Finite Group Databasebeta

← #84120⋊C3 #11921381

Subgroups — #84120⋊C3 #11921381

Order 1152.

412 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C3×#5945C211yesmaximal, characteristic
2Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
3#5819⋊C3C211yesmaximal, characteristic, complemented
4#84120C311yesmaximal, characteristic, complemented
5#2901⋊C3—31nomaximal
6C3×2−1+4⋊C3—21no—
7C3×2−1+4⋊C3C411yescharacteristic, complemented
8C3×#994C411yescharacteristic, complemented
9C2×C3×#41⋊C3—21no—
10C2×C3×#41⋊C3C2211yescharacteristic
11C3⋊(#131⋊C3)—21no—
12Dic3×SL(2,𝔽3)—21no—
13C3×#837—41nomaximal
14C3×#585C2×C311yescharacteristic
15#5945—21no—
16#5945D311yescharacteristic
17Dic3×#41C2×C311yescharacteristic, complemented
18#5819C2×C311yescharacteristic, complemented
19#568⋊C3—31no—
20C4×#41⋊C3—31no—
21#5236—31no—
22#5236—31no—
23C2×#837—31no—
24C2×#837—31no—
25C32×#41—41no—
26C3×#41⋊C3—21no—
27C3×#41⋊C3C2×C411yescharacteristic
28C3×#41⋊C3D411yescharacteristic
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)D411yescharacteristic
31C3×C4×Dic3—41no—
32C3×#208—41no—
33#2901—31noSylow
34C3×2−1+4—21no—
35C3×2−1+4—31no—
36C3×2−1+4C3×C411yescharacteristic, complemented
37C3×H(5,𝔽2)—31no—
38C3×H(5,𝔽2)—61no—
39C3×H(5,𝔽2)C3×C411yescharacteristic, complemented
40C2×C3×#41—21no—
41C2×C3×#41—31no—
42C2×C3×#41—31no—
43C2×C3×#41—61no—
44C2×C3×#41C22×C311yescharacteristic
452−1+4⋊C3—22no—
462−1+4⋊C3—21no—
472−1+4⋊C3—21no—
482−1+4⋊C3Dic311yescharacteristic, complemented
49#994—21no—
50#994Dic311yescharacteristic, complemented
51C2×#41⋊C3—22no—
52C2×#41⋊C3—21no—
53C2×#41⋊C3—21no—
54C2×#41⋊C3C2×D311yescharacteristic
55#948—21no—
56#948—61no—
57Q8×Dic3—21no—
58D4×Dic3—61no—
59#926—31no—
60C2×C4×Dic3—31no—
61C2×#208—31no—
62C2×#208—31no—
63#131⋊C3—61no—
64C4×SL(2,𝔽3)—61no—
65C3×#104—121no—
66#837—32no—
67#837—31no—
68#837—31no—
69#837—31no—
70#837A411yescharacteristic
71C32×Q8—41no—
72C32×D4—41no—
73C32×D4—81no—
74C2×C32×C4—41no—
75C2×C32×C4—81no—
76C2×C3×Dic3—81no—
77C3×SL(2,𝔽3)C22⋊C411yescharacteristic
78C3×C3⋊C8—81no—
79#585C3×D311yescharacteristic
80#568—31no—
81C4×#41—31no—
82#421—91no—
83#421—91no—
84C2×#104—91no—
85C2×#104—91no—
86C3×#41—21no—
87C3×#41—31no—
88C3×#41—31no—
89C3×#41—32no—
90C3×#41—31no—
91C3×#41—31no—
92C3×#41—31no—
93C3×#41—41no—
94C3×#41—61no—
95C3×#41—61no—
96C3×#41—61no—
97C3×#41—61no—
98C3×#41—62no—
99C3×#41—61no—
100C3×#41—61no—
101C3×#41—61no—
102C3×#41—81no—
103C3×#41C2×C3×C411yescharacteristic
104C3×#41C3×D411yescharacteristic
105C3×#41C2×A411yescharacteristic
106C2×C3×Q8—21no—
107C2×C3×Q8—31no—
108C2×C3×Q8—31no—
109C2×C3×Q8—61no—
110C2×C3×Q8C3×D411yescharacteristic
111C2×C3×D4—31no—
112C2×C3×D4—31no—
113C2×C3×D4—31no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×C3×D4—62no—
117C2×C3×D4—61no—
118C2×C3×D4—61no—
119C22×C3×C4—31no—
120C22×C3×C4—61no—
121C22×C3×C4—61no—
122C22×Dic3—61no—
123#41⋊C3—22no—
124#41⋊C3—21no—
125#41⋊C3—21no—
126#41⋊C3—21no—
127#41⋊C3C2×Dic311yescharacteristic
128#41⋊C3#6711yescharacteristic
129C2×SL(2,𝔽3)—22no—
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)—21no—
132C2×SL(2,𝔽3)#6711yescharacteristic
133C3×M16—121no—
134C3×C42—121no—
135#217—61no—
136C4⋊Dic3—61no—
137C4×Dic3—31no—
138C4×Dic3—61no—
139C4×Dic3C2×A411yescharacteristic
140#208—31no—
141#208—61no—
142#208—61no—
143#208C2×A411yescharacteristic
144C2×C3⋊C8—61no—
145C2×C3⋊C8—61no—
146C22×C32—41no—
147C22×C32—81no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4—81no—
151C3×Dic3—82no—
1522−1+4—21no—
1532−1+4—31no—
1542−1+4C3×Dic311yescharacteristic, complemented
155H(5,𝔽2)—31no—
156H(5,𝔽2)—61no—
157H(5,𝔽2)C3×Dic311yescharacteristic, complemented
158C2×#41—21no—
159C2×#41—31no—
160C2×#41—31no—
161C2×#41—61no—
162C2×#41C2×C3×D311yescharacteristic
163#131—61no—
164#131—181no—
165C2×M16—91no—
166C2×M16—91no—
167C4×Q8—61no—
168C4×D4—181no—
169#117—91no—
170C2×C42—91no—
171#104—31no—
172#104—91no—
173#104—91no—
174#104—92no—
175#104—91no—
176C23×C3—31no—
177C23×C3—61no—
178C23×C3—61no—
179C3×Q8—31no—
180C3×Q8—31no—
181C3×Q8—41no—
182C3×Q8—61no—
183C3×Q8—61no—
184C3×Q8—81no—
185C3×Q8C3×C22⋊C411yescharacteristic
186C3×Q8C4×A411yescharacteristic
187C3×D4—21no—
188C3×D4—31no—
189C3×D4—31no—
190C3×D4—32no—
191C3×D4—31no—
192C3×D4—41no—
193C3×D4—61no—
194C3×D4—61no—
195C3×D4—61no—
196C3×D4—61no—
197C3×D4—62no—
198C3×D4—61no—
199C3×D4—81no—
200C3×D4—82no—
201C3×D4—81no—
202C3×D4C4×A411yescharacteristic
203C2×C3×C4—21no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—41no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—62no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—81no—
217C2×C3×C4—82no—
218C2×C3×C4—81no—
219C2×C3×C4—241no—
220C2×C3×C4C22×A411yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—61no—
223C2×Dic3—122no—
224SL(2,𝔽3)—21no—
225SL(2,𝔽3)#21711yescharacteristic
226C3×C8—241no—
227C3⋊C8—21no—
228C3⋊C8—61no—
229C2×C32—41no—
230C2×C32—81no—
231C2×C32—161no—
232#41—21no—
233#41—31no—
234#41—31no—
235#41—32no—
236#41—31no—
237#41—31no—
238#41—31no—
239#41—61no—
240#41—61no—
241#41—61no—
242#41—61no—
243#41—62no—
244#41—61no—
245#41—61no—
246#41—61no—
247#41C2×C3×Dic311yescharacteristic
248#41C3×#6711yescharacteristic
249#41D3×A411yescharacteristic
250C2×Q8—21no—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—61no—
254C2×Q8C3×#6711yescharacteristic
255C2×D4—31no—
256C2×D4—31no—
257C2×D4—31no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—62no—
261C2×D4—61no—
262C2×D4—61no—
263C22×C4—31no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—181no—
267M16—31no—
268M16—91no—
269M16—181no—
270M16—181no—
271C2×C8—181no—
272C2×C8—181no—
273C4⋊C4—181no—
274C22⋊C4—181no—
275C42—31no—
276C42—91no—
277C42—181no—
278C22×C3—21no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—41no—
282C22×C3—61no—
283C22×C3—81no—
284C22×C3—82no—
285C22×C3—81no—
286C22×C3—121no—
287C22×C3—241no—
288C22×C3—241no—
289C22×C3D4×A411yescharacteristic
290C3×C4—21no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—61no—
296C3×C4—81no—
297C3×C4—82no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—242no—
301C3×C4C2×C4×A411yescharacteristic
302C3×C4D4×A411yescharacteristic
303Dic3—22no—
304Dic3—121no—
305C32—41noSylow
306C23—31no—
307C23—61no—
308C23—61no—
309Q8—31no—
310Q8—31no—
311Q8—61no—
312Q8—61no—
313Q8C3×#21711yescharacteristic
314Q8Dic3×A411yescharacteristic
315D4—21no—
316D4—31no—
317D4—31no—
318D4—32no—
319D4—31no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—61no—
324D4—62no—
325D4—61no—
326D4Dic3×A411yescharacteristic
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—62no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—181no—
341C2×C4—362no—
342C2×C4C2×D3×A411yescharacteristic
343C8—61no—
344C8—181no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3—121no—
353C2×C3—161no—
354C2×C3—161no—
355C2×C3—162no—
356C2×C3A4×C22⋊C411yescharacteristic
357C22—21no—
358C22—31no—
359C22—31no—
360C22—61no—
361C22—121no—
362C22—241no—
363C22—241no—
364C22A4×#6711yescharacteristic
365C4—21no—
366C4—31no—
367C4—31no—
368C4—62no—
369C4—61no—
370C4—361no—
371C4C2×Dic3×A411yescharacteristic
372C4A4×#6711yescharacteristic
373C3—41no—
374C3—81no—
375C3#2901⋊C311yescharacteristic, complemented
376C2—21no—
377C2—41no—
378C2—61no—
379C2—61no—
380C2—121no—
381C2A4×#21711yescharacteristic
subgroupG/H
centerC4A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#585C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC3×#5945C2
cosocleC2×C3×#41C22×C3
hypercenter#41D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. Dic3×#41⋊C3C2semidirect
3. #5819⋊C3C2semidirect
4. #84120C3semidirect
7. C3×2−1+4⋊C3C4semidirect
8. C3×#994C4semidirect
17. Dic3×#41C2×C3semidirect
18. #5819C2×C3semidirect
36. C3×2−1+4C3×C4semidirect
39. C3×H(5,𝔽2)C3×C4semidirect
48. 2−1+4⋊C3Dic3semidirect
50. #994Dic3semidirect
154. 2−1+4C3×Dic3semidirect
157. H(5,𝔽2)C3×Dic3semidirect
375. C3#2901⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#59452
Dic3×#41⋊C32
#5819⋊C32
#841203
#2901⋊C33
C3×#8374