Finite Group Databasebeta

← #10592463⋊C3 #13141779

Subgroups — #10592463⋊C3 #13141779

Order 1728.

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

normal subgroups only

This group has 435 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#64149⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592463C311yesmaximal, characteristic, complemented
9C3×#4764—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64149C2×C311yescharacteristic
22#64099C2×C311yescharacteristic, complemented
23C3×#940—31no—
24C3×#890—31no—
25C32×#102—21no—
26C32×#102D311yescharacteristic, complemented
27C32×#102C2×C311yescharacteristic, complemented
28#63659C2×C311yescharacteristic, complemented
29#63648C2×C311yescharacteristic, complemented
30#63642C2×C311yescharacteristic, complemented
31#63636C2×C311yescharacteristic, complemented
32C3×#832—31no—
33C3×#821—31no—
34C3×#815—31no—
35C3×#809—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—42no—
43#638⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#490—91no—
52#4764—31no—
53#4764—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic
57C2×#638—21no—
58C2×#638—21no—
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×C3×#67—61no—
61C2×C3×#67—61no—
62C2×C3×D12—31no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71#3750—21no—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×Dic3⋊C4—61no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—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—21no—
91(C3⋊Dic3)⋊C3—41no—
92C3×#127—91no—
93C3×#125—91no—
94#940—31no—
95#940—91no—
96#890—31no—
97#890—91no—
98C3×#102—31no—
99C3×#102—31no—
100C3×#102—61no—
101C3×#102—91no—
102C3×#102—91no—
103C3×#102—91no—
104C3×#102C3×D311yescharacteristic
105C3×#97—91no—
106#832—31no—
107#832—91no—
108#821—31no—
109#821—91no—
110#815—31no—
111#815—91no—
112#809—31no—
113#809—91no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C22×C3⋊D3—21no—
118C22×C3×D3—61no—
119C32×D4—22no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131#638—42no—
132#638—42no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3C3×D411yescharacteristic
135C2×C3⋊Dic3C3×D411yescharacteristic
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C3×#67—122no—
139C3×#67—122no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—61no—
143C3×D12—61no—
144C3×D12—61no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—41no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—61no—
150#490—91noSylow
151C2×H(3,𝔽3)—81no—
152C2×H(3,𝔽3)#12411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155(C3⋊D3)⋊C3—81no—
156C2×C3×D4—31no—
157C2×C3×D4—31no—
158C2×C3×D4—61no—
159C2×C3×D4—91no—
160C2×C3×D4—181no—
161C2×C3×D4—181no—
162C2×C3×D4C2×C3×D311yescharacteristic
163C2×#67—61no—
164C2×#67—61no—
165C2×#67—181no—
166C2×#67—181no—
167C2×D12—31no—
168C2×D12—91no—
169C2×C3×C8—31no—
170C2×C3×C8—31no—
171C2×C3×C8—61no—
172C2×C3×C8—91no—
173C2×C3×C8C2×C3×D311yescharacteristic
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4C2×C3×D311yescharacteristic
181C3×C42—91no—
182C4⋊Dic3—31no—
183C4⋊Dic3—61no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186Dic3⋊C4—61no—
187Dic3⋊C4—181no—
188C4×Dic3—31no—
189C4×Dic3—91no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32—42no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—81no—
215C32×C4#23511yescharacteristic
216C32×C4#23711yescharacteristic
217C32×C4C3×#4111yescharacteristic
218C32×C4C3×#4111yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—21no—
221C3⋊Dic3—41no—
222C3×Dic3—61no—
223C3×Dic3—61no—
224C3×Dic3—121no—
225#127—91no—
226#125—91no—
227#102—91no—
228#102—91no—
229#102—91no—
230#102(C3⋊D3)⋊C311yescharacteristic, complemented
231#97—91no—
232H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C23×C3—181no—
238C22×D3—61no—
239C22×D3—181no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—61no—
245C3×D4—61no—
246C3×D4—62no—
247C3×D4—121no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—362no—
251C3×D4—362no—
252C2×C3×C4—21no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—121no—
261C2×C3×C4—181no—
262C2×C3×C4C22×C3×D311yescharacteristic
263#67—122no—
264#67—122no—
265#67—362no—
266#67—362no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—181no—
273D12—61no—
274D12—61no—
275D12—181no—
276D12—181no—
277C3×C8—21no—
278C3×C8—61no—
279C3×C8—61no—
280C3×C8—121no—
281C3×C8—181no—
282C3⋊C8—61no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—82no—
288C2×C32—81no—
289C2×C32—81no—
290C2×C32#88511yescharacteristic
291C2×C32#91111yescharacteristic
292C2×C32#91411yescharacteristic
293C2×C32C3×#12411yescharacteristic
294C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
295C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
296C3⋊D3—81no—
297C3×D3—241no—
298C2×D4—91no—
299C2×D4—181no—
300C2×D4—181no—
301C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
302C2×C8—91no—
303C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
304C4⋊C4—91no—
305C4⋊C4—181no—
306C4⋊C4—181no—
307C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
308C42—91no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—61no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—122no—
322C22×C3—241no—
323C22×C3—241no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3C3×D3×D411yescharacteristic
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—361no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—41no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—181no—
344C3×C4—181no—
345C3×C4—241no—
346C3×C4—361no—
347C3×C4C3×#23511yescharacteristic
348C3×C4C3×#23711yescharacteristic
349Dic3—61no—
350Dic3—61no—
351Dic3—121no—
352Dic3—181no—
353Dic3—181no—
354Dic3—361no—
355C32—21no—
356C32#476411yescharacteristic, complemented
357C32C3×#49011yescharacteristic, complemented
358C23—21no—
359C23—181no—
360D4—21no—
361D4—21no—
362D4—181no—
363D4—181no—
364D4—362no—
365D4—362no—
366C2×C4—21no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—181no—
370C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
371C8—21no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—81no—
383C2×C3—241no—
384C2×C3—241no—
385C2×C3—242no—
386C2×C3—721no—
387C2×C3C3×#88511yescharacteristic
388C2×C3C3×#91111yescharacteristic
389C2×C3C3×#91411yescharacteristic
390D3—241no—
391D3—721no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22D4×(C3⋊D3)⋊C311yescharacteristic
399C4—41no—
400C4—181no—
401C4—181no—
402C4—361no—
403C4#3828⋊C311yescharacteristic
404C4#3830⋊C311yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#476411yescharacteristic
409C2—81no—
410C2—721no—
411C2#64094⋊C311yescharacteristic
412C2#64120⋊C311yescharacteristic
413C2#64123⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radical#64149⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #64099⋊C3C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #63659⋊C3C2semidirect
5. #63648⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592463C3semidirect
22. #64099C2×C3semidirect
26. C32×#102D3semidirect
27. C32×#102C2×C3semidirect
28. #63659C2×C3semidirect
29. #63648C2×C3semidirect
30. #63642C2×C3semidirect
31. #63636C2×C3semidirect
230. #102(C3⋊D3)⋊C3semidirect
232. H(3,𝔽3)#490semidirect
356. C32#4764semidirect
357. C32C3×#490semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
#64099⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63648⋊C32
#63642⋊C32
#63636⋊C32
#105924633
C3×#47643