Finite Group Databasebeta

← #11922858

Subgroups — #11922858

Order 1152.

1042 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×C4×C3⋊S4C212yesmaximal, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4#10593647C212yesmaximal, complemented
5C4×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6#85308—33nomaximal
7#82163—31nomaximal
8C22×C3⋊S4C2212yes—
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
13C4⋊(C3⋊S4)C414yescomplemented
14C4×C3⋊S4—24no—
15#64102—41nomaximal
16C2×C4⋊S4—33no—
17C2×C4×S4—36no—
18C22×C3×C4⋊C4D311yescharacteristic, complemented
19C2×#930—31no—
20C2×C4×#67—32no—
21C2×#893—31no—
22A4×C4⋊C4D313yescomplemented
23(C2×S4)⋊C4—36no—
24C4×SL(2,ℤ/4)—33no—
25#5213—32no—
26C4×#217—31no—
27#4990—32no—
28#4989—32no—
29#4988—31no—
30Dic3×C4⋊C4—31no—
31#4972—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3⋊S4—22no—
34C2×C3⋊S4—22no—
35C2×C3⋊S4C2×C414yescomplemented
36C2×C4⋊(C3⋊D3)—41no—
37C2×C4×C3⋊D3—42no—
38C3×C4×A4—22no—
39C3×C4×A4C2×C412yescomplemented
40C3⋊SL(2,ℤ/4)—21no—
41C3⋊SL(2,ℤ/4)D412yescomplemented
42C32×C4⋊C4—41no—
43#3752—42no—
44C4×C3⋊Dic3—41no—
45#2361—91noSylow
46C22×S4—36no—
47C23×C3×C4C2×D312yes—
48C23×C3×C4C2×D311yescharacteristic
49C22×#67—32no—
50C22×D12—31no—
51C22×C4×D3—32no—
52C2×C4×A4C2×D316yes—
53C2×C4×A4C2×D313yes—
54C2×SL(2,ℤ/4)—33no—
55C2×SL(2,ℤ/4)—33no—
56C4⋊S4—312no—
57C4×S4—612no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—31no—
60C2×C3×C4⋊C4—32no—
61C2×#217—31no—
62C2×#217—31no—
63#930—34no—
64#930—34no—
65C4×#67—64no—
66C4×#67—64no—
67C2×(C2×D3)⋊C4—32no—
68C2×(C2×D3)⋊C4—32no—
69C2×(C2×D3)⋊C4—32no—
70C2×(C2×D3)⋊C4—32no—
71C2×C4⋊Dic3—31no—
72C2×Dic3⋊C4—32no—
73C2×C4×Dic3—31no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—32no—
76#893—62no—
77#893—61no—
78#893—61no—
79#893—123no—
80C3⋊#95—31no—
81C3⋊#95—31no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C22×C3⋊D3—42no—
85C2×C3×A4C22×C411yescharacteristic
86C2×C3×A4C2×D411yescharacteristic
87C2×C3×A4#4111yescharacteristic
88C3⋊S4—24no—
89C2×C32×C4—41no—
90C2×C32×C4—42no—
91C2×C3⋊Dic3—41no—
92C2×C3⋊Dic3—41no—
93C4⋊(C3⋊D3)—44no—
94C4×C3⋊D3—84no—
95C2×#121—91no—
96C2×C4×D4—91no—
97C2×C4×D4—92no—
98C22×C4⋊C4C3⋊D311yescharacteristic, complemented
99#390—91no—
100#389—91no—
101#388—92no—
102#386—92no—
103#385—92no—
104C4×C4⋊C4—91no—
105C4×C22⋊C4—91no—
106C24×C3C22×D311yescharacteristic
107C23×D3—32no—
108C22×A4C22×D313yes—
109C2×S4—312no—
110C2×S4—66no—
111C2×S4—66no—
112C22×C3×C4—22no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—32no—
116C22×C3×C4—32no—
117C22×C3×C4—32no—
118C22×C3×C4—62no—
119C22×C3×C4C4×D312yescomplemented
120C2×#67—34no—
121C2×#67—34no—
122C2×#67—62no—
123C2×#67—62no—
124C2×#67—62no—
125C2×#67—62no—
126C22×Dic3—32no—
127C22×Dic3—31no—
128C22×Dic3—31no—
129C22×Dic3—31no—
130C22×Dic3—31no—
131C2×D12—34no—
132C2×D12—61no—
133C2×D12—62no—
134C2×D12—61no—
135C2×D12—123no—
136C2×C4×D3—64no—
137C2×C4×D3—62no—
138C2×C4×D3—62no—
139C2×C4×D3—126no—
140C2×C4×D3—122no—
141C4×A4—26no—
142C4×A4C4×D316yescomplemented
143SL(2,ℤ/4)—36no—
144SL(2,ℤ/4)—63no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—43no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4S411yescharacteristic, complemented
150#217—32no—
151#217—32no—
152#217—61no—
153#217—61no—
154(C2×D3)⋊C4—38no—
155(C2×D3)⋊C4—64no—
156(C2×D3)⋊C4—62no—
157(C2×D3)⋊C4—62no—
158(C2×D3)⋊C4—126no—
159(C2×D3)⋊C4—122no—
160C4⋊Dic3—34no—
161Dic3⋊C4—64no—
162C4×Dic3—64no—
163C4×Dic3—61no—
164C4×Dic3—62no—
165C4×Dic3—61no—
166C4×Dic3—123no—
167C22×C32—41no—
168C2×C3⋊D3—44no—
169C2×C3⋊D3—82no—
170C2×C3⋊D3—82no—
171C3×A4C4×D411yescharacteristic, complemented
172C32×C4—42no—
173C32×C4—82no—
174C3⋊Dic3—42no—
175C3⋊Dic3—81no—
176C22×D4—91no—
177C22×D4—92no—
178C23×C4—92no—
179C23×C4C2×C3⋊D312yes—
180C23×C4C2×C3⋊D311yescharacteristic
181#121—94no—
182#121—94no—
183C4×D4—184no—
184C4×D4—184no—
185C4×D4—182no—
186C4×D4—181no—
187C4×D4—181no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4—92no—
193C2×C22⋊C4—92no—
194C2×C22⋊C4—91no—
195C2×C22⋊C4—92no—
196C2×C22⋊C4—92no—
197C2×C22⋊C4—91no—
198C2×C22⋊C4—92no—
199C2×C42—91no—
200C2×C42—92no—
201C2×C42—91no—
202#95—91no—
203#95—91no—
204#95—92no—
205#95—92no—
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—31no—
210C23×C3C2×C4×D311yescharacteristic
211C23×C3D3×D411yescharacteristic
212C23×C3#23911yescharacteristic
213C22×D3—34no—
214C22×D3—62no—
215C22×D3—62no—
216C22×D3—62no—
217C22×D3—62no—
218C22×D3—62no—
219C22×D3—62no—
220C22×D3—126no—
221C2×A4C2×C4×D313yes—
222C2×A4D3×D413yes—
223C2×A4#23913yes—
224S4—612no—
225C2×C3×C4—31no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—43no—
231C2×C3×C4—46no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4C2×S412yes—
238C2×C3×C4C2×S411yescharacteristic
239#67—64no—
240#67—64no—
241#67—64no—
242#67—64no—
243C2×Dic3—32no—
244C2×Dic3—32no—
245C2×Dic3—34no—
246C2×Dic3—62no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—123no—
259C2×Dic3—123no—
260D12—64no—
261D12—64no—
262D12—1212no—
263C4×D3—124no—
264C4×D3—124no—
265C4×D3—2412no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C3⋊D3—84no—
270C24—92no—
271C24C22×C3⋊D311yescharacteristic
272C2×D4—94no—
273C2×D4—94no—
274C2×D4—94no—
275C2×D4—181no—
276C2×D4—182no—
277C2×D4—182no—
278C2×D4—182no—
279C2×D4—182no—
280C2×D4—182no—
281C2×D4—181no—
282C22×C4—22no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—62no—
289C22×C4—92no—
290C22×C4—91no—
291C22×C4—91no—
292C22×C4—91no—
293C22×C4—91no—
294C22×C4—182no—
295C22×C4—184no—
296C22×C4—182no—
297C22×C4—362no—
298C22×C4C4×C3⋊D312yescomplemented
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—61no—
302C4⋊C4—94no—
303C4⋊C4—184no—
304C4⋊C4C3⋊S411yescharacteristic, complemented
305C22⋊C4—98no—
306C22⋊C4—92no—
307C22⋊C4—92no—
308C22⋊C4—182no—
309C22⋊C4—184no—
310C22⋊C4—181no—
311C22⋊C4—181no—
312C22⋊C4—182no—
313C22⋊C4—362no—
314C42—181no—
315C42—182no—
316C42—184no—
317C42—181no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—43no—
328C22×C3—61no—
329C22×C3#89311yescharacteristic, complemented
330C22×C3C22×S411yescharacteristic
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—64no—
334C2×D3—1212no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—246no—
340C2×D3—246no—
341A4#89313yescomplemented
342C3×C4—22no—
343C3×C4—32no—
344C3×C4—46no—
345C3×C4—62no—
346C3×C4—86no—
347C3×C4C4×S412yescomplemented
348Dic3—62no—
349Dic3—64no—
350Dic3—62no—
351Dic3—126no—
352Dic3—121no—
353Dic3—121no—
354Dic3—243no—
355C32—41noSylow
356C23—31no—
357C23—31no—
358C23—31no—
359C23—31no—
360C23—94no—
361C23—182no—
362C23—182no—
363C23—182no—
364C23—182no—
365C23—182no—
366C23—182no—
367C23C2×C4×C3⋊D311yescharacteristic
368C23D4×C3⋊D311yescharacteristic
369C23#383211yescharacteristic
370D4—184no—
371D4—184no—
372D4—184no—
373D4—184no—
374D4—184no—
375D4—184no—
376C2×C4—31no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—61no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—92no—
387C2×C4—94no—
388C2×C4—92no—
389C2×C4—182no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—182no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4—364no—
402C2×C4—364no—
403C2×C4C2×C3⋊S412yes—
404C2×C4C2×C3⋊S411yescharacteristic
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—43no—
410C2×C3—43no—
411C2×C3—43no—
412C2×C3C2×C4×S411yescharacteristic
413C2×C3D4×S411yescharacteristic
414C2×C3#591911yescharacteristic
415D3—124no—
416D3—2412no—
417C22—31no—
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—61no—
427C22—184no—
428C22—184no—
429C22—184no—
430C22—362no—
431C22—362no—
432C22—362no—
433C22—362no—
434C22#6410211yescharacteristic, complemented
435C22C22×C3⋊S411yescharacteristic
436C4—22no—
437C4—32no—
438C4—62no—
439C4—184no—
440C4—182no—
441C4—182no—
442C4—361no—
443C4—361no—
444C4C4×C3⋊S412yescomplemented
445C3—43no—
446C3#8530811yescharacteristic, complemented
447C2—31no—
448C2—31no—
449C2—31no—
450C2—31no—
451C2—364no—
452C2C2×C4×C3⋊S411yescharacteristic
453C2D4×C3⋊S411yescharacteristic
454C2#1059658211yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutatorC2×C3×A4C22×C4
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
cosocleC22×C3×A4C23
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4C4×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊S4C2semidirect
3. C3×A4×C4⋊C4C2semidirect
4. #10593647C2semidirect
5. C4×C3⋊SL(2,ℤ/4)C2semidirect
13. C4⋊(C3⋊S4)C4semidirect
18. C22×C3×C4⋊C4D3semidirect
22. A4×C4⋊C4D3semidirect
35. C2×C3⋊S4C2×C4semidirect
39. C3×C4×A4C2×C4semidirect
41. C3⋊SL(2,ℤ/4)D4semidirect
98. C22×C4⋊C4C3⋊D3semidirect
119. C22×C3×C4C4×D3semidirect
142. C4×A4C4×D3semidirect
149. C3×C4⋊C4S4semidirect
171. C3×A4C4×D4semidirect
298. C22×C4C4×C3⋊D3semidirect
304. C4⋊C4C3⋊S4semidirect
329. C22×C3#893semidirect
341. A4#893semidirect
347. C3×C4C4×S4semidirect
434. C22#64102semidirect
444. C4C4×C3⋊S4semidirect
446. C3#85308semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊S4)2
C2×C4×C3⋊S42
C3×A4×C4⋊C42
#105936472
C4×C3⋊SL(2,ℤ/4)2
#853083
#821633
#641024