3)×#107C2semidirect
4. #63648⋊C3C2semidirect
5. #63647⋊C3C2semidirect
6. #63636⋊C3C2semidirect
7. #63635⋊C3C2semidirect
8. #10592584C3semidirect
22. #64101C2×C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
28. #63648C2×C3semidirect
29. #63647C2×C3semidirect
30. #63636C2×C3semidirect
31. #63635C2×C3semidirect
220. #107(C3⋊D3)⋊C3semidirect
225. H(3,𝔽3)#482semidirect
341. C32#4885semidirect
342. C32C3×#482semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
#64101⋊C32
H(3,𝔽3)×#1072
#63648⋊C32
#63647⋊C32
#63636⋊C32
#63635⋊C32
#105925843
C3×#48853