Finite Group Databasebeta

← #11921959

Subgroups — #11921959

Order 1152.

648 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 79 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
5SL(2,ℤ/4)⋊Dic3C212yesmaximal
6#85303—31nomaximal
7#73493—31nomaximal
8C2×C3×C4×A4C2211yescharacteristic
9C2×Dic3×A4C2212yes—
10C2×C3⋊SL(2,ℤ/4)C2212yes—
11C2×C3×SL(2,ℤ/4)C2211yescharacteristic
12C2×C3×SL(2,ℤ/4)C2211yescharacteristic
13#64204—24no—
14#63918—41nomaximal
15C2×A4⋊Q8—32no—
16C22×C4⋊Dic3D311yescharacteristic
17C2×#878—32no—
18C2×#869—31no—
19A4×C4⋊C4—31no—
20C4⋊SL(2,ℤ/4)—31no—
21C4⋊SL(2,ℤ/4)—61no—
22SL(2,ℤ/4)⋊C4—32no—
23C4×SL(2,ℤ/4)D311yescharacteristic
24C3×C4×C22⋊C4—31no—
25#5212—31no—
26#4950—32no—
27#4935—31no—
28#4928—31no—
29#4928—31no—
30#4661—32no—
31#4656—32no—
32C22×C3×A4C2311yescharacteristic
33C3×C4×A4Q812yescomplemented
34C2×#627—42no—
35Dic3×A4—22no—
36C3⋊SL(2,ℤ/4)—22no—
37C3×SL(2,ℤ/4)D412yes—
38C3×SL(2,ℤ/4)Q812yescomplemented
39C4⋊(C3⋊Dic3)—41no—
40C3×C4⋊Dic3—41no—
41C3×C4×Dic3—41no—
42Dic3⋊Dic3—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)—32no—
51C2×SL(2,ℤ/4)—62no—
52C2×SL(2,ℤ/4)C2×D311yescharacteristic
53C2×SL(2,ℤ/4)C2×D311yescharacteristic
54A4⋊Q8—64no—
55C2×C3×C22⋊C4—31no—
56C2×C3×C22⋊C4—31no—
57C2×C3×C42—31no—
58C2×C3×C42—31no—
59C2×#217—32no—
60C2×C4⋊Dic3—31no—
61C2×C4⋊Dic3—32no—
62C2×C4⋊Dic3—31no—
63C2×C4⋊Dic3—32no—
64C2×C4⋊Dic3—32no—
65C2×C4⋊Dic3—31no—
66C2×C4⋊Dic3—31no—
67C2×C4⋊Dic3—32no—
68C2×Dic3⋊C4—32no—
69C2×Dic3⋊C4—32no—
70#888—121no—
71#878—64no—
72#878—64no—
73#869—61no—
74#869—62no—
75#869—61no—
76C3×#95—31no—
77C3×#95—31no—
78C3⋊#95—32no—
79C3⋊#95—32no—
80C3⋊#95—32no—
81C2×C3×A4C2×D411yescharacteristic
82C2×C3×A4C2×Q811yescharacteristic
83C2×C3×A4C2×Q811yescharacteristic
84C2×C32×C4—41no—
85C2×C3⋊Dic3—42no—
86C2×C3×Dic3—41no—
87C2×C3×Dic3—41no—
88C2×C3×Dic3—42no—
89#627—84no—
90C2×C4⋊Q8—91no—
91C2×C22⋊Q8—92no—
92C22×C4⋊C4—31no—
93#398—92no—
94#395—92no—
95#389—91no—
96#389—91no—
97#385—92no—
98#384—91no—
99#380—91no—
100C4×C22⋊C4—31no—
101C24×C3C22×D311yescharacteristic
102C22×A4—21no—
103C22×A4C22×D311yescharacteristic
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—32no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4Dic612yescomplemented
112C22×Dic3—22no—
113C22×Dic3—32no—
114C22×Dic3—32no—
115C22×Dic3—32no—
116C22×Dic3—32no—
117C22×Dic3—62no—
118C2×Dic6—62no—
119C2×Dic6—64no—
120C2×Dic6—62no—
121C2×Dic6—122no—
122C2×Dic6—122no—
123C4×A4—22no—
124C4×A4—62no—
125C4×A4Dic612yescomplemented
126SL(2,ℤ/4)—62no—
127SL(2,ℤ/4)—122no—
128SL(2,ℤ/4)Dic612yescomplemented
129SL(2,ℤ/4)D1212yes—
130C3×C4⋊C4—121no—
131C3×C22⋊C4—32no—
132C3×C22⋊C4—32no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—32no—
135C3×C42—61no—
136C3×C42—62no—
137C3×C42—61no—
138#217—62no—
139#217—62no—
140C4⋊Dic3—31no—
141C4⋊Dic3—32no—
142C4⋊Dic3—61no—
143C4⋊Dic3—62no—
144C4⋊Dic3—61no—
145C4⋊Dic3—64no—
146C4⋊Dic3—62no—
147C4⋊Dic3—61no—
148C4⋊Dic3—62no—
149C4⋊Dic3—121no—
150C4⋊Dic3—122no—
151C4⋊Dic3—241no—
152C4⋊Dic3S411yescharacteristic
153Dic3⋊C4—64no—
154Dic3⋊C4—64no—
155Dic3⋊C4—122no—
156C4×Dic3—41no—
157C22×C32—41no—
158C3×A4C4⋊Q811yescharacteristic, complemented
159C32×C4—42no—
160C3⋊Dic3—82no—
161C3×Dic3—42no—
162C3×Dic3—42no—
163C3×Dic3—82no—
164C22×Q8—92no—
165C23×C4—32no—
166C23×C4D3211yescharacteristic
167C4⋊Q8—181no—
168C4⋊Q8—182no—
169C4⋊Q8—181no—
170C22⋊Q8—184no—
171C22⋊Q8—184no—
172C2×C4⋊C4—91no—
173C2×C4⋊C4—91no—
174C2×C4⋊C4—92no—
175C2×C4⋊C4—92no—
176C2×C4⋊C4—92no—
177C2×C4⋊C4—92no—
178C2×C4⋊C4—91no—
179C2×C4⋊C4—92no—
180C2×C4⋊C4—91no—
181C2×C4⋊C4—92no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—31no—
184C2×C22⋊C4—92no—
185C2×C42—31no—
186C2×C42—31no—
187#95—31no—
188#95—31no—
189#95—92no—
190#95—92no—
191#95—92no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—31no—
195C23×C3—31no—
196C23×C3C2×Dic611yescharacteristic
197C23×C3D3×D411yescharacteristic
198C23×C3D3×Q811yescharacteristic
199C2×A4—21no—
200C2×A4—21no—
201C2×A4—21no—
202C2×A4C2×Dic611yescharacteristic
203C2×A4C2×Dic611yescharacteristic
204C2×A4C2×D1211yescharacteristic
205C2×C3×C4—31no—
206C2×C3×C4—32no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—41no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—81no—
224C2×C3×C4—122no—
225C2×C3×C4C2×S411yescharacteristic
226C2×Dic3—32no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—122no—
240C2×Dic3—242no—
241C2×Dic3C2×S412yes—
242Dic6—124no—
243Dic6—124no—
244Dic6—244no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C24C2×D3211yescharacteristic
249C2×Q8—182no—
250C2×Q8—184no—
251C2×Q8—182no—
252C2×Q8—362no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—62no—
261C22×C4—92no—
262C22×C4—92no—
263C22×C4—92no—
264C22×C4—92no—
265C22×C4—182no—
266C22×C4#62712yescomplemented
267C4⋊C4—31no—
268C4⋊C4—92no—
269C4⋊C4—91no—
270C4⋊C4—182no—
271C4⋊C4—181no—
272C4⋊C4—184no—
273C4⋊C4—184no—
274C4⋊C4—184no—
275C4⋊C4—181no—
276C4⋊C4—182no—
277C4⋊C4—182no—
278C4⋊C4—181no—
279C4⋊C4—362no—
280C22⋊C4—32no—
281C22⋊C4—32no—
282C22⋊C4—32no—
283C22⋊C4—32no—
284C22⋊C4—182no—
285C22⋊C4—182no—
286C42—61no—
287C42—62no—
288C42—61no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—41no—
299C22×C3—61no—
300C22×C3—81no—
301C22×C3#88811yescharacteristic, complemented
302C22×C3C22×S411yescharacteristic
303A4—21no—
304A4#86911yescharacteristic, complemented
305C3×C4—32no—
306C3×C4—42no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—62no—
310C3×C4—62no—
311C3×C4—82no—
312C3×C4—242no—
313C3×C4A4⋊Q812yescomplemented
314Dic3—22no—
315Dic3—42no—
316Dic3—42no—
317Dic3—62no—
318Dic3—122no—
319Dic3—122no—
320Dic3—242no—
321Dic3—482no—
322C32—41noSylow
323C23—31no—
324C23—31no—
325C23—31no—
326C23—31no—
327C23D3×Dic611yescharacteristic
328C23D3×D1211yescharacteristic
329C23C2×#62711yescharacteristic
330Q8—364no—
331Q8—364no—
332C2×C4—31no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—92no—
351C2×C4—182no—
352C2×C4—182no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C4D3×S411yescharacteristic
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—81no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3C2×A4⋊Q811yescharacteristic
373C2×C3D4×S411yescharacteristic
374C2×C3Q8×S411yescharacteristic
375C22—31no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—31no—
384C22—61no—
385C22#6391811yescharacteristic, complemented
386C22C2×D3×S411yescharacteristic
387C4—32no—
388C4—62no—
389C4—62no—
390C4—62no—
391C4—62no—
392C4—62no—
393C4—182no—
394C4—362no—
395C4—362no—
396C4#6420412yescomplemented
397C3—41no—
398C3—81no—
399C3#8530311yescharacteristic, complemented
400C2—31no—
401C2—31no—
402C2—31no—
403C2—31no—
404C2Dic6×S411yescharacteristic
405C2D12×S411yescharacteristic
406C2C2×#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
38. C3×SL(2,ℤ/4)Q8semidirect
111. C22×C3×C4Dic6semidirect
125. C4×A4Dic6semidirect
128. SL(2,ℤ/4)Dic6semidirect
158. C3×A4C4⋊Q8semidirect
266. C22×C4#627semidirect
301. C22×C3#888semidirect
304. A4#869semidirect
313. C3×C4A4⋊Q8semidirect
385. C22#63918semidirect
396. C4#64204semidirect
399. 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
SL(2,ℤ/4)⋊Dic32
#853033
#734933
#639184