Finite Group Databasebeta

← #11921962

Subgroups — #11921962

Order 1152.

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

normal subgroups only

This group has 652 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×#64204C212yesmaximal
2A4×C4⋊Dic3C211yesmaximal, characteristic
3C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4C3×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
5#10593238C212yesmaximal
6#85303—31nomaximal
7#73899—31nomaximal
8C2×C3×C4×A4C2211yescharacteristic
9C2×Dic3×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2212yes—
13#64204—24no—
14#63919—41nomaximal
15C2×A4⋊Q8—32no—
16C22×C4⋊Dic3D311yescharacteristic
17C2×#888—31no—
18C2×#878—32no—
19A4×C4⋊C4—31no—
20C4⋊SL(2,ℤ/4)D311yescharacteristic
21SL(2,ℤ/4)⋊C4—32no—
22C4×SL(2,ℤ/4)—31no—
23C4×SL(2,ℤ/4)—61no—
24C3×#380—31no—
25C4×#217—31no—
26#4984—31no—
27#4975—31no—
28#4972—31no—
29#4946—32no—
30#4657—32no—
31#4656—32no—
32C22×C3×A4C2311yescharacteristic
33C3×C4×A4Q812yescomplemented
34C2×#627—42no—
35Dic3×A4—22no—
36C3⋊SL(2,ℤ/4)D412yes—
37C3⋊SL(2,ℤ/4)Q812yescomplemented
38C3×SL(2,ℤ/4)—22no—
39C4×C3⋊Dic3—41no—
40C3×C4⋊Dic3—41no—
41C3×C4⋊Dic3—41no—
42#3724—42no—
43#2440—91noSylow
44C23×C3×C4C2×D311yescharacteristic
45C23×Dic3C2×D312yes—
46C22×Dic6—32no—
47C2×C4×A4—21no—
48C2×C4×A4—32no—
49C2×C4×A4C2×D311yescharacteristic
50C2×SL(2,ℤ/4)—31no—
51C2×SL(2,ℤ/4)—31no—
52C2×SL(2,ℤ/4)—61no—
53C2×SL(2,ℤ/4)—61no—
54C2×SL(2,ℤ/4)C2×D312yes—
55A4⋊Q8—64no—
56C2×C3×C4⋊C4—31no—
57C2×C3×C4⋊C4—31no—
58C2×C3×C22⋊C4—32no—
59C2×#217—31no—
60C2×#217—31no—
61C2×C4⋊Dic3—31no—
62C2×C4⋊Dic3—32no—
63C2×C4⋊Dic3—31no—
64C2×C4⋊Dic3—32no—
65C2×Dic3⋊C4—32no—
66C2×Dic3⋊C4—32no—
67C2×Dic3⋊C4—32no—
68C2×Dic3⋊C4—32no—
69C2×C4×Dic3—31no—
70C2×C4×Dic3—31no—
71#888—62no—
72#888—61no—
73#888—61no—
74#888—121no—
75#878—64no—
76#878—64no—
77C3×#95—32no—
78C3⋊#95—31no—
79C3⋊#95—32no—
80C3⋊#95—31no—
81C3⋊#95—32no—
82C2×C3×A4C2×D411yescharacteristic
83C2×C3×A4C2×Q811yescharacteristic
84C2×C3×A4C2×Q811yescharacteristic
85C2×C32×C4—41no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3×Dic3—42no—
89C2×C3×Dic3—42no—
90#627—84no—
91C2×C4⋊Q8—91no—
92C2×C22⋊Q8—92no—
93C22×C4⋊C4—31no—
94#398—92no—
95#395—92no—
96#389—91no—
97#389—91no—
98#385—92no—
99#384—91no—
100#380—31no—
101C4×C22⋊C4—91no—
102C24×C3C22×D311yescharacteristic
103C22×A4—21no—
104C22×A4C22×D311yescharacteristic
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—32no—
108C22×C3×C4—32no—
109C22×C3×C4—32no—
110C22×C3×C4Dic612yescomplemented
111C22×Dic3—22no—
112C22×Dic3—31no—
113C22×Dic3—31no—
114C22×Dic3—31no—
115C22×Dic3—31no—
116C22×Dic3—32no—
117C22×Dic3—32no—
118C22×Dic3—62no—
119C2×Dic6—64no—
120C2×Dic6—62no—
121C2×Dic6—62no—
122C2×Dic6—122no—
123C2×Dic6—122no—
124C4×A4—22no—
125C4×A4—62no—
126C4×A4Dic612yescomplemented
127SL(2,ℤ/4)—22no—
128SL(2,ℤ/4)—32no—
129SL(2,ℤ/4)—32no—
130SL(2,ℤ/4)—62no—
131SL(2,ℤ/4)—62no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—121no—
136C3×C22⋊C4—62no—
137C3×C22⋊C4—62no—
138#217—32no—
139#217—32no—
140#217—32no—
141#217—32no—
142C4⋊Dic3—31no—
143C4⋊Dic3—32no—
144C4⋊Dic3—41no—
145C4⋊Dic3—61no—
146C4⋊Dic3—64no—
147C4⋊Dic3S411yescharacteristic
148Dic3⋊C4—64no—
149Dic3⋊C4—64no—
150Dic3⋊C4—62no—
151Dic3⋊C4—62no—
152Dic3⋊C4—122no—
153Dic3⋊C4—122no—
154C4×Dic3—61no—
155C4×Dic3—62no—
156C4×Dic3—61no—
157C4×Dic3—121no—
158C4×Dic3—241no—
159C22×C32—41no—
160C3×A4C4⋊Q811yescharacteristic, complemented
161C32×C4—42no—
162C3⋊Dic3—42no—
163C3⋊Dic3—42no—
164C3×Dic3—82no—
165C3×Dic3—82no—
166C22×Q8—92no—
167C23×C4—32no—
168C23×C4D3211yescharacteristic
169C4⋊Q8—181no—
170C4⋊Q8—182no—
171C4⋊Q8—181no—
172C22⋊Q8—184no—
173C22⋊Q8—184no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—91no—
177C2×C4⋊C4—91no—
178C2×C4⋊C4—92no—
179C2×C4⋊C4—92no—
180C2×C4⋊C4—92no—
181C2×C4⋊C4—92no—
182C2×C4⋊C4—92no—
183C2×C4⋊C4—92no—
184C2×C22⋊C4—32no—
185C2×C22⋊C4—91no—
186C2×C22⋊C4—91no—
187C2×C42—91no—
188C2×C42—91no—
189#95—32no—
190#95—91no—
191#95—91no—
192#95—92no—
193#95—92no—
194C23×C3—31no—
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—31no—
198C23×C3C2×Dic611yescharacteristic
199C23×C3D3×D411yescharacteristic
200C23×C3D3×Q811yescharacteristic
201C2×A4—21no—
202C2×A4—21no—
203C2×A4—21no—
204C2×A4C2×Dic611yescharacteristic
205C2×A4D3×D411yescharacteristic
206C2×A4D3×Q811yescharacteristic
207C2×C3×C4—31no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—41no—
212C2×C3×C4—61no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—81no—
220C2×C3×C4—122no—
221C2×C3×C4C2×S411yescharacteristic
222C2×Dic3—32no—
223C2×Dic3—32no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—42no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—121no—
241C2×Dic3—121no—
242C2×Dic3—241no—
243C2×Dic3—241no—
244C2×Dic3C2×S412yes—
245Dic6—124no—
246Dic6—124no—
247Dic6—244no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C24C2×D3211yescharacteristic
252C2×Q8—182no—
253C2×Q8—184no—
254C2×Q8—182no—
255C2×Q8—362no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—62no—
262C22×C4—92no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—91no—
267C22×C4—92no—
268C22×C4—182no—
269C22×C4#62712yescomplemented
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—62no—
273C4⋊C4—61no—
274C4⋊C4—92no—
275C4⋊C4—91no—
276C4⋊C4—182no—
277C4⋊C4—181no—
278C4⋊C4—184no—
279C4⋊C4—184no—
280C4⋊C4—184no—
281C4⋊C4—182no—
282C4⋊C4—362no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—92no—
286C22⋊C4—92no—
287C22⋊C4—92no—
288C22⋊C4—92no—
289C42—181no—
290C42—182no—
291C42—181no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—41no—
302C22×C3—61no—
303C22×C3—81no—
304C22×C3#88811yescharacteristic, complemented
305C22×C3C22×S411yescharacteristic
306A4—21no—
307A4#88811yescharacteristic, complemented
308C3×C4—32no—
309C3×C4—42no—
310C3×C4—82no—
311C3×C4—122no—
312C3×C4—122no—
313C3×C4—242no—
314C3×C4A4⋊Q812yescomplemented
315Dic3—22no—
316Dic3—62no—
317Dic3—62no—
318Dic3—62no—
319Dic3—62no—
320Dic3—62no—
321Dic3—82no—
322Dic3—122no—
323Dic3—122no—
324Dic3—242no—
325Dic3—242no—
326C32—41noSylow
327C23—31no—
328C23—31no—
329C23—31no—
330C23—31no—
331C23#379711yescharacteristic
332C23#380211yescharacteristic
333C23C2×#62711yescharacteristic
334Q8—364no—
335Q8—364no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—61no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—92no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—92no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4D3×S411yescharacteristic
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3—81no—
376C2×C3C2×A4⋊Q811yescharacteristic
377C2×C3D4×S411yescharacteristic
378C2×C3Q8×S411yescharacteristic
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—31no—
385C22—31no—
386C22—31no—
387C22—31no—
388C22—61no—
389C22#6391911yescharacteristic, complemented
390C22C2×D3×S411yescharacteristic
391C4—32no—
392C4—62no—
393C4—122no—
394C4—122no—
395C4—182no—
396C4—182no—
397C4—182no—
398C4—182no—
399C4—182no—
400C4#6420412yescomplemented
401C3—41no—
402C3—81no—
403C3#8530311yescharacteristic, complemented
404C2—31no—
405C2—31no—
406C2—31no—
407C2—31no—
408C2(C3⋊S4)⋊Q811yescharacteristic
409C2#1059643211yescharacteristic
410C2C2×#6420411yescharacteristic
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×C4×A4Q8semidirect
37. C3⋊SL(2,ℤ/4)Q8semidirect
110. C22×C3×C4Dic6semidirect
126. C4×A4Dic6semidirect
160. C3×A4C4⋊Q8semidirect
269. C22×C4#627semidirect
304. C22×C3#888semidirect
307. A4#888semidirect
314. C3×C4A4⋊Q8semidirect
389. C22#63919semidirect
400. C4#64204semidirect
403. C3#85303semidirect

Maximal subgroups

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

subgroupindex
C2×#642042
A4×C4⋊Dic32
C4×C3⋊SL(2,ℤ/4)2
C3×C4⋊SL(2,ℤ/4)2
#105932382
#853033
#738993
#639194