Finite Group Databasebeta

← C3×#68115 #11778249

Subgroups — C3×#68115 #11778249

Order 1152.

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

normal subgroups only

This group has 561 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
1C3×#5483C211yesmaximal, characteristic
2C3×#4947C211yesmaximal, characteristic
3C3×#4739C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic
5C3×#4467C212yesmaximal, complemented
6C3×#1602—31nomaximal
7#68115C311yesmaximal, characteristic, direct product factor
8C22×C3×Dic6C2211yescharacteristic
9C2×C32×C22⋊C4C2211yescharacteristic
10C2×C3×#217—21no—
11C2×C3×#217—21no—
12C3×#924—21no—
13C3×#924C2212yes—
14C2×C3×Dic12—22no—
15C3×#878—22no—
16C3×#878—22no—
17C32×C22⋊C8C2212yes—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2211yescharacteristic
20C3×#816—22no—
21C3×#543—31no—
22#5483C2×C311yescharacteristic
23C3×#451—32no—
24C3×#393—31no—
25#4947C2×C311yescharacteristic
26#4739C2×C312yescomplemented
27C3×#328—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic
30C3×#323C2×C311yescharacteristic
31#4467C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C22×C3×Dic3—21no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—42no—
42C2×C3×Dic6D412yes—
43C2×C32×C8—22no—
44C32×C22⋊C4—22no—
45C3×#217—22no—
46C3×#217—22no—
47C3×#217—42no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3D412yes—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×Dic12—44no—
53#1602—31noSylow
54C22×C3×Q8—31no—
55C22×Dic6C22×C311yescharacteristic
56C2×C3×Q16—62no—
57C3×C22⋊Q8—32no—
58C3×C22⋊Q8—62no—
59C3×C22⋊Q8—62no—
60C3×C22⋊Q8—61no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—61no—
63C2×C3×C22⋊C4—61no—
64C2×C3×C22⋊C4C2×D311yescharacteristic
65C2×C3×C22⋊C4C22×C311yescharacteristic
66C2×#217—21no—
67C2×#217—21no—
68#924—21no—
69#924C22×C312yes—
70C2×Dic12—22no—
71#878—22no—
72#878—22no—
73C3×#103—62no—
74C3×C22⋊C8—22no—
75C3×C22⋊C8C2×D312yes—
76C3×C22⋊C8C22×C312yes—
77C3×#95—31no—
78C3×#95—61no—
79C3⋊#95—21no—
80C3⋊#95C22×C311yescharacteristic
81#816—22no—
82C23×C32—21no—
83C23×C32—21no—
84C23×C32—41no—
85C23×C32—41no—
86C23×C32Q1612yescomplemented
87C23×C32C2×D411yescharacteristic
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4C2×D412yes—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—81no—
101C2×C3×Dic3—81no—
102C3×Dic6—24no—
103C3×Dic6—42no—
104C3×Dic6—42no—
105C3×Dic6—81no—
106C32×C8—42no—
107#543—31no—
108#451—32no—
109#393—31no—
110#328—32no—
111#323C3×D311yescharacteristic
112C24×C3—21no—
113C24×C3D1211yescharacteristic
114C24×C3C3×D411yescharacteristic
115C2×C3×Q8—32no—
116C2×C3×Q8—61no—
117C2×C3×Q8—62no—
118C2×C3×Q8—122no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—61no—
122C22×C3×C4—61no—
123C22×C3×C4—61no—
124C22×C3×C4—61no—
125C22×C3×C4D1211yescharacteristic
126C22×C3×C4C3×D411yescharacteristic
127C22×C3×C4C22×D311yescharacteristic
128C22×C3×C4C23×C311yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C2×Dic6—22no—
134C2×Dic6—21no—
135C2×Dic6—42no—
136C2×Dic6C3×D412yes—
137C3×Q16—124no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—42no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—62no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—62no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—62no—
149C3×C22⋊C4—62no—
150C3×C22⋊C4—122no—
151#217—22no—
152#217—22no—
153#217—42no—
154C4⋊Dic3—21no—
155C4⋊Dic3C3×D412yes—
156Dic3⋊C4—22no—
157Dic3⋊C4—22no—
158Dic12—44no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—42no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32—81no—
166C22×C32#12011yescharacteristic
167C22×C32C2×Q1611yescharacteristic
168C22×C32#13711yescharacteristic
169C32×C4—22no—
170C32×C4—41no—
171C3×Dic3—42no—
172C3×Dic3—42no—
173C3×Dic3—42no—
174C3×Dic3—81no—
175C22×Q8—31no—
176C2×Q16—62no—
177C22⋊Q8—32no—
178C22⋊Q8—62no—
179C22⋊Q8—62no—
180C22⋊Q8—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4C2×C3×D311yescharacteristic
184#103—62no—
185C22⋊C8C2×C3×D312yes—
186#95—31no—
187#95—61no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—22no—
193C23×C3—21no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—81no—
201C23×C3—81no—
202C23×C3Dic1212yescomplemented
203C23×C3C3×Q1612yescomplemented
204C23×C3C2×D1211yescharacteristic
205C23×C3C2×C3×D411yescharacteristic
206C3×Q8—64no—
207C3×Q8—122no—
208C3×Q8—122no—
209C3×Q8—241no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4—121no—
224C2×C3×C4—121no—
225C2×C3×C4—121no—
226C2×C3×C4—121no—
227C2×C3×C4—121no—
228C2×C3×C4—241no—
229C2×C3×C4—241no—
230C2×C3×C4D3×D412yes—
231C2×C3×C4C2×C3×D412yes—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242Dic6—24no—
243Dic6—42no—
244Dic6—42no—
245Dic6—81no—
246C3×C8—42no—
247C3×C8—42no—
248C3×C8—82no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—42no—
252C2×C32#45111yescharacteristic
253C2×C32#45511yescharacteristic
254C2×C32B(4,𝔽2)11yescharacteristic
255C24C3×D1211yescharacteristic
256C2×Q8—32no—
257C2×Q8—61no—
258C2×Q8—62no—
259C2×Q8—122no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4C3×D1211yescharacteristic
265C22×C4C22×C3×D311yescharacteristic
266Q16—124no—
267C2×C8—22no—
268C4⋊C4—32no—
269C4⋊C4—62no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C22⋊C4—22no—
273C22⋊C4—62no—
274C22⋊C4—62no—
275C22⋊C4—122no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—82no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—82no—
300C22×C3#88211yescharacteristic
301C22×C3C2×Dic1211yescharacteristic
302C22×C3#90911yescharacteristic
303C22×C3C3×#12011yescharacteristic
304C22×C3C2×C3×Q1611yescharacteristic
305C22×C3C3×#13711yescharacteristic
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—81no—
312C3×C4—122no—
313C3×C4—122no—
314C3×C4—122no—
315C3×C4—241no—
316Dic3—42no—
317Dic3—42no—
318Dic3—42no—
319Dic3—81no—
320C32#160211yescharacteristic, complemented, Sylow
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23C3×Dic1212yescomplemented
326C23C2×C3×D1211yescharacteristic
327Q8—64no—
328Q8—122no—
329Q8—122no—
330Q8—241no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—62no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4C3×D3×D412yes—
345C8—42no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—42no—
356C2×C3—42no—
357C2×C3—82no—
358C2×C3#473911yescharacteristic
359C2×C3#474111yescharacteristic
360C2×C3#474611yescharacteristic
361C2×C3C3×#45111yescharacteristic
362C2×C3C3×#45511yescharacteristic
363C2×C3C3×B(4,𝔽2)11yescharacteristic
364C22—21no—
365C22—21no—
366C22—22no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22C3×#88211yescharacteristic
372C22C2×C3×Dic1211yescharacteristic
373C22C3×#90911yescharacteristic
374C4—22no—
375C4—41no—
376C4—122no—
377C4—122no—
378C4—122no—
379C4—241no—
380C3—21no—
381C3#6811511yescharacteristic, direct product factor
382C3C3×#160211yescharacteristic, complemented
383C2—21no—
384C2—21no—
385C2—42no—
386C2C3×#473911yescharacteristic
387C2C3×#474111yescharacteristic
388C2C3×#474611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC24×C32D4
hypercenterC3×#323D3
hypocenterC3C3×#1602
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
3. C3×#4739C2semidirect
5. C3×#4467C2semidirect
7. #68115C3direct (N × H)
26. #4739C2×C3semidirect
31. #4467C2×C3semidirect
86. C23×C32Q16semidirect
202. C23×C3Dic12semidirect
203. C23×C3C3×Q16semidirect
320. C32#1602semidirect
325. C23C3×Dic12semidirect
381. C3#68115direct (N × H)
382. C3C3×#1602semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
C3×#49472
C3×#47392
C32×#3232
C3×#44672
C3×#16023
#681153