Finite Group Databasebeta

← C2×C3×#5933 #11923754

Subgroups — C2×C3×#5933 #11923754

Order 1152.

576 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 36 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×#997C211yesmaximal, characteristic
2C3×#5933C212yesmaximal, direct product factor
3C2×#5933C311yesmaximal, characteristic, direct product factor
4C2×C3×#455—31nomaximal
5C3×#997C2211yescharacteristic
6C2×C3×SL(2,ℤ/4)—41nomaximal
7C2×C3×#226—42nomaximal
8C2×C3×H(5,𝔽2)D311yescharacteristic
9C2×#997—21no—
10C2×#997C2×C311yescharacteristic
11#5933C2×C312yesdirect product factor
12C2×C3×#137—32no—
13C2×C3×#124—31no—
14C3×#455—32no—
15C3×#455—34no—
16C3×#455—32no—
17C2×C3×#104—32no—
18C2×C3×#100—31no—
19C22×C3×A4—41no—
20C22×C3×A4—81no—
21C2×C3×SL(2,𝔽3)—42no—
22C3×SL(2,ℤ/4)—42no—
23C3×#226—44no—
24C2×#455—31noSylow
25C3×H(5,𝔽2)—31no—
26C3×H(5,𝔽2)—32no—
27C3×H(5,𝔽2)—61no—
28C3×H(5,𝔽2)C2×D311yescharacteristic
29C2×C3×#41—32no—
30C22×C3×Q8—31no—
31C22×C3×D4—31no—
32C22×C3×D4—61no—
33#997—21no—
34#997C22×C311yescharacteristic
35C2×SL(2,ℤ/4)—41no—
36C2×#226—42no—
37C3×#137—34no—
38C3×#137—34no—
39C3×#137—62no—
40C3×#137—62no—
41C2×C3×Q16—62no—
42C2×C3×SD16—62no—
43C2×C3×M16—32no—
44C3×#124—32no—
45C3×#124—32no—
46C3×#124—61no—
47C3×#124—61no—
48C2×C3×C22⋊C4—61no—
49C2×C3×C22⋊C4—61no—
50C2×C3×C42—31no—
51C3×#104—34no—
52C3×#104—34no—
53C3×#100—32no—
54C3×#100—32no—
55C2×C3×A4—42no—
56C2×C3×A4—41no—
57C2×C3×A4—81no—
58C2×C3×A4—82no—
59C2×C3×Dic3—161no—
60C3×SL(2,𝔽3)—42no—
61C2×H(5,𝔽2)C3×D311yescharacteristic
62C2×#137—32no—
63C2×#124—31no—
64#455—32no—
65#455—34no—
66#455—32no—
67C2×#104—32no—
68C2×#100—31no—
69C24×C3—21no—
70C24×C3—31no—
71C24×C3S411yescharacteristic
72C22×A4—41no—
73C22×A4—81no—
74C22×A4—82no—
75C22×A4—81no—
76C3×#41—32no—
77C3×#41—32no—
78C3×#41—32no—
79C3×#41—32no—
80C3×#41—62no—
81C3×#41—62no—
82C2×C3×Q8—32no—
83C2×C3×Q8—61no—
84C2×C3×Q8—62no—
85C2×C3×Q8—62no—
86C2×C3×Q8S412yes—
87C2×C3×D4—32no—
88C2×C3×D4—31no—
89C2×C3×D4—31no—
90C2×C3×D4—31no—
91C2×C3×D4—31no—
92C2×C3×D4—62no—
93C2×C3×D4—62no—
94C2×C3×D4—61no—
95C2×C3×D4—61no—
96C2×C3×D4—62no—
97C2×C3×D4—62no—
98C2×C3×D4—61no—
99C2×C3×D4—61no—
100C22×C3×C4—31no—
101C22×C3×C4—61no—
102C22×C3×C4—61no—
103C22×C3×C4—61no—
104C2×SL(2,𝔽3)—42no—
105C2×SL(2,𝔽3)—82no—
106SL(2,ℤ/4)—42no—
107#226—44no—
108C3×Q16—64no—
109C3×Q16—64no—
110C3×SD16—64no—
111C3×SD16—64no—
112C3×M16—34no—
113C3×M16—62no—
114C2×C3×C8—62no—
115C3×C22⋊C4—62no—
116C3×C22⋊C4—62no—
117C3×C22⋊C4—62no—
118C3×C22⋊C4—62no—
119C3×C42—32no—
120C3×C42—61no—
121C22×C32—161no—
122C3×A4—41no—
123C3×A4—81no—
124C3×Dic3—162no—
125H(5,𝔽2)—31no—
126H(5,𝔽2)—32no—
127H(5,𝔽2)—61no—
128H(5,𝔽2)C2×C3×D311yescharacteristic
129C2×#41—32no—
130C22×Q8—31no—
131C22×D4—31no—
132C22×D4—61no—
133#137—34no—
134#137—34no—
135#137—62no—
136#137—62no—
137C2×Q16—62no—
138C2×SD16—62no—
139C2×M16—32no—
140#124—32no—
141#124—32no—
142#124—61no—
143#124—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C42—31no—
147#104—34no—
148#104—34no—
149#100—32no—
150#100—32no—
151C23×C3—21no—
152C23×C3—31no—
153C23×C3—31no—
154C23×C3—31no—
155C23×C3—31no—
156C23×C3—42no—
157C23×C3—61no—
158C23×C3—61no—
159C23×C3—61no—
160C23×C3—82no—
161C23×C3—122no—
162C23×C3C2×S411yescharacteristic
163C2×A4—42no—
164C2×A4—41no—
165C2×A4—81no—
166C2×A4—82no—
167C2×A4—82no—
168C2×A4—82no—
169C2×A4—84no—
170C2×A4—81no—
171C3×Q8—32no—
172C3×Q8—64no—
173C3×Q8—122no—
174C3×Q8C2×S412yes—
175C3×D4—32no—
176C3×D4—32no—
177C3×D4—62no—
178C3×D4—62no—
179C3×D4—62no—
180C3×D4—62no—
181C3×D4—62no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—61no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—121no—
192C2×C3×C4—122no—
193C2×C3×C4—121no—
194C2×Dic3—161no—
195SL(2,𝔽3)—42no—
196SL(2,𝔽3)—82no—
197C3×C8—64no—
198C2×C32—162no—
199C2×C32—161no—
200C24—21no—
201C24—31no—
202C24C3×S411yescharacteristic
203#41—32no—
204#41—32no—
205#41—32no—
206#41—32no—
207#41—62no—
208#41—62no—
209C2×Q8—32no—
210C2×Q8—61no—
211C2×Q8—62no—
212C2×Q8—62no—
213C2×Q8C3×S412yes—
214C2×D4—32no—
215C2×D4—31no—
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—62no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C22×C4—31no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231Q16—64no—
232Q16—64no—
233SD16—64no—
234SD16—64no—
235M16—34no—
236M16—62no—
237C2×C8—62no—
238C22⋊C4—62no—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C42—32no—
243C42—61no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—41no—
247C22×C3—62no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—81no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3—122no—
255C22×C3—161no—
256C22×C3—241no—
257C22×C3—241no—
258C22×C3—321no—
259C22×C3#102011yescharacteristic
260A4—41no—
261A4—81no—
262A4—81no—
263A4—82no—
264C3×C4—32no—
265C3×C4—32no—
266C3×C4—122no—
267C3×C4—122no—
268Dic3—162no—
269C32—161noSylow
270C23—21no—
271C23—31no—
272C23—31no—
273C23—31no—
274C23—31no—
275C23—42no—
276C23—61no—
277C23—61no—
278C23—61no—
279C23—82no—
280C23—122no—
281C23C2×C3×S411yescharacteristic
282Q8—32no—
283Q8—64no—
284Q8—122no—
285Q8C2×C3×S412yes—
286D4—32no—
287D4—32no—
288D4—62no—
289D4—62no—
290D4—62no—
291D4—62no—
292D4—62no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—61no—
298C2×C4—62no—
299C2×C4—61no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—121no—
305C8—64no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—121no—
309C2×C3—121no—
310C2×C3—161no—
311C2×C3—162no—
312C2×C3—321no—
313C2×C3—322no—
314C2×C3#593312yesdirect product factor
315C2×C3C2×#102011yescharacteristic
316C22—31no—
317C22—31no—
318C22—41no—
319C22—62no—
320C22—61no—
321C22—61no—
322C22—81no—
323C22—121no—
324C22—121no—
325C22—121no—
326C22—122no—
327C22—241no—
328C22—241no—
329C22C3×#102011yescharacteristic
330C4—32no—
331C4—32no—
332C4—122no—
333C4—122no—
334C3—161no—
335C3—321no—
336C3C2×#593311yescharacteristic, direct product factor
337C2—61no—
338C2—61no—
339C2—121no—
340C2—121no—
341C2C3×#593312yesdirect product factor
342C2C2×C3×#102011yescharacteristic
subgroupG/H
center, socle, hypercenterC22×C3#1020
commutator, cosocle, hypocenter#997C22×C3
FrattiniC2C2×C3×#1020
FittingC2×C3×H(5,𝔽2)D3
prime radicalC2×C3×#997C2
LCSG ▷ #997
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#5933C2direct (N × H)
3. C2×#5933C3direct (N × H)
11. #5933C2×C3direct (N × H)
314. C2×C3#5933direct (N × H)
336. C3C2×#5933direct (N × H)
341. C2C3×#5933direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9972
C3×#59332
C2×#59333
C2×C3×#4553
C2×C3×SL(2,ℤ/4)4
C2×C3×#2264