Finite Group Databasebeta

← C3×#73153 #11920931

Subgroups — C3×#73153 #11920931

Order 1152.

455 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×C3×#994C211yesmaximal, characteristic
2C3×#5426C211yesmaximal, characteristic, complemented
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73153C311yesmaximal, characteristic, direct product factor
5C3×#1792—31nomaximal
6C3×#994—21no—
7C3×#994C412yescomplemented
8C2×C3×#41⋊C3—21no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C3×GU(2,𝔽3)—21no—
11C3×#859—21no—
12C3×#835—41nomaximal
13C2×C3×H(5,𝔽2)D311yescharacteristic
14C2×#994—21no—
15C2×#994C2×C311yescharacteristic
16#5426C2×C311yescharacteristic, complemented
17C2×GU(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C3×#104—31no—
19C2×C3×#104—31no—
20C3×#417—31no—
21C3×#417—31no—
22C2×C3×#100—31no—
23C3×#390—31no—
24C2×C32×Q8—41no—
25C3×#41⋊C3—22no—
26C3×#41⋊C3C2×C411yescharacteristic
27C3×#41⋊C3D411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C3×C4⋊Dic3—41no—
30C2×C3×C3⋊C8—41no—
31#1792—31noSylow
32C3×H(5,𝔽2)—21no—
33C3×H(5,𝔽2)—32no—
34C3×H(5,𝔽2)—61no—
35C3×H(5,𝔽2)Dic312yescomplemented
36C2×C3×#41—21no—
37C2×C3×#41—31no—
38C2×C3×#41C2×D311yescharacteristic
39C22×C3×D4—31no—
40C22×C3×D4—61no—
41#994—22no—
42#994—21no—
43#994—22no—
44#994C3×C412yescomplemented
45C2×#41⋊C3—22no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C22×C311yescharacteristic
49C2×C3×M16—31no—
50C2×C3×M16—31no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C22⋊C4—62no—
53C2×C3×C42—31no—
54GU(2,𝔽3)—21no—
55#859—21no—
56C3×#104—61no—
57C3×#104—62no—
58C3×#104—61no—
59C3×#103—61no—
60C3×#103—61no—
61C3×#102—61no—
62C3×#102—61no—
63C3×#100—62no—
64#835—41no—
65C32×Q8—42no—
66C32×Q8—81no—
67C2×C32×C4—41no—
68C2×C32×C4—81no—
69C2×C3×Dic3—81no—
70C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
71C3×C3⋊C8—81no—
72C2×H(5,𝔽2)C3×D311yescharacteristic
73C2×#104—31no—
74C2×#104—31no—
75#417—31no—
76#417—31no—
77C2×#100—31no—
78#390—31no—
79C24×C3—32no—
80C3×#41—22no—
81C3×#41—31no—
82C3×#41—31no—
83C3×#41—32no—
84C3×#41—32no—
85C3×#41—61no—
86C3×#41—61no—
87C3×#41—62no—
88C3×#41C2×Dic311yescharacteristic
89C3×#41#6711yescharacteristic
90C2×C3×Q8—41no—
91C2×C3×Q8—81no—
92C2×C3×Q8#6711yescharacteristic
93C2×C3×Q8S411yescharacteristic
94C2×C3×D4—31no—
95C2×C3×D4—32no—
96C2×C3×D4—32no—
97C2×C3×D4—31no—
98C2×C3×D4—61no—
99C2×C3×D4—61no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C2×C3×D4—62no—
103C2×C3×D4—62no—
104C2×C3×D4—64no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C22×C3×C4—61no—
108C22×C3×C4—61no—
109#41⋊C3—24no—
110#41⋊C3—22no—
111#41⋊C3—21no—
112#41⋊C3—21no—
113#41⋊C3C2×C3×C411yescharacteristic
114#41⋊C3C3×D411yescharacteristic
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C3×D411yescharacteristic
117C3×M16—61no—
118C3×M16—61no—
119C3×M16—61no—
120C3×M16—61no—
121C2×C3×C8—61no—
122C2×C3×C8—61no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—61no—
125C3×C22⋊C4—124no—
126C3×C42—61no—
127C3×C42—61no—
128C4⋊Dic3—41no—
129C2×C3⋊C8—41no—
130C22×C32—41no—
131C32×C4—41no—
132C32×C4—41no—
133C32×C4—82no—
134C3×Dic3—161no—
135H(5,𝔽2)—21no—
136H(5,𝔽2)—32no—
137H(5,𝔽2)—61no—
138H(5,𝔽2)C3×Dic312yescomplemented
139C2×#41—21no—
140C2×#41—31no—
141C2×#41C2×C3×D311yescharacteristic
142C22×D4—31no—
143C22×D4—61no—
144C2×M16—31no—
145C2×M16—31no—
146C2×C4⋊C4—31no—
147C2×C22⋊C4—62no—
148C2×C42—31no—
149#104—61no—
150#104—62no—
151#104—61no—
152#103—61no—
153#103—61no—
154#102—61no—
155#102—61no—
156#100—62no—
157C23×C3—31no—
158C23×C3—34no—
159C23×C3—62no—
160C23×C3—61no—
161C23×C3—122no—
162C23×C3—122no—
163C3×Q8—21no—
164C3×Q8—31no—
165C3×Q8—42no—
166C3×Q8—81no—
167C3×Q8—82no—
168C3×Q8—82no—
169C3×Q8#21711yescharacteristic
170C3×Q8SL(2,ℤ/4)12yes—
171C3×D4—31no—
172C3×D4—31no—
173C3×D4—32no—
174C3×D4—62no—
175C3×D4—62no—
176C3×D4—61no—
177C3×D4—62no—
178C3×D4—62no—
179C3×D4—61no—
180C2×C3×C4—21no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—41no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—81no—
191C2×C3×C4—82no—
192C2×C3×C4—81no—
193C2×C3×C4—121no—
194C2×C3×C4—121no—
195C2×C3×C4—122no—
196C2×C3×C4—121no—
197C2×C3×C4C2×S411yescharacteristic
198C2×Dic3—81no—
199SL(2,𝔽3)—21no—
200SL(2,𝔽3)C3×C22⋊C411yescharacteristic, complemented
201C3×C8—121no—
202C3×C8—121no—
203C3⋊C8—81no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C24—32no—
208#41—22no—
209#41—31no—
210#41—31no—
211#41—32no—
212#41—32no—
213#41—61no—
214#41—61no—
215#41—62no—
216#41C2×C3×Dic311yescharacteristic
217#41C3×#6711yescharacteristic
218C2×Q8C3×#6711yescharacteristic
219C2×Q8C3×S411yescharacteristic
220C2×D4—31no—
221C2×D4—32no—
222C2×D4—32no—
223C2×D4—31no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—64no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235M16—61no—
236M16—61no—
237M16—61no—
238M16—61no—
239C2×C8—61no—
240C2×C8—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C22⋊C4—124no—
244C42—61no—
245C42—61no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—41no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—62no—
252C22×C3—81no—
253C22×C3—124no—
254C22×C3—121no—
255C22×C3—121no—
256C22×C3—242no—
257C22×C3GL(2,ℤ/4)11yescharacteristic
258C3×C4—22no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—82no—
264C3×C4—84no—
265C3×C4—81no—
266C3×C4—81no—
267C3×C4—122no—
268C3×C4—241no—
269C3×C4C2×SL(2,ℤ/4)11yescharacteristic
270C3×C4GL(2,ℤ/4)11yescharacteristic
271Dic3—161no—
272C32—41noSylow
273C23—31no—
274C23—34no—
275C23—62no—
276C23—61no—
277C23—122no—
278C23—122no—
279Q8—21no—
280Q8—31no—
281Q8C3×#21711yescharacteristic
282Q8C3×SL(2,ℤ/4)12yes—
283D4—31no—
284D4—31no—
285D4—32no—
286D4—62no—
287D4—62no—
288D4—61no—
289D4—62no—
290D4—62no—
291D4—61no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4C2×C3×S411yescharacteristic
306C8—121no—
307C8—121no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—122no—
317C2×C3#542811yescharacteristic
318C2×C3#542911yescharacteristic
319C2×C3#543211yescharacteristic
320C22—31no—
321C22—31no—
322C22—61no—
323C22—61no—
324C22—62no—
325C22—124no—
326C22—121no—
327C22—121no—
328C22—242no—
329C22C3×GL(2,ℤ/4)11yescharacteristic
330C4—22no—
331C4—31no—
332C4—31no—
333C4—122no—
334C4—241no—
335C4C2×C3×SL(2,ℤ/4)11yescharacteristic
336C4C3×GL(2,ℤ/4)11yescharacteristic
337C3—41no—
338C3—81no—
339C3#7315311yescharacteristic, direct product factor
340C2—61no—
341C2—61no—
342C2—122no—
343C2C3×#542811yescharacteristic
344C2C3×#542911yescharacteristic
345C2C3×#543211yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator#41⋊C3C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC2×C3×H(5,𝔽2)D3
prime radicalC2×C3×#994C2
cosocleC2×#41⋊C3C22×C3
hypercenterC2×C3×Q8S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5426C2semidirect
3. C2×C3×GU(2,𝔽3)C2semidirect
4. #73153C3direct (N × H)
7. C3×#994C4semidirect
16. #5426C2×C3semidirect
17. C2×GU(2,𝔽3)C2×C3semidirect
35. C3×H(5,𝔽2)Dic3semidirect
44. #994C3×C4semidirect
70. C3×SL(2,𝔽3)C22⋊C4semidirect
138. H(5,𝔽2)C3×Dic3semidirect
200. SL(2,𝔽3)C3×C22⋊C4semidirect
339. C3#73153direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C3×#54262
C2×C3×GU(2,𝔽3)2
#731533
C3×#17923
C3×#8354