Finite Group Databasebeta

← C2×C3×#5416 #11922781

Subgroups — C2×C3×#5416 #11922781

Order 1152.

640 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 88 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×Q8×A4C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic
3C3×#5416C214yesmaximal, direct product factor
4C2×C3×A4⋊C8C211yesmaximal, characteristic
5C2×#5416C311yesmaximal, characteristic, direct product factor
6C2×C3×#451—31nomaximal
7C3×Q8×A4—21no—
8C3×Q8×A4C2212yes—
9C2×C3×C4×A4—21no—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3×SL(2,ℤ/4)—21no—
12C3×A4⋊Q8—21no—
13C3×A4⋊Q8C2212yes—
14C2×C3×#216—41nomaximal
15C3×A4⋊C8C2212yes—
16C23×C3×Q8D311yescharacteristic
17C2×Q8×A4—21no—
18C2×Q8×A4C2×C311yescharacteristic
19C2×A4⋊Q8C2×C311yescharacteristic
20C22×C3×Q16—31no—
21C22×C3×Q16—31no—
22C2×C3×C22⋊Q8—31no—
23#5416C2×C314yesdirect product factor
24C2×A4⋊C8C2×C311yescharacteristic
25C3×#451—34no—
26C3×#451—34no—
27C2×C3×#103—31no—
28C2×C3×#103—31no—
29C2×C3×C22⋊C8—31no—
30C22×C3×A4D411yescharacteristic
31C2×C32×Q8—41no—
32C2×C3×Dic6—41no—
33C3×C4×A4—22no—
34C3×C4×A4D411yescharacteristic
35C3×C4×A4C2311yescharacteristic
36C3×SL(2,ℤ/4)—22no—
37C3×#216—44no—
38C2×C3×C3⋊C8—41no—
39C2×#451—31noSylow
40C22×C3×Q8—21no—
41C22×C3×Q8—31no—
42C22×C3×Q8—31no—
43C22×C3×Q8—32no—
44C22×C3×Q8—31no—
45C22×C3×Q8—61no—
46C22×C3×Q8—61no—
47C22×C3×Q8C2×D312yes—
48C23×C3×C4—21no—
49C23×C3×C4C2×D311yescharacteristic
50Q8×A4—22no—
51Q8×A4—21no—
52Q8×A4—22no—
53Q8×A4C22×C312yes—
54C2×C4×A4—22no—
55C2×C4×A4—21no—
56C2×C4×A4—21no—
57C2×C4×A4C22×C311yescharacteristic
58C2×SL(2,ℤ/4)—21no—
59A4⋊Q8—21no—
60A4⋊Q8C22×C312yes—
61C2×C3×Q16—34no—
62C2×C3×Q16—34no—
63C2×C3×Q16—62no—
64C2×C3×Q16—62no—
65C2×C3×Q16—62no—
66C2×C3×Q16—61no—
67C2×C3×Q16—61no—
68C22×C3×C8—31no—
69C22×C3×C8—31no—
70C3×C22⋊Q8—32no—
71C3×C22⋊Q8—32no—
72C3×C22⋊Q8—61no—
73C3×C22⋊Q8—61no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—61no—
76C2×C3×C22⋊C4—61no—
77C2×#216—41no—
78A4⋊C8C22×C312yes—
79C3×#103—34no—
80C3×#103—34no—
81C3×C22⋊C8—32no—
82C3×C22⋊C8—32no—
83C2×C3×A4Q1612yescomplemented
84C2×C3×A4C2×D411yescharacteristic
85C32×Q8—42no—
86C32×Q8—81no—
87C2×C32×C4—41no—
88C2×C32×C4—81no—
89C2×C3×Dic3—81no—
90C3×Dic6—42no—
91C3×Dic6—81no—
92C3×C3⋊C8—42no—
93C23×Q8C3×D311yescharacteristic
94C22×Q16—31no—
95C22×Q16—31no—
96C2×C22⋊Q8—31no—
97#451—34no—
98#451—34no—
99C2×#103—31no—
100C2×#103—31no—
101C2×C22⋊C8—31no—
102C24×C3#6711yescharacteristic
103C22×A4—21no—
104C22×A4C3×D411yescharacteristic
105C2×C3×Q8—32no—
106C2×C3×Q8—32no—
107C2×C3×Q8—32no—
108C2×C3×Q8—32no—
109C2×C3×Q8—32no—
110C2×C3×Q8—32no—
111C2×C3×Q8—31no—
112C2×C3×Q8—41no—
113C2×C3×Q8—61no—
114C2×C3×Q8—61no—
115C2×C3×Q8—61no—
116C2×C3×Q8—62no—
117C2×C3×Q8—61no—
118C2×C3×Q8—61no—
119C2×C3×Q8—62no—
120C2×C3×Q8—61no—
121C2×C3×Q8—61no—
122C2×C3×Q8—61no—
123C2×C3×Q8—61no—
124C2×C3×Q8—64no—
125C2×C3×Q8—61no—
126C2×C3×Q8—81no—
127C2×C3×Q8S411yescharacteristic
128C22×C3×C4—22no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C22×C3×C4—62no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C22×C3×C4—61no—
138C22×C3×C4#6711yescharacteristic
139C22×C3×C4C22×D311yescharacteristic
140C2×Dic6—41no—
141C4×A4—24no—
142C4×A4—22no—
143C4×A4—21no—
144C4×A4—21no—
145C4×A4C3×D411yescharacteristic
146C4×A4C23×C311yescharacteristic
147SL(2,ℤ/4)—22no—
148C3×Q16—68no—
149C3×Q16—64no—
150C3×Q16—64no—
151C2×C3×C8—32no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C2×C3×C8—61no—
155C2×C3×C8—61no—
156C2×C3×C8—61no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—62no—
160C3×C4⋊C4—62no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163#216—44no—
164C2×C3⋊C8—41no—
165C22×C32—41no—
166C3×A4C2×Q1611yescharacteristic, complemented
167C32×C4—41no—
168C32×C4—41no—
169C32×C4—82no—
170C3×Dic3—82no—
171C22×Q8—21no—
172C22×Q8—31no—
173C22×Q8—31no—
174C22×Q8—32no—
175C22×Q8—31no—
176C22×Q8—61no—
177C22×Q8—61no—
178C22×Q8C2×C3×D312yes—
179C23×C4—21no—
180C23×C4C2×C3×D311yescharacteristic
181C2×Q16—34no—
182C2×Q16—34no—
183C2×Q16—62no—
184C2×Q16—62no—
185C2×Q16—62no—
186C2×Q16—61no—
187C2×Q16—61no—
188C22×C8—31no—
189C22×C8—31no—
190C22⋊Q8—32no—
191C22⋊Q8—32no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C22⋊C4—61no—
197#103—34no—
198#103—34no—
199C22⋊C8—32no—
200C22⋊C8—32no—
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—32no—
204C23×C3#21612yescomplemented
205C23×C3C2×#6711yescharacteristic
206C2×A4—22no—
207C2×A4—21no—
208C2×A4C3×Q1612yescomplemented
209C2×A4C2×C3×D411yescharacteristic
210C3×Q8—21no—
211C3×Q8—34no—
212C3×Q8—32no—
213C3×Q8—42no—
214C3×Q8—62no—
215C3×Q8—62no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—61no—
219C3×Q8—62no—
220C3×Q8—81no—
221C3×Q8—82no—
222C3×Q8—82no—
223C3×Q8—121no—
224C3×Q8—121no—
225C3×Q8C2×S412yes—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—41no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—81no—
244C2×C3×C4—82no—
245C2×C3×C4—81no—
246C2×C3×C4—122no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4C2×S411yescharacteristic
250C2×Dic3—81no—
251Dic6—42no—
252Dic6—81no—
253C3×C8—62no—
254C3×C8—62no—
255C3⋊C8—42no—
256C2×C32—42no—
257C2×C32—41no—
258C24C3×#6711yescharacteristic
259C2×Q8—32no—
260C2×Q8—32no—
261C2×Q8—32no—
262C2×Q8—32no—
263C2×Q8—32no—
264C2×Q8—32no—
265C2×Q8—31no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—62no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—62no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—64no—
278C2×Q8—61no—
279C2×Q8C3×S411yescharacteristic
280C22×C4—22no—
281C22×C4—31no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4C3×#6711yescharacteristic
291C22×C4C22×C3×D311yescharacteristic
292Q16—68no—
293Q16—64no—
294Q16—64no—
295C2×C8—32no—
296C2×C8—32no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—61no—
301C4⋊C4—32no—
302C4⋊C4—61no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22×C3—32no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—41no—
314C22×C3—61no—
315C22×C3—81no—
316C22×C3C2×#21611yescharacteristic, complemented
317C22×C3GL(2,ℤ/4)11yescharacteristic
318A4—21no—
319A4C2×C3×Q1611yescharacteristic, complemented
320C3×C4—22no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—62no—
326C3×C4—82no—
327C3×C4—84no—
328C3×C4—81no—
329C3×C4—81no—
330C3×C4—122no—
331C3×C4—122no—
332C3×C4GL(2,ℤ/4)11yescharacteristic
333C3×C4C22×S411yescharacteristic
334Dic3—82no—
335C32—41noSylow
336C23—31no—
337C23—31no—
338C23—32no—
339C23C3×#21612yescomplemented
340C23C2×C3×#6711yescharacteristic
341Q8—21no—
342Q8—34no—
343Q8—32no—
344Q8—62no—
345Q8—62no—
346Q8—62no—
347Q8—62no—
348Q8—61no—
349Q8—62no—
350Q8—121no—
351Q8—121no—
352Q8C2×C3×S412yes—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—61no—
364C2×C4—62no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—121no—
372C2×C4C2×C3×S411yescharacteristic
373C8—62no—
374C8—62no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—32no—
378C2×C3—42no—
379C2×C3—41no—
380C2×C3—82no—
381C2×C3—81no—
382C2×C3#541612yesdirect product factor
383C2×C3C2×GL(2,ℤ/4)11yescharacteristic
384C22—32no—
385C22—31no—
386C22—32no—
387C22—32no—
388C22—31no—
389C22—31no—
390C22—61no—
391C22C2×C3×#21611yescharacteristic, complemented
392C22C3×GL(2,ℤ/4)11yescharacteristic
393C4—22no—
394C4—31no—
395C4—31no—
396C4—62no—
397C4—122no—
398C4—122no—
399C4C3×GL(2,ℤ/4)11yescharacteristic
400C4C22×C3×S411yescharacteristic
401C3—41no—
402C3—81no—
403C3C2×#541611yescharacteristic, direct product factor
404C2—31no—
405C2—31no—
406C2—32no—
407C2C3×#541612yesdirect product factor
408C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3GL(2,ℤ/4)
commutator, cosocleC4×A4C23×C3
FrattiniC4C22×C3×S4
FittingC23×C3×Q8D3
socleC24×C3#67
prime radicalC22×C3×A4D4
hypercenterC2×C3×Q8S4
hypocenterA4C2×C3×Q16
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#5416C2direct (N × H)
5. C2×#5416C3direct (N × H)
23. #5416C2×C3direct (N × H)
83. C2×C3×A4Q16semidirect
166. C3×A4C2×Q16semidirect
204. C23×C3#216semidirect
208. C2×A4C3×Q16semidirect
316. C22×C3C2×#216semidirect
319. A4C2×C3×Q16semidirect
339. C23C3×#216semidirect
382. C2×C3#5416direct (N × H)
391. C22C2×C3×#216semidirect
403. C3C2×#5416direct (N × H)
407. C2C3×#5416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C2×C3×A4⋊Q82
C3×#54162
C2×C3×A4⋊C82
C2×#54163
C2×C3×#4513
C2×C3×#2164