Finite Group Databasebeta

← C3×#72113 #11798300

Subgroups — C3×#72113 #11798300

Order 1152.

563 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C32×#575C211yesmaximal, characteristic
2C3×#5195C212yesmaximal, complemented
3C3×#5137C211yesmaximal, characteristic, complemented
4C3×#4566C212yesmaximal, complemented
5C3×#4553C211yesmaximal, characteristic, complemented
6C3×#2191—31nomaximal
7#72113C311yesmaximal, characteristic, direct product factor
8C2×C32×#41—21no—
9C32×#137—22no—
10C32×AGL(1,ℤ/8)—22no—
11C32×#135—21no—
12C32×#135C2212yes—
13C2×C32×M16C2211yescharacteristic
14C3×#924—21no—
15C2×C3×Dic12C2211yescharacteristic, complemented
16C3×#829—22no—
17C3×#827—22no—
18C3×#817C2211yescharacteristic, complemented
19C3×#816—21no—
20C3×#812C2212yescomplemented
21C3×#575—21no—
22C3×#575D311yescharacteristic
23C3×#575C2×C311yescharacteristic
24C3×#510—32no—
25C3×#470—31no—
26#5195C2×C312yescomplemented
27#5137C2×C311yescharacteristic, complemented
28C3×#360—32no—
29C3×#349—31no—
30#4566C2×C312yescomplemented
31#4553C2×C311yescharacteristic, complemented
32C32×#41—22no—
33C32×#41—22no—
34C32×#41—41no—
35C32×#41—41no—
36C2×C32×Q8—21no—
37C2×C32×D4—21no—
38C2×C32×D4—41no—
39C22×C32×C4—41no—
40C22×C32×C4D411yescharacteristic
41C2×C3×Dic6—21no—
42C32×Q16—21no—
43C32×Q16D412yes—
44C32×SD16—22no—
45C32×SD16—22no—
46C32×D8—21no—
47C32×D8D412yes—
48C32×M16—22no—
49C2×C32×C8D411yescharacteristic
50C2×C32×C8C2311yescharacteristic
51C3×#217—41no—
52C3×C4⋊Dic3—21no—
53C3×Dic3⋊C4—41no—
54C3×Dic12—22no—
55C3×C3⋊C16—22no—
56#2191—31noSylow
57C2×C3×#41—22no—
58C2×C3×#41—21no—
59C2×C3×#41—21no—
60C3×#137—24no—
61C3×#137—22no—
62C3×#137—22no—
63C3×AGL(1,ℤ/8)—24no—
64C3×AGL(1,ℤ/8)—22no—
65C3×AGL(1,ℤ/8)—22no—
66C3×#135—22no—
67C3×#135—21no—
68C3×#135—21no—
69C3×#135—22no—
70C3×#135C2×D312yes—
71C3×#135C22×C312yes—
72C2×C3×Q16—31no—
73C2×C3×M16—21no—
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×M16C22×C311yescharacteristic
76C3×C22⋊Q8—61no—
77#924—21no—
78C2×Dic12C22×C311yescharacteristic, complemented
79C3×Q32—62no—
80C3×SD32—62no—
81C3×M32—32no—
82C3×#106—31no—
83C3×#103—61no—
84#829—22no—
85#827—22no—
86#817C22×C311yescharacteristic, complemented
87#816—21no—
88#812C22×C312yescomplemented
89C23×C32—41no—
90C23×C32D811yescharacteristic
91C32×Q8—22no—
92C32×Q8—41no—
93C32×D4—22no—
94C32×D4—41no—
95C32×D4—42no—
96C32×D4—42no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—42no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C32×C4D811yescharacteristic
103C2×C32×C4C2×D411yescharacteristic
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C32×C8—21no—
109C32×C8C2×D412yes—
110#575C3×D311yescharacteristic
111#510—32no—
112#470—31no—
113#360—32no—
114#349—31no—
115C3×#41—24no—
116C3×#41—24no—
117C3×#41—22no—
118C3×#41—22no—
119C3×#41—22no—
120C3×#41—22no—
121C3×#41—42no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—42no—
125C3×#41—41no—
126C3×#41—41no—
127C2×C3×Q8—22no—
128C2×C3×Q8—21no—
129C2×C3×Q8—21no—
130C2×C3×Q8—61no—
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—42no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4#6711yescharacteristic
142C22×C3×C4C3×D411yescharacteristic
143C2×Dic6—21no—
144C3×Q16—22no—
145C3×Q16—21no—
146C3×Q16—21no—
147C3×Q16—22no—
148C3×Q16—62no—
149C3×Q16#6712yes—
150C3×Q16C3×D412yes—
151C3×SD16—24no—
152C3×SD16—22no—
153C3×SD16—22no—
154C3×SD16—24no—
155C3×SD16—22no—
156C3×SD16—22no—
157C3×D8—22no—
158C3×D8—21no—
159C3×D8—21no—
160C3×D8—22no—
161C3×D8#6712yes—
162C3×D8C3×D412yes—
163C3×M16—24no—
164C3×M16—22no—
165C3×M16—22no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8#6711yescharacteristic
169C2×C3×C8C3×D411yescharacteristic
170C2×C3×C8C22×D311yescharacteristic
171C2×C3×C8C23×C311yescharacteristic
172C3×C4⋊C4—61no—
173C3×C4⋊C4—121no—
174C3×C22⋊C4—121no—
175#217—41no—
176C4⋊Dic3—21no—
177Dic3⋊C4—41no—
178Dic12—22no—
179C3×C16—62no—
180C3⋊C16—22no—
181C22×C32—21no—
182C22×C32—42no—
183C22×C32—41no—
184C22×C32—82no—
185C22×C32C2×D811yescharacteristic
186C32×C4—21no—
187C32×C4—42no—
188C32×C4#12011yescharacteristic
189C32×C4AGL(1,ℤ/8)11yescharacteristic
190C3×Dic3—81no—
191C3×Dic3—81no—
192C2×#41—21no—
193#137—22no—
194AGL(1,ℤ/8)—22no—
195#135—21no—
196#135C2×C3×D312yes—
197C2×Q16—31no—
198C2×M16C2×C3×D311yescharacteristic
199C22⋊Q8—61no—
200Q32—62no—
201SD32—62no—
202M32—32no—
203#106—31no—
204#103—61no—
205C23×C3—21no—
206C23×C3—42no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3#21311yescharacteristic
210C23×C3C3×D811yescharacteristic
211C3×Q8—24no—
212C3×Q8—22no—
213C3×Q8—22no—
214C3×Q8—42no—
215C3×Q8—41no—
216C3×Q8—41no—
217C3×Q8—121no—
218C3×Q8—121no—
219C3×D4—24no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—42no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—44no—
226C3×D4—42no—
227C3×D4—42no—
228C3×D4—44no—
229C3×D4—42no—
230C3×D4—42no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4#21311yescharacteristic
251C2×C3×C4C3×D811yescharacteristic
252C2×C3×C4C2×#6711yescharacteristic
253C2×C3×C4C2×C3×D411yescharacteristic
254C2×Dic3—41no—
255C2×Dic3—41no—
256Dic6—41no—
257Dic6—41no—
258C3×C8—22no—
259C3×C8—21no—
260C3×C8—21no—
261C3×C8—22no—
262C3×C8C2×#6712yes—
263C3×C8C2×C3×D412yes—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—82no—
267C2×C32#44711yescharacteristic
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—61no—
274C2×D4—21no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4C3×#6711yescharacteristic
278Q16—21no—
279Q16—62no—
280Q16C3×#6712yes—
281SD16—22no—
282SD16—22no—
283D8—21no—
284D8C3×#6712yes—
285M16—22no—
286C2×C8C3×#6711yescharacteristic
287C2×C8C22×C3×D311yescharacteristic
288C4⋊C4—61no—
289C4⋊C4—121no—
290C22⋊C4—121no—
291C16—62no—
292C22×C3—22no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—81no—
302C22×C3—84no—
303C22×C3—82no—
304C22×C3—82no—
305C22×C3C2×#21311yescharacteristic
306C22×C3C2×C3×D811yescharacteristic
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—44no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4—241no—
316C3×C4—241no—
317C3×C4#93211yescharacteristic
318C3×C4#95311yescharacteristic
319C3×C4C3×#12011yescharacteristic
320C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
321Dic3—81no—
322Dic3—81no—
323C32#219111yescharacteristic, complemented, Sylow
324C23—41no—
325C23C3×#21311yescharacteristic
326Q8—22no—
327Q8—41no—
328Q8—121no—
329Q8—121no—
330D4—22no—
331D4—41no—
332D4—42no—
333D4—42no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4C3×#21311yescharacteristic
342C2×C4C2×C3×#6711yescharacteristic
343C8—21no—
344C8C2×C3×#6712yes—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—42no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—84no—
353C2×C3—82no—
354C2×C3—82no—
355C2×C3#521711yescharacteristic
356C2×C3C3×#44711yescharacteristic
357C22—21no—
358C22—41no—
359C22—42no—
360C22—82no—
361C22C2×C3×#21311yescharacteristic
362C4—21no—
363C4—42no—
364C4—241no—
365C4—241no—
366C4C3×#93211yescharacteristic
367C4C3×#95311yescharacteristic
368C3—21no—
369C3#7211311yescharacteristic, direct product factor
370C3C3×#219111yescharacteristic, complemented
371C2—21no—
372C2—41no—
373C2—82no—
374C2C3×#521711yescharacteristic
subgroupG/H
centerC2×C3#5217
commutator, cosocleC2×C3×C8C23×C3
FrattiniC2×C8C22×C3×D3
Fitting, prime radicalC32×#575C2
socleC2×C32#447
hypercenterC3×#575D3
hypocenterC3C3×#2191
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ C3×#575

Complements

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

NH ≅ G/Ntype
2. C3×#5195C2semidirect
3. C3×#5137C2semidirect
4. C3×#4566C2semidirect
5. C3×#4553C2semidirect
7. #72113C3direct (N × H)
15. C2×C3×Dic12C22semidirect
18. C3×#817C22semidirect
20. C3×#812C22semidirect
26. #5195C2×C3semidirect
27. #5137C2×C3semidirect
30. #4566C2×C3semidirect
31. #4553C2×C3semidirect
78. C2×Dic12C22×C3semidirect
86. #817C22×C3semidirect
88. #812C22×C3semidirect
323. C32#2191semidirect
369. C3#72113direct (N × H)
370. C3C3×#2191semidirect

Maximal subgroups

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

subgroupindex
C32×#5752
C3×#51952
C3×#51372
C3×#45662
C3×#45532
C3×#21913
#721133