Finite Group Databasebeta

← C2×A4×#216 #11923368

Subgroups — C2×A4×#216 #11923368

Order 1152.

726 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 99 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×A4×Dic6C211yesmaximal, characteristic
3A4×#216C214yesmaximal, direct product factor
4C2×A4×C3⋊C8C211yesmaximal, characteristic
5C23×#216C311yesmaximal, characteristic, complemented
6C2×A4×Q16—31nomaximal
7C3×Q8×A4—21no—
8C3×Q8×A4C2212yes—
9C2×C3×C4×A4—21no—
10C2×C3×C4×A4C2211yescharacteristic
11C2×Dic3×A4—21no—
12A4×Dic6—21no—
13A4×Dic6C2212yes—
14C2×C3×#216—41nomaximal
15A4×C3⋊C8C2212yes—
16C23×C3×Q8C2×C311yescharacteristic
17C23×Dic6C2×C311yescharacteristic
18C2×Q8×A4—21no—
19C2×Q8×A4—31no—
20C2×Q8×A4D311yescharacteristic
21C22×#216—31no—
22C22×#216—31no—
23C22×#216—31no—
24C22×#216—31no—
25C22×#216—34no—
26C22×#216C2×C314yescomplemented
27C23×C3⋊C8C2×C311yescharacteristic
28A4×Q16—34no—
29C2×C8×A4—31no—
30C22×C3×A4D411yescharacteristic
31C2×C32×Q8—41no—
32C2×C3×Dic6—41no—
33C3×C4×A4—22no—
34C3×C4×A4D411yescharacteristic
35C3×C4×A4C2311yescharacteristic
36Dic3×A4—22no—
37C3×#216—44no—
38C2×C3×C3⋊C8—41no—
39C23×Q16—31noSylow
40C22×C3×Q8—21no—
41C22×C3×Q8—31no—
42C22×C3×Q8—32no—
43C22×C3×Q8—31no—
44C22×C3×Q8—61no—
45C22×C3×Q8—61no—
46C22×C3×Q8C22×C312yes—
47C23×C3×C4—21no—
48C23×C3×C4C22×C311yescharacteristic
49C23×Dic3—21no—
50C22×Dic6—21no—
51C22×Dic6—31no—
52C22×Dic6—31no—
53C22×Dic6—32no—
54C22×Dic6—61no—
55C22×Dic6—61no—
56C22×Dic6C22×C312yes—
57Q8×A4—22no—
58Q8×A4—21no—
59Q8×A4—22no—
60Q8×A4—32no—
61Q8×A4—61no—
62Q8×A4C2×D312yes—
63C2×C4×A4—22no—
64C2×C4×A4—21no—
65C2×C4×A4—21no—
66C2×C4×A4—61no—
67C2×C4×A4C2×D311yescharacteristic
68C2×C3×Q16—121no—
69C2×#216—34no—
70C2×#216—34no—
71C2×#216—32no—
72C2×#216—34no—
73C2×#216—31no—
74C2×#216—31no—
75C2×#216—34no—
76C2×#216—34no—
77C2×#216—34no—
78C2×#216—34no—
79C2×#216—34no—
80C2×#216—31no—
81C2×#216A411yescharacteristic, direct product factor
82C22×C3⋊C8—31no—
83C22×C3⋊C8—31no—
84C22×C3⋊C8—32no—
85C22×C3⋊C8C22×C312yes—
86C8×A4—32no—
87C2×C3×A4Q1612yescomplemented
88C2×C3×A4C2×D411yescharacteristic
89C32×Q8—42no—
90C32×Q8—81no—
91C2×C32×C4—41no—
92C2×C32×C4—81no—
93C2×C3×Dic3—81no—
94C3×Dic6—42no—
95C3×Dic6—81no—
96C3×C3⋊C8—42no—
97C23×Q8—31no—
98C23×Q8C3×D311yescharacteristic
99C22×Q16—34no—
100C22×Q16—91no—
101C22×Q16—91no—
102C22×Q16—94no—
103C22×Q16—91no—
104C22×Q16—91no—
105C23×C8—31no—
106C24×C3C3×D411yescharacteristic
107C22×A4—21no—
108C22×A4#6711yescharacteristic
109C2×C3×Q8—32no—
110C2×C3×Q8—32no—
111C2×C3×Q8—32no—
112C2×C3×Q8—32no—
113C2×C3×Q8—31no—
114C2×C3×Q8—41no—
115C2×C3×Q8—61no—
116C2×C3×Q8—61no—
117C2×C3×Q8—61no—
118C2×C3×Q8—62no—
119C2×C3×Q8—61no—
120C2×C3×Q8—62no—
121C2×C3×Q8—61no—
122C2×C3×Q8—64no—
123C2×C3×Q8—61no—
124C2×C3×Q8—81no—
125C2×C3×Q8—121no—
126C2×C3×Q8C2×A411yescharacteristic
127C22×C3×C4—22no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C22×C3×C4—62no—
134C22×C3×C4—61no—
135C22×C3×C4C3×D411yescharacteristic
136C22×C3×C4C23×C311yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—61no—
139C22×Dic3—62no—
140C22×Dic3—61no—
141C2×Dic6—32no—
142C2×Dic6—31no—
143C2×Dic6—32no—
144C2×Dic6—32no—
145C2×Dic6—32no—
146C2×Dic6—61no—
147C2×Dic6—61no—
148C2×Dic6—61no—
149C2×Dic6—64no—
150C2×Dic6—61no—
151C2×Dic6—62no—
152C2×Dic6—61no—
153C2×Dic6—61no—
154C2×Dic6—62no—
155C2×Dic6C2×A411yescharacteristic
156C4×A4—24no—
157C4×A4—22no—
158C4×A4—21no—
159C4×A4—21no—
160C4×A4—62no—
161C4×A4#6711yescharacteristic
162C4×A4C22×D311yescharacteristic
163C3×Q16—124no—
164C2×C3×C8—121no—
165#216—38no—
166#216—34no—
167#216—34no—
168#216—34no—
169#216C2×A414yesdirect product factor
170C2×C3⋊C8—32no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—32no—
175C2×C3⋊C8C2×A411yescharacteristic
176C22×C32—41no—
177C3×A4C2×Q1611yescharacteristic, complemented
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—82no—
181C3×Dic3—82no—
182C22×Q8—21no—
183C22×Q8—31no—
184C22×Q8—32no—
185C22×Q8—31no—
186C22×Q8—32no—
187C22×Q8—61no—
188C22×Q8—61no—
189C22×Q8—61no—
190C22×Q8—91no—
191C22×Q8—91no—
192C22×Q8—92no—
193C22×Q8—181no—
194C22×Q8—181no—
195C22×Q8C2×C3×D312yes—
196C23×C4—21no—
197C23×C4—61no—
198C23×C4C2×C3×D311yescharacteristic
199C2×Q16—31no—
200C2×Q16—91no—
201C2×Q16—94no—
202C2×Q16—94no—
203C2×Q16—91no—
204C2×Q16—94no—
205C2×Q16—94no—
206C2×Q16—94no—
207C2×Q16—92no—
208C2×Q16—91no—
209C2×Q16—94no—
210C2×Q16—94no—
211C2×Q16—94no—
212C22×C8—32no—
213C22×C8—91no—
214C22×C8—92no—
215C22×C8—91no—
216C23×C3—31no—
217C23×C3—32no—
218C23×C3—31no—
219C23×C3C3×Q1612yescomplemented
220C23×C3C2×C3×D411yescharacteristic
221C2×A4—22no—
222C2×A4—21no—
223C2×A4#21612yesdirect product factor
224C2×A4C2×#6711yescharacteristic
225C3×Q8—21no—
226C3×Q8—32no—
227C3×Q8—42no—
228C3×Q8—62no—
229C3×Q8—62no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—61no—
233C3×Q8—81no—
234C3×Q8—82no—
235C3×Q8—82no—
236C3×Q8—122no—
237C3×Q8—241no—
238C3×Q8C22×A412yes—
239C2×C3×C4—21no—
240C2×C3×C4—32no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—41no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4—62no—
254C2×C3×C4—81no—
255C2×C3×C4—82no—
256C2×C3×C4—81no—
257C2×C3×C4—241no—
258C2×C3×C4C22×A411yescharacteristic
259C2×Dic3—21no—
260C2×Dic3—62no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—61no—
264C2×Dic3—62no—
265Dic6—21no—
266Dic6—32no—
267Dic6—62no—
268Dic6—62no—
269Dic6—62no—
270Dic6—61no—
271Dic6—62no—
272Dic6C22×A412yes—
273C3×C8—122no—
274C3⋊C8—32no—
275C3⋊C8C22×A412yes—
276C2×C32—41no—
277C2×C32—42no—
278C24C3×#6711yescharacteristic
279C2×Q8—32no—
280C2×Q8—32no—
281C2×Q8—32no—
282C2×Q8—32no—
283C2×Q8—31no—
284C2×Q8—31no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—62no—
289C2×Q8—61no—
290C2×Q8—62no—
291C2×Q8—61no—
292C2×Q8—64no—
293C2×Q8—61no—
294C2×Q8—92no—
295C2×Q8—92no—
296C2×Q8—92no—
297C2×Q8—92no—
298C2×Q8—91no—
299C2×Q8—181no—
300C2×Q8—181no—
301C2×Q8—181no—
302C2×Q8—182no—
303C2×Q8—181no—
304C2×Q8—182no—
305C2×Q8—181no—
306C2×Q8—184no—
307C2×Q8—181no—
308C2×Q8D3×A411yescharacteristic
309C22×C4—22no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—61no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—62no—
318C22×C4—181no—
319C22×C4—182no—
320C22×C4—181no—
321C22×C4C3×#6711yescharacteristic
322C22×C4C22×C3×D311yescharacteristic
323Q16—34no—
324Q16—94no—
325Q16—94no—
326Q16—98no—
327Q16—94no—
328C2×C8—31no—
329C2×C8—92no—
330C2×C8—92no—
331C2×C8—92no—
332C2×C8—91no—
333C2×C8—92no—
334C22×C3—32no—
335C22×C3—31no—
336C22×C3—32no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—32no—
340C22×C3—32no—
341C22×C3—41no—
342C22×C3—81no—
343C22×C3C2×C3×Q1611yescharacteristic, complemented
344C22×C3D4×A411yescharacteristic
345A4—21no—
346A4C2×#21611yescharacteristic, direct product factor
347C3×C4—22no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—62no—
353C3×C4—82no—
354C3×C4—84no—
355C3×C4—81no—
356C3×C4—81no—
357C3×C4—242no—
358C3×C4D4×A411yescharacteristic
359C3×C4C23×A411yescharacteristic
360Dic3—22no—
361Dic3—62no—
362C32—41noSylow
363C23—31no—
364C23—32no—
365C23—31no—
366C23C3×#21612yescomplemented
367C23C2×C3×#6711yescharacteristic
368Q8—21no—
369Q8—32no—
370Q8—32no—
371Q8—62no—
372Q8—62no—
373Q8—62no—
374Q8—62no—
375Q8—61no—
376Q8—61no—
377Q8—92no—
378Q8—182no—
379Q8—182no—
380Q8—182no—
381Q8—182no—
382Q8—181no—
383Q8C2×D3×A412yes—
384C2×C4—21no—
385C2×C4—32no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—62no—
399C2×C4—182no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4—181no—
403C2×C4—182no—
404C2×C4C2×D3×A411yescharacteristic
405C8—32no—
406C8—92no—
407C2×C3—32no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—41no—
411C2×C3—42no—
412C2×C3—81no—
413C2×C3—82no—
414C2×C3A4×Q1612yescomplemented
415C2×C3C2×D4×A411yescharacteristic
416C22—32no—
417C22—31no—
418C22—32no—
419C22—31no—
420C22—31no—
421C22—32no—
422C22—32no—
423C22C2×C3×#21611yescharacteristic, complemented
424C22A4×#6711yescharacteristic
425C4—22no—
426C4—31no—
427C4—31no—
428C4—62no—
429C4—62no—
430C4—182no—
431C4A4×#6711yescharacteristic
432C4C22×D3×A411yescharacteristic
433C3—41no—
434C3—81no—
435C3C2×A4×Q1611yescharacteristic, complemented
436C2—31no—
437C2—32no—
438C2—31no—
439C2A4×#21612yesdirect product factor
440C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC22×C3×C4C23×C3
FrattiniC4C22×D3×A4
FittingC23×C3×Q8C2×C3
socleC24×C3C3×D4
prime radicalC22×C3×A4D4
hypercenterC2×Q8D3×A4
hypocenterC22×C3C2×C3×Q16
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. A4×#216C2direct (N × H)
5. C23×#216C3semidirect
26. C22×#216C2×C3semidirect
81. C2×#216A4direct (N × H)
87. C2×C3×A4Q16semidirect
169. #216C2×A4direct (N × H)
177. C3×A4C2×Q16semidirect
219. C23×C3C3×Q16semidirect
223. C2×A4#216direct (N × H)
343. C22×C3C2×C3×Q16semidirect
346. A4C2×#216direct (N × H)
366. C23C3×#216semidirect
414. C2×C3A4×Q16semidirect
423. C22C2×C3×#216semidirect
435. C3C2×A4×Q16semidirect
439. C2A4×#216direct (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×A4×Dic62
A4×#2162
C2×A4×C3⋊C82
C23×#2163
C2×A4×Q163
C2×C3×#2164