Finite Group Databasebeta

← #11921899

Subgroups — #11921899

Order 1152.

708 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 75 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×A4×Dic6C211yesmaximal, characteristic
2C2×C3×A4⋊Q8C211yesmaximal, characteristic
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4#10593238C214yesmaximal
5#85502—31nomaximal
6#82760—31nomaximal
7C2×C3×C4×A4C2211yescharacteristic
8C2×Dic3×A4C2212yes—
9A4×Dic6—22no—
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×A4⋊Q8—22no—
13#63873—41nomaximal
14C23×Dic6D311yescharacteristic
15C2×Q8×A4—31no—
16C2×A4⋊Q8D311yescharacteristic
17C2×C3×C22⋊Q8—31no—
18C2×#944—31no—
19C2×#944—31no—
20SL(2,ℤ/4)⋊C4—34no—
21C4×SL(2,ℤ/4)—31no—
22C4×SL(2,ℤ/4)—61no—
23C4×#217—31no—
24#4979—34no—
25#4973—31no—
26#4973—31no—
27#4946—34no—
28C22×C3×A4C2311yescharacteristic
29C2×C3×Dic6—41no—
30C2×C3×Dic6—41no—
31C3×C4×A4D412yes—
32Dic3×A4—22no—
33C3⋊SL(2,ℤ/4)Q814yescomplemented
34C3×SL(2,ℤ/4)—22no—
35C4×C3⋊Dic3—41no—
36#3724—44no—
37#2434—91noSylow
38C22×C3×Q8—31no—
39C23×C3×C4C2×D311yescharacteristic
40C23×Dic3C2×D312yes—
41C22×Dic6—22no—
42C22×Dic6—31no—
43C22×Dic6—31no—
44C22×Dic6—32no—
45C22×Dic6—62no—
46Q8×A4—62no—
47C2×C4×A4—21no—
48C2×C4×A4—32no—
49C2×C4×A4C2×D311yescharacteristic
50C2×SL(2,ℤ/4)—32no—
51C2×SL(2,ℤ/4)—62no—
52C2×SL(2,ℤ/4)C2×D312yes—
53A4⋊Q8—22no—
54C3×C22⋊Q8—62no—
55C3×C22⋊Q8—62no—
56C2×C3×C4⋊C4—31no—
57C2×C3×C4⋊C4—32no—
58C2×C3×C22⋊C4—32no—
59C2×#217—32no—
60#944—64no—
61#944—62no—
62#944—61no—
63#944—61no—
64#944—121no—
65C2×Dic3⋊C4—34no—
66C2×Dic3⋊C4—34no—
67C2×C4×Dic3—31no—
68C2×C4×Dic3—31no—
69C3⋊#95—34no—
70C3⋊#95—34no—
71C3⋊#95—32no—
72C2×C3×A4C2×D411yescharacteristic
73C2×C3×A4C2×Q812yes—
74C2×C32×C4—41no—
75C2×C3⋊Dic3—42no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—42no—
78C3×Dic6—82no—
79C3×Dic6—82no—
80C23×Q8—31no—
81C2×C4⋊Q8—91no—
82C2×C4⋊Q8—91no—
83C2×C22⋊Q8—31no—
84#395—94no—
85#391—91no—
86#391—91no—
87#385—94no—
88C4×C22⋊C4—91no—
89C24×C3C22×D311yescharacteristic
90C22×A4—21no—
91C22×A4C22×D311yescharacteristic
92C2×C3×Q8—61no—
93C2×C3×Q8—62no—
94C2×C3×Q8—62no—
95C2×C3×Q8—61no—
96C2×C3×Q8—121no—
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—32no—
100C22×C3×C4—32no—
101C22×C3×C4—32no—
102C22×C3×C4#6712yes—
103C22×Dic3—22no—
104C22×Dic3—32no—
105C22×Dic3—32no—
106C22×Dic3—32no—
107C22×Dic3—32no—
108C22×Dic3—62no—
109C2×Dic6—34no—
110C2×Dic6—31no—
111C2×Dic6—32no—
112C2×Dic6—41no—
113C2×Dic6—62no—
114C2×Dic6—62no—
115C2×Dic6—61no—
116C2×Dic6—64no—
117C2×Dic6—64no—
118C2×Dic6—62no—
119C2×Dic6S411yescharacteristic
120C4×A4—22no—
121C4×A4—62no—
122C4×A4#6712yes—
123SL(2,ℤ/4)—22no—
124SL(2,ℤ/4)—34no—
125SL(2,ℤ/4)—64no—
126C3×C4⋊C4—62no—
127C3×C4⋊C4—64no—
128C3×C22⋊C4—62no—
129C3×C22⋊C4—62no—
130#217—34no—
131#217—34no—
132Dic3⋊C4—68no—
133Dic3⋊C4—64no—
134Dic3⋊C4—64no—
135Dic3⋊C4—124no—
136C4×Dic3—62no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C4×Dic3—121no—
140C4×Dic3—241no—
141C22×C32—41no—
142C3×A4C4⋊Q811yescharacteristic, complemented
143C32×C4—42no—
144C3⋊Dic3—44no—
145C3×Dic3—82no—
146C3×Dic3—82no—
147C22×Q8—31no—
148C22×Q8—62no—
149C22×Q8—91no—
150C22×Q8—91no—
151C22×Q8—92no—
152C22×Q8—182no—
153C23×C4—32no—
154C23×C4D3211yescharacteristic
155C4⋊Q8—181no—
156C4⋊Q8—182no—
157C4⋊Q8—184no—
158C4⋊Q8—181no—
159C22⋊Q8—62no—
160C22⋊Q8—62no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—94no—
164C2×C4⋊C4—94no—
165C2×C22⋊C4—32no—
166C2×C22⋊C4—92no—
167C2×C42—91no—
168C2×C42—91no—
169#95—92no—
170#95—94no—
171#95—94no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3—31no—
175C23×C3D3×Q812yes—
176C23×C3C2×#6711yescharacteristic
177C2×A4—22no—
178C2×A4—21no—
179C2×A4D3×Q812yes—
180C2×A4C2×#6711yescharacteristic
181C3×Q8—122no—
182C3×Q8—122no—
183C3×Q8—242no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—41no—
189C2×C3×C4—61no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—64no—
194C2×C3×C4—62no—
195C2×C3×C4—81no—
196C2×C3×C4—122no—
197C2×C3×C4C2×S411yescharacteristic
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—34no—
201C2×Dic3—32no—
202C2×Dic3—42no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—122no—
211C2×Dic3—242no—
212C2×Dic3C2×S412yes—
213Dic6—22no—
214Dic6—64no—
215Dic6—64no—
216Dic6—62no—
217Dic6—82no—
218C2×C32—42no—
219C2×C32—41no—
220C24C2×D3211yescharacteristic
221C2×Q8—31no—
222C2×Q8—62no—
223C2×Q8—61no—
224C2×Q8—62no—
225C2×Q8—61no—
226C2×Q8—92no—
227C2×Q8—94no—
228C2×Q8—91no—
229C2×Q8—181no—
230C2×Q8—182no—
231C2×Q8—182no—
232C2×Q8—182no—
233C2×Q8—184no—
234C2×Q8—184no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—32no—
238C22×C4—32no—
239C22×C4—32no—
240C22×C4—62no—
241C22×C4—92no—
242C22×C4—92no—
243C22×C4—92no—
244C22×C4—92no—
245C22×C4—182no—
246C22×C4#62512yes—
247C4⋊C4—62no—
248C4⋊C4—64no—
249C4⋊C4—184no—
250C4⋊C4—188no—
251C4⋊C4—184no—
252C22⋊C4—62no—
253C22⋊C4—62no—
254C22⋊C4—94no—
255C22⋊C4—94no—
256C42—181no—
257C42—182no—
258C42—181no—
259C22×C3—32no—
260C22×C3—31no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—41no—
266C22×C3—61no—
267C22×C3—81no—
268C22×C3#94411yescharacteristic, complemented
269C22×C3C22×S411yescharacteristic
270A4—21no—
271A4#94411yescharacteristic, complemented
272C3×C4—32no—
273C3×C4—42no—
274C3×C4—82no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—242no—
278C3×C4GL(2,ℤ/4)12yes—
279Dic3—22no—
280Dic3—64no—
281Dic3—64no—
282Dic3—62no—
283Dic3—82no—
284Dic3—124no—
285Dic3—244no—
286C32—41noSylow
287C23—31no—
288C23—31no—
289C23—32no—
290C23#379712yes—
291C23C2×#62511yescharacteristic
292Q8—62no—
293Q8—122no—
294Q8—122no—
295Q8—184no—
296Q8—184no—
297Q8—182no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—61no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—64no—
308C2×C4—62no—
309C2×C4—94no—
310C2×C4—94no—
311C2×C4—94no—
312C2×C4—92no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—182no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4D3×S411yescharacteristic
321C2×C3—32no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—42no—
325C2×C3—41no—
326C2×C3—82no—
327C2×C3—81no—
328C2×C3Q8×S412yes—
329C2×C3C2×GL(2,ℤ/4)11yescharacteristic
330C22—32no—
331C22—31no—
332C22—32no—
333C22—32no—
334C22—31no—
335C22—31no—
336C22—61no—
337C22#6387311yescharacteristic, complemented
338C22C2×D3×S411yescharacteristic
339C4—32no—
340C4—62no—
341C4—122no—
342C4—122no—
343C4—184no—
344C4—184no—
345C4—182no—
346C4#6420912yes—
347C3—41no—
348C3—81no—
349C3#8550211yescharacteristic, complemented
350C2—31no—
351C2—32no—
352C2—31no—
353C2(C3⋊S4)⋊Q812yes—
354C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, prime radical, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
hypercenterC2×C4D3×S4
hypocenterC3×A4C4⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
33. C3⋊SL(2,ℤ/4)Q8semidirect
142. C3×A4C4⋊Q8semidirect
268. C22×C3#944semidirect
271. A4#944semidirect
337. C22#63873semidirect
349. C3#85502semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic62
C2×C3×A4⋊Q82
C4×C3⋊SL(2,ℤ/4)2
#105932382
#855023
#827603
#638734