Finite Group Databasebeta

← #11922853

Subgroups — #11922853

Order 1152.

758 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 103 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×#64259C212yesmaximal
2C3×A4×C4⋊C4C211yesmaximal, characteristic
3C4⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
4#10593645C212yesmaximal
5C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
6#85303—33nomaximal
7#82171—31nomaximal
8C2×C3×C4×A4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13#64259—24no—
14#64097—41nomaximal
15C2×A4⋊Q8—36no—
16C22×C3×C4⋊C4D311yescharacteristic
17C2×#924—32no—
18C2×#888—31no—
19A4×C4⋊C4D313yes—
20C4⋊SL(2,ℤ/4)—33no—
21SL(2,ℤ/4)⋊C4—36no—
22C4×SL(2,ℤ/4)—33no—
23#5212—31no—
24#5210—32no—
25C4×#217—31no—
26#4984—31no—
27#4982—32no—
28#4979—32no—
29#4975—31no—
30#4972—31no—
31C22×C3×A4C2311yescharacteristic
32C2×#634—42no—
33C3×C4×A4—22no—
34C3×C4×A4Q812yescomplemented
35C3⋊SL(2,ℤ/4)—22no—
36C3⋊SL(2,ℤ/4)D412yes—
37C3⋊SL(2,ℤ/4)Q812yescomplemented
38C32×C4⋊C4—41no—
39C4⋊(C3⋊Dic3)—41no—
40#3750—42no—
41C4×C3⋊Dic3—41no—
42#2440—91noSylow
43C23×C3×C4C2×D312yes—
44C23×C3×C4C2×D311yescharacteristic
45C22×Dic6—32no—
46C2×C4×A4C2×D316yes—
47C2×C4×A4C2×D313yes—
48C2×SL(2,ℤ/4)—33no—
49C2×SL(2,ℤ/4)—33no—
50C2×SL(2,ℤ/4)—36no—
51A4⋊Q8—612no—
52C2×C3×C4⋊C4—31no—
53C2×C3×C4⋊C4—31no—
54C2×C3×C4⋊C4—32no—
55C2×#217—31no—
56C2×#217—31no—
57C2×#217—32no—
58C2×C4⋊Dic3—31no—
59C2×C4⋊Dic3—31no—
60C2×C4⋊Dic3—32no—
61#924—64no—
62#924—64no—
63C2×Dic3⋊C4—32no—
64C2×Dic3⋊C4—32no—
65C2×Dic3⋊C4—32no—
66C2×Dic3⋊C4—32no—
67C2×C4×Dic3—31no—
68C2×C4×Dic3—31no—
69#888—62no—
70#888—61no—
71#888—61no—
72#888—123no—
73C3⋊#95—31no—
74C3⋊#95—31no—
75C3⋊#95—32no—
76C3⋊#95—32no—
77C3⋊#95—32no—
78C2×C3×A4C2×D411yescharacteristic
79C2×C3×A4C2×Q811yescharacteristic
80C2×C3×A4C2×Q811yescharacteristic
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C3⋊Dic3—41no—
84C2×C3⋊Dic3—41no—
85C2×C3⋊Dic3—42no—
86#634—84no—
87C2×C4⋊Q8—91no—
88C2×C22⋊Q8—92no—
89C22×C4⋊C4C3⋊D311yescharacteristic
90#398—92no—
91#395—92no—
92#389—91no—
93#389—91no—
94#385—92no—
95#384—91no—
96#380—91no—
97C4×C22⋊C4—91no—
98C24×C3C22×D311yescharacteristic
99C22×A4C22×D313yes—
100C22×C3×C4—22no—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×C3×C4—32no—
105C22×C3×C4—32no—
106C22×C3×C4—62no—
107C22×C3×C4Dic612yescomplemented
108C22×Dic3—31no—
109C22×Dic3—31no—
110C22×Dic3—31no—
111C22×Dic3—31no—
112C22×Dic3—32no—
113C22×Dic3—32no—
114C2×Dic6—64no—
115C2×Dic6—62no—
116C2×Dic6—62no—
117C2×Dic6—126no—
118C2×Dic6—122no—
119C4×A4—26no—
120C4×A4Dic616yescomplemented
121SL(2,ℤ/4)—36no—
122SL(2,ℤ/4)—36no—
123SL(2,ℤ/4)—66no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—43no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4S411yescharacteristic
129#217—32no—
130#217—32no—
131#217—32no—
132#217—32no—
133#217—62no—
134#217—62no—
135C4⋊Dic3—64no—
136C4⋊Dic3—61no—
137C4⋊Dic3—62no—
138C4⋊Dic3—61no—
139C4⋊Dic3—123no—
140Dic3⋊C4—64no—
141Dic3⋊C4—64no—
142Dic3⋊C4—62no—
143Dic3⋊C4—62no—
144Dic3⋊C4—126no—
145Dic3⋊C4—122no—
146C4×Dic3—61no—
147C4×Dic3—62no—
148C4×Dic3—61no—
149C4×Dic3—123no—
150C22×C32—41no—
151C3×A4C4⋊Q811yescharacteristic, complemented
152C32×C4—42no—
153C32×C4—82no—
154C3⋊Dic3—42no—
155C3⋊Dic3—42no—
156C3⋊Dic3—82no—
157C22×Q8—92no—
158C23×C4C2×C3⋊D312yes—
159C23×C4C2×C3⋊D311yescharacteristic
160C4⋊Q8—182no—
161C4⋊Q8—181no—
162C4⋊Q8—181no—
163C22⋊Q8—184no—
164C22⋊Q8—184no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—92no—
169C2×C4⋊C4—92no—
170C2×C4⋊C4—92no—
171C2×C4⋊C4—91no—
172C2×C4⋊C4—92no—
173C2×C4⋊C4—91no—
174C2×C4⋊C4—92no—
175C2×C22⋊C4—91no—
176C2×C22⋊C4—91no—
177C2×C22⋊C4—92no—
178C2×C42—91no—
179C2×C42—91no—
180#95—91no—
181#95—91no—
182#95—92no—
183#95—92no—
184#95—92no—
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—31no—
188C23×C3—31no—
189C23×C3C2×Dic611yescharacteristic
190C23×C3D3×D411yescharacteristic
191C23×C3D3×Q811yescharacteristic
192C2×A4C2×Dic613yes—
193C2×A4D3×D413yes—
194C2×A4D3×Q813yes—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—43no—
201C2×C3×C4—46no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—61no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4C2×S412yes—
208C2×C3×C4C2×S411yescharacteristic
209C2×Dic3—32no—
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—123no—
228C2×Dic3—123no—
229C2×Dic3—126no—
230Dic6—124no—
231Dic6—124no—
232Dic6—2412no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C24C22×C3⋊D311yescharacteristic
237C2×Q8—182no—
238C2×Q8—184no—
239C2×Q8—182no—
240C2×Q8—362no—
241C22×C4—22no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—32no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—62no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4#63412yescomplemented
255C4⋊C4—32no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—182no—
259C4⋊C4—181no—
260C4⋊C4—182no—
261C4⋊C4—184no—
262C4⋊C4—184no—
263C4⋊C4—184no—
264C4⋊C4—182no—
265C4⋊C4—181no—
266C4⋊C4—362no—
267C4⋊C4C3⋊S411yescharacteristic
268C22⋊C4—92no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C22⋊C4—92no—
272C22⋊C4—182no—
273C22⋊C4—182no—
274C42—181no—
275C42—182no—
276C42—181no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—43no—
287C22×C3—61no—
288C22×C3#88811yescharacteristic, complemented
289C22×C3C22×S411yescharacteristic
290A4#88813yescomplemented
291C3×C4—22no—
292C3×C4—32no—
293C3×C4—46no—
294C3×C4—62no—
295C3×C4—86no—
296C3×C4A4⋊Q812yescomplemented
297Dic3—62no—
298Dic3—62no—
299Dic3—62no—
300Dic3—62no—
301Dic3—126no—
302Dic3—126no—
303Dic3—122no—
304Dic3—122no—
305Dic3—246no—
306C32—41noSylow
307C23—31no—
308C23—31no—
309C23—31no—
310C23—31no—
311C23C2×#63411yescharacteristic
312C23D4×C3⋊D311yescharacteristic
313C23Q8×C3⋊D311yescharacteristic
314Q8—364no—
315Q8—364no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—92no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—92no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—182no—
339C2×C4—182no—
340C2×C4—182no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4C2×C3⋊S412yes—
345C2×C4C2×C3⋊S411yescharacteristic
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—43no—
351C2×C3—43no—
352C2×C3—43no—
353C2×C3C2×A4⋊Q811yescharacteristic
354C2×C3D4×S411yescharacteristic
355C2×C3Q8×S411yescharacteristic
356C22—31no—
357C22—31no—
358C22—31no—
359C22—31no—
360C22—31no—
361C22—31no—
362C22—31no—
363C22—31no—
364C22—31no—
365C22—61no—
366C22#6409711yescharacteristic, complemented
367C22C22×C3⋊S411yescharacteristic
368C4—22no—
369C4—32no—
370C4—62no—
371C4—182no—
372C4—182no—
373C4—182no—
374C4—182no—
375C4—362no—
376C4—362no—
377C4#6425912yescomplemented
378C3—43no—
379C3#8530311yescharacteristic, complemented
380C2—31no—
381C2—31no—
382C2—31no—
383C2—31no—
384C2C2×#6425911yescharacteristic
385C2D4×C3⋊S411yescharacteristic
386C2Q8×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, prime radical, cosocleC22×C3×A4C23
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4C4⋊Q8
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
34. C3×C4×A4Q8semidirect
37. C3⋊SL(2,ℤ/4)Q8semidirect
107. C22×C3×C4Dic6semidirect
120. C4×A4Dic6semidirect
151. C3×A4C4⋊Q8semidirect
254. C22×C4#634semidirect
288. C22×C3#888semidirect
290. A4#888semidirect
296. C3×C4A4⋊Q8semidirect
366. C22#64097semidirect
377. C4#64259semidirect
379. C3#85303semidirect

Maximal subgroups

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

subgroupindex
C2×#642592
C3×A4×C4⋊C42
C4⋊(C3⋊SL(2,ℤ/4))2
#105936452
C4×C3⋊SL(2,ℤ/4)2
#853033
#821713
#640974