Finite Group Databasebeta

← C3×#83222 #11878888

Subgroups — C3×#83222 #11878888

Order 1152.

913 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊Q8C211yesmaximal, characteristic
2C3×#5470C211yesmaximal, characteristic
3C3×#5228C214yesmaximal
4C3×#5092C212yesmaximal, complemented
5C3×#5090C214yesmaximal, complemented
6C3×#5079C212yesmaximal, complemented
7C3×#4925C211yesmaximal, characteristic
8C3×#3209—31nomaximal
9#83222C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2212yes—
11C32×C4⋊Q8—21no—
12C32×C4⋊Q8—21no—
13C32×C4⋊Q8C2212yes—
14C32×C4⋊Q8C2212yes—
15C2×C32×C4⋊C4—21no—
16C2×C32×C4⋊C4—21no—
17C2×C32×C42C2211yescharacteristic
18C2×C3×#216C2218yes—
19C3×#926—21no—
20C3×#924—21no—
21C3×#924C2212yes—
22C3×#870—21no—
23C3×#869C2211yescharacteristic
24C3×#869C2211yescharacteristic
25C3×C4×Dic6—21no—
26C3×C4×Dic6C2212yes—
27C3×#835C2218yes—
28C3×#830C2212yes—
29C3×C4⋊(C3⋊C8)C2212yes—
30C3×C4×C3⋊C8C2212yes—
31C3×#533—31no—
32C2×C3×C4⋊Q8—21no—
33C2×C3×C4⋊Q8D311yescharacteristic
34C2×C3×C4⋊Q8C2×C311yescharacteristic
35#5470C2×C311yescharacteristic
36C3×#494—32no—
37C3×#487—32no—
38C3×#462—34no—
39C3×#451—34no—
40C3×#431—31no—
41#5228C2×C314yes—
42#5092C2×C312yescomplemented
43#5090C2×C314yescomplemented
44#5079C2×C312yescomplemented
45#4925C2×C311yescharacteristic
46C2×C32×Q8—22no—
47C2×C32×Q8—44no—
48C2×C32×Q8C2314yes—
49C22×C32×C4—22no—
50C22×C32×C4D411yescharacteristic
51C22×C32×C4D411yescharacteristic
52C22×C32×C4C2311yescharacteristic
53C2×C3×Dic6—21no—
54C2×C3×Dic6C2312yes—
55C32×C4⋊C4—22no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4—22no—
58C32×C4⋊C4—22no—
59C32×C42D411yescharacteristic
60C32×C42D411yescharacteristic
61C32×C42C2311yescharacteristic
62C32×C42C2311yescharacteristic
63C3×#217—22no—
64C3×#216—48no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3C2312yes—
69C3×Dic3⋊C4—22no—
70C3×Dic3⋊C4—22no—
71C3×C4×Dic3—22no—
72C2×C3×C3⋊C8C2314yes—
73#3209—31noSylow
74C22×C3×Q8—22no—
75C22×C3×Q8C2×D312yes—
76C22×C3×Q8C22×C312yes—
77C2×C3×Q16—38no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—31no—
87C3×C4⋊Q8—31no—
88C3×C4⋊Q8C2×D312yes—
89C3×C4⋊Q8C2×D312yes—
90C3×C4⋊Q8C22×C312yes—
91C3×C4⋊Q8C22×C312yes—
92C3×#125—61no—
93C3×C22⋊Q8—32no—
94C3×C22⋊Q8—61no—
95C3×C4×Q8—32no—
96C3×C4×Q8—61no—
97C3×#117—61no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C42—21no—
105C2×C3×C42C2×D311yescharacteristic
106C2×C3×C42C22×C311yescharacteristic
107C2×#216C22×C318yes—
108#926—21no—
109#924—21no—
110#924C22×C312yes—
111#870—21no—
112#869C22×C311yescharacteristic
113#869C22×C311yescharacteristic
114C4×Dic6—21no—
115C4×Dic6C22×C312yes—
116C3×C4⋊C8—32no—
117C3×#103—38no—
118C3×C22⋊C8—32no—
119C3×C4×C8—32no—
120#835C22×C318yes—
121#830C22×C312yes—
122C4⋊(C3⋊C8)C22×C312yes—
123C4×C3⋊C8C22×C312yes—
124C23×C32C2×D411yescharacteristic
125C32×Q8—28no—
126C32×Q8—44no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—24no—
130C2×C32×C4—44no—
131C2×C32×C4Q1614yescomplemented
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2411yescharacteristic
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C3×Dic6—24no—
141C3×Dic6—42no—
142C3×C3⋊C8—24no—
143#533—31no—
144C2×C4⋊Q8C3×D311yescharacteristic
145#494—32no—
146#487—32no—
147#462—34no—
148#451—34no—
149#431—31no—
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—24no—
154C2×C3×Q8—32no—
155C2×C3×Q8—44no—
156C2×C3×Q8—44no—
157C2×C3×Q8—48no—
158C2×C3×Q8—61no—
159C2×C3×Q8C22×D314yes—
160C2×C3×Q8C23×C314yes—
161C22×C3×C4—24no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4#6711yescharacteristic
168C22×C3×C4#6711yescharacteristic
169C22×C3×C4C3×D411yescharacteristic
170C22×C3×C4C3×D411yescharacteristic
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C23×C311yescharacteristic
173C2×Dic6—21no—
174C2×Dic6C23×C312yes—
175C3×Q16—128no—
176C2×C3×C8—34no—
177C3×C4⋊C4—24no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—62no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C22⋊C4—62no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—62no—
201C3×C42#6711yescharacteristic
202C3×C42#6711yescharacteristic
203C3×C42C3×D411yescharacteristic
204C3×C42C3×D411yescharacteristic
205C3×C42C22×D311yescharacteristic
206C3×C42C22×D311yescharacteristic
207C3×C42C23×C311yescharacteristic
208C3×C42C23×C311yescharacteristic
209#217—22no—
210#216—48no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C23×C312yes—
215Dic3⋊C4—22no—
216Dic3⋊C4—22no—
217C4×Dic3—22no—
218C2×C3⋊C8C23×C314yes—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32C2×Q1612yes—
222C22×C32C22×D411yescharacteristic
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—44no—
226C32×C4C2×Q1612yes—
227C32×C4C2×Q1612yes—
228C32×C4#13712yes—
229C32×C4H(5,𝔽2)12yes—
230C3×Dic3—42no—
231C3×Dic3—42no—
232C22×Q8C2×C3×D312yes—
233C2×Q16—38no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—31no—
237C4⋊Q8—31no—
238C4⋊Q8C2×C3×D312yes—
239C4⋊Q8C2×C3×D312yes—
240#125—61no—
241C22⋊Q8—32no—
242C22⋊Q8—61no—
243C4×Q8—32no—
244C4×Q8—61no—
245#117—61no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42C2×C3×D311yescharacteristic
249C4⋊C8—32no—
250#103—38no—
251C22⋊C8—32no—
252C4×C8—32no—
253C23×C3—21no—
254C23×C3C2×#6711yescharacteristic
255C23×C3C2×C3×D411yescharacteristic
256C3×Q8—28no—
257C3×Q8—28no—
258C3×Q8—48no—
259C3×Q8—48no—
260C3×Q8—44no—
261C3×Q8—44no—
262C3×Q8—64no—
263C3×Q8—122no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—28no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—48no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4#21614yescomplemented
286C2×C3×C4C3×Q1614yescomplemented
287C2×C3×C4C2×#6711yescharacteristic
288C2×C3×C4C2×#6711yescharacteristic
289C2×C3×C4C2×#6711yescharacteristic
290C2×C3×C4C2×#6711yescharacteristic
291C2×C3×C4C2×#6711yescharacteristic
292C2×C3×C4C2×C3×D411yescharacteristic
293C2×C3×C4C2×C3×D411yescharacteristic
294C2×C3×C4C2×C3×D411yescharacteristic
295C2×C3×C4C2×C3×D411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×C3×C4C23×D311yescharacteristic
298C2×C3×C4C24×C311yescharacteristic
299C2×Dic3—22no—
300C2×Dic3—22no—
301Dic6—24no—
302Dic6—42no—
303C3×C8—64no—
304C3⋊C8—24no—
305C2×C32—22no—
306C2×C32#53511yescharacteristic
307C2×C32C22×Q1611yescharacteristic
308C2×C32C2×#13711yescharacteristic
309C2×Q8—22no—
310C2×Q8—32no—
311C2×Q8—44no—
312C2×Q8—61no—
313C2×Q8C22×C3×D314yes—
314C22×C4—22no—
315C22×C4C3×#6711yescharacteristic
316C22×C4C3×#6711yescharacteristic
317C22×C4C22×C3×D311yescharacteristic
318Q16—128no—
319C2×C8—34no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—32no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—62no—
328C4⋊C4—62no—
329C4⋊C4—61no—
330C22⋊C4—62no—
331C42—62no—
332C42C3×#6711yescharacteristic
333C42C3×#6711yescharacteristic
334C42C22×C3×D311yescharacteristic
335C42C22×C3×D311yescharacteristic
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3C2×#21612yes—
345C22×C3C2×C3×Q1612yes—
346C22×C3C22×#6711yescharacteristic
347C22×C3C22×C3×D411yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—48no—
359C3×C4—44no—
360C3×C4—44no—
361C3×C4—122no—
362C3×C4—122no—
363C3×C4#94212yes—
364C3×C4C2×#21612yes—
365C3×C4C2×#21612yes—
366C3×C4C2×C3×Q1612yes—
367C3×C4C2×C3×Q1612yes—
368C3×C4C3×#13712yes—
369C3×C4#100412yes—
370C3×C4C3×H(5,𝔽2)12yes—
371Dic3—42no—
372Dic3—42no—
373C32#320911yescharacteristic, complemented, Sylow
374C23C2×C3×#6711yescharacteristic
375Q8—28no—
376Q8—44no—
377Q8—64no—
378Q8—122no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4C3×#21614yescomplemented
386C2×C4C2×C3×#6711yescharacteristic
387C2×C4C2×C3×#6711yescharacteristic
388C2×C4C2×C3×#6711yescharacteristic
389C2×C4C2×C3×#6711yescharacteristic
390C2×C4C2×C3×#6711yescharacteristic
391C2×C4C23×C3×D311yescharacteristic
392C8—64no—
393C2×C3—24no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#580511yescharacteristic
400C2×C3C2×#94211yescharacteristic
401C2×C3C22×#21611yescharacteristic
402C2×C3C3×#53511yescharacteristic
403C2×C3C22×C3×Q1611yescharacteristic
404C2×C3C2×C3×#13711yescharacteristic
405C22—21no—
406C22—21no—
407C22C2×C3×#21612yes—
408C22C22×C3×#6711yescharacteristic
409C4—21no—
410C4—21no—
411C4—44no—
412C4—122no—
413C4—122no—
414C4C3×#94212yes—
415C4C2×C3×#21612yes—
416C4C2×C3×#21612yes—
417C4C3×#100412yes—
418C3—21no—
419C3#8322211yescharacteristic, direct product factor
420C3C3×#320911yescharacteristic, complemented
421C2—22no—
422C2C3×#580511yescharacteristic
423C2C2×C3×#94211yescharacteristic
424C2C22×C3×#21611yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4⋊Q8C2
socleC22×C32C22×D4
prime radicalC23×C32C2×D4
hypercenterC2×C3×C4⋊Q8D3
hypocenterC3C3×#3209
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. C3×#5092C2semidirect
5. C3×#5090C2semidirect
6. C3×#5079C2semidirect
9. #83222C3direct (N × H)
42. #5092C2×C3semidirect
43. #5090C2×C3semidirect
44. #5079C2×C3semidirect
131. C2×C32×C4Q16semidirect
285. C2×C3×C4#216semidirect
286. C2×C3×C4C3×Q16semidirect
373. C32#3209semidirect
385. C2×C4C3×#216semidirect
419. C3#83222direct (N × H)
420. C3C3×#3209semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊Q82
C3×#54702
C3×#52282
C3×#50922
C3×#50902
C3×#50792
C3×#49252
C3×#32093
#832223