Finite Group Databasebeta

← #10592456⋊C3 #13141784

Subgroups — #10592456⋊C3 #13141784

Order 1728.

470 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 112 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
1D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic
3#64096⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592456C311yesmaximal, characteristic, complemented
9C3×#4757—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3754⋊C3C414yescomplemented
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22D4×C3⋊Dic3C2×C311yescharacteristic
23C2×#3754C2×C311yescharacteristic
24#64096C2×C311yescharacteristic, complemented
25C3×D4×Dic3—31no—
26C2×C3×#214—31no—
27C3×#887—31no—
28C32×#102—21no—
29C32×#102D311yescharacteristic
30C32×#102C2×C311yescharacteristic
31#63643C2×C311yescharacteristic, complemented
32#63642C2×C311yescharacteristic, complemented
33#63634C2×C311yescharacteristic, complemented
34C3×#816—31no—
35C3×#815—31no—
36C3×#807—31no—
37C23×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51C3×#440—91no—
52#4757—31no—
53#4757—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic
57C22×C3⋊Dic3—21no—
58C2×#634C22×C311yescharacteristic
59C22×C3×Dic3—61no—
60C2×C3×Dic6—31no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C22×C311yescharacteristic
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67#3757—21no—
68#3754C3×C414yescomplemented
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70#3750—21no—
71C4×C3⋊Dic3—21no—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×#217—61no—
75C3×#214—34no—
76C3×C4⋊Dic3—31no—
77C3×Dic3⋊C4—61no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×SD16—91no—
93C3×C4×Q8—91no—
94C3×C4×D4—91no—
95D4×Dic3—31no—
96D4×Dic3—91no—
97C2×#214—31no—
98C2×#214—91no—
99#887—31no—
100#887—91no—
101C3×#107—91no—
102C3×#103—91no—
103C3×#102—31no—
104C3×#102—31no—
105C3×#102—61no—
106C3×#102C3×D311yescharacteristic
107C3×#97—91no—
108#816—31no—
109#816—91no—
110#815—31no—
111#815—91no—
112#807—31no—
113#807—91no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C32×D4—21no—
118C32×D4—21no—
119C32×D4—22no—
120C32×D4—41no—
121C32×D4C4×D312yes—
122C32×D4C2×C3×C412yes—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—42no—
132C2×C3⋊Dic3C3×D411yescharacteristic
133C2×C3⋊Dic3C3×D411yescharacteristic
134#634—21no—
135#634C2×C3×C412yescomplemented
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—122no—
141C3×Dic6—32no—
142C3×Dic6—61no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)C2×C3×C412yescomplemented
147C3×C3⋊C8—32no—
148#440—91noSylow
149C2×H(3,𝔽3)—42no—
150C2×H(3,𝔽3)C4×D411yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152C2×H(3,𝔽3)#13711yescharacteristic
153C2×C3×Q8—91no—
154C2×C3×D4—31no—
155C2×C3×D4—31no—
156C2×C3×D4—61no—
157C2×C3×D4C2×C3×D311yescharacteristic
158C22×C3×C4—181no—
159C22×Dic3—61no—
160C22×Dic3—181no—
161C2×Dic6—31no—
162C2×Dic6—91no—
163C3×SD16—94no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—91no—
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4C2×C3×D311yescharacteristic
175C3×C22⋊C4—181no—
176C3×C42—91no—
177C3×C42—181no—
178#217—61no—
179#217—181no—
180#214—34no—
181#214—94no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184Dic3⋊C4—61no—
185Dic3⋊C4—181no—
186C4×Dic3—31no—
187C4×Dic3—61no—
188C4×Dic3—91no—
189C4×Dic3—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—91no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—42no—
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
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—81no—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4C22×C3×C411yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—22no—
214C3⋊Dic3—21no—
215C3⋊Dic3—21no—
216C3⋊Dic3—41no—
217C3×Dic3—62no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221C2×SD16—91no—
222C4×Q8—91no—
223C4×D4—91no—
224#107—91no—
225#103—91no—
226#102(C3⋊D3)⋊C311yescharacteristic
227#97—91no—
228H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
229C23×C3—21no—
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—121no—
233C3×Q8—92no—
234C3×Q8—181no—
235C3×D4—21no—
236C3×D4—32no—
237C3×D4—32no—
238C3×D4—61no—
239C3×D4—61no—
240C3×D4—62no—
241C3×D4—121no—
242C3×D4C3×C4×D312yes—
243C2×C3×C4—21no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—121no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—122no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265C2×Dic3—362no—
266Dic6—32no—
267Dic6—61no—
268Dic6—92no—
269Dic6—181no—
270C3×C8—21no—
271C3×C8—61no—
272C3×C8—61no—
273C3×C8—92no—
274C3×C8—121no—
275C3⋊C8—32no—
276C3⋊C8—92no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32—42no—
282C2×C32—82no—
283C2×C32#67⋊C411yescharacteristic
284C2×C32#91111yescharacteristic
285C2×C32#91511yescharacteristic
286C2×C32C3×C4×D411yescharacteristic
287C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
288C2×C32C3×#13711yescharacteristic
289C2×Q8—91no—
290C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
291C22×C4—181no—
292SD16—94no—
293C2×C8—91no—
294C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
295C4⋊C4—91no—
296C4⋊C4—181no—
297C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
298C22⋊C4—181no—
299C42—91no—
300C42—181no—
301C22×C3—22no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—122no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—241no—
315C22×C3—241no—
316C22×C3C3×D3×D411yescharacteristic
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—41no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—121no—
325C3×C4—121no—
326C3×C4—181no—
327C3×C4—181no—
328C3×C4—182no—
329C3×C4—241no—
330C3×C4—361no—
331C3×C4C2×C3×C4×D311yescharacteristic
332C3×C4C3×#23711yescharacteristic
333Dic3—62no—
334Dic3—61no—
335Dic3—61no—
336Dic3—121no—
337Dic3—182no—
338Dic3—181no—
339Dic3—181no—
340Dic3—361no—
341C32—21no—
342C32#475711yescharacteristic, complemented
343C32C3×#44011yescharacteristic, complemented
344C23—21no—
345Q8—92no—
346Q8—181no—
347D4—21no—
348D4C4×(C3⋊D3)⋊C312yes—
349C2×C4—21no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—362no—
355C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
356C8—21no—
357C8—92no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—42no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—122no—
369C2×C3—122no—
370C2×C3—242no—
371C2×C3C3×#67⋊C411yescharacteristic
372C2×C3C3×#91111yescharacteristic
373C2×C3C3×#91511yescharacteristic
374C22—22no—
375C22—41no—
376C22—41no—
377C22D4×(C3⋊D3)⋊C311yescharacteristic
378C4—41no—
379C4—181no—
380C4—181no—
381C4—182no—
382C4—361no—
383C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
384C4#3830⋊C311yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3C3×#475711yescharacteristic
389C2—42no—
390C2#64090⋊C311yescharacteristic
391C2#64120⋊C311yescharacteristic
392C2#64124⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C32×C4C23×C3
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
3. #64096⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592456C3semidirect
16. #3754⋊C3C4semidirect
24. #64096C2×C3semidirect
31. #63643C2×C3semidirect
32. #63642C2×C3semidirect
33. #63634C2×C3semidirect
48. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
68. #3754C3×C4semidirect
135. #634C2×C3×C4semidirect
146. C3⋊(C3⋊C8)C2×C3×C4semidirect
228. H(3,𝔽3)#440semidirect
342. C32#4757semidirect
343. C32C3×#440semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
C2×#3754⋊C32
#64096⋊C32
H(3,𝔽3)×#1022
#63643⋊C32
#63642⋊C32
#63634⋊C32
#105924563
C3×#47573