Finite Group Databasebeta

← #11922861

Subgroups — #11922861

Order 1152.

822 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 101 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⋊S4C211yesmaximal, characteristic
2C2×#64259C211yesmaximal, characteristic, complemented
3C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4#10593647C212yesmaximal
5C4⋊(C3⋊SL(2,ℤ/4))C212yesmaximal, complemented
6#85312—33nomaximal
7#82194—31nomaximal
8C22×C3⋊S4C2211yescharacteristic
9C2×C3×C4×A4C2212yes—
10C2×C3×C4×A4C2211yescharacteristic
11C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
12C2×C3⋊SL(2,ℤ/4)C2212yes—
13C4×C3⋊S4—22no—
14#64259—22no—
15#64106—41nomaximal
16C2×C4×S4—33no—
17C2×A4⋊Q8—33no—
18C22×C3×C4⋊C4D311yescharacteristic, complemented
19C2×C4×#67—31no—
20C2×#924—31no—
21C2×#897—31no—
22A4×C4⋊C4D313yescomplemented
23(C2×S4)⋊C4—36no—
24C4⋊SL(2,ℤ/4)—36no—
25#5213—32no—
26#5212—32no—
27#4994—32no—
28#4987—31no—
29#4984—31no—
30#4981—32no—
31#4973—31no—
32C22×C3×A4C2311yescharacteristic
33C2×C3⋊S4—21no—
34C2×C3⋊S4—21no—
35C2×C3⋊S4Q812yescomplemented
36C2×C4×C3⋊D3—41no—
37C2×#634—41no—
38C3×C4×A4—22no—
39C3×C4×A4D412yes—
40C3⋊SL(2,ℤ/4)—22no—
41C3⋊SL(2,ℤ/4)—21no—
42C32×C4⋊C4—41no—
43#3752—42no—
44C4⋊(C3⋊Dic3)—42no—
45#2417—91noSylow
46C22×S4—33no—
47C23×C3×C4C2×D312yes—
48C23×C3×C4C2×D311yescharacteristic
49C22×#67—31no—
50C22×C4×D3—31no—
51C22×Dic6—31no—
52C2×C4×A4C2×D316yes—
53C2×C4×A4C2×D313yes—
54C2×SL(2,ℤ/4)—36no—
55C2×SL(2,ℤ/4)—33no—
56C4×S4—66no—
57A4⋊Q8—66no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—31no—
60C2×C3×C4⋊C4—32no—
61C2×#217—31no—
62C2×#217—32no—
63C4×#67—62no—
64C4×#67—62no—
65C2×(C2×D3)⋊C4—32no—
66C2×(C2×D3)⋊C4—31no—
67C2×(C2×D3)⋊C4—32no—
68C2×C4⋊Dic3—32no—
69C2×C4⋊Dic3—31no—
70C2×C4⋊Dic3—32no—
71#924—62no—
72#924—62no—
73C2×Dic3⋊C4—31no—
74C2×Dic3⋊C4—32no—
75C2×C4×Dic3—31no—
76#897—61no—
77#897—62no—
78#897—61no—
79#897—123no—
80C3⋊#95—32no—
81C3⋊#95—32no—
82C3⋊#95—32no—
83C3⋊#95—32no—
84C22×C3⋊D3—41no—
85C2×C3×A4C2×D411yescharacteristic
86C2×C3×A4C2×Q811yescharacteristic
87C2×C3×A4#4111yescharacteristic
88C3⋊S4—22no—
89C2×C32×C4—41no—
90C2×C32×C4—42no—
91C2×C3⋊Dic3—41no—
92C2×C3⋊Dic3—42no—
93C4×C3⋊D3—82no—
94#634—82no—
95C2×C22⋊Q8—91no—
96C2×C22⋊Q8—91no—
97C2×C4×D4—91no—
98C22×C4⋊C4C3⋊D311yescharacteristic, complemented
99#398—92no—
100#396—92no—
101#391—91no—
102#389—91no—
103#386—92no—
104#380—91no—
105#380—92no—
106C24×C3C22×D311yescharacteristic
107C23×D3—31no—
108C22×A4C22×D313yes—
109C2×S4—36no—
110C2×S4—63no—
111C2×S4—63no—
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×C4D1212yes—
120C2×#67—32no—
121C2×#67—32no—
122C2×#67—61no—
123C2×#67—61no—
124C2×#67—61no—
125C2×#67—61no—
126C22×Dic3—31no—
127C22×Dic3—31no—
128C22×Dic3—32no—
129C22×Dic3—31no—
130C22×Dic3—32no—
131C2×C4×D3—61no—
132C2×C4×D3—62no—
133C2×C4×D3—62no—
134C2×C4×D3—61no—
135C2×C4×D3—123no—
136C2×Dic6—61no—
137C2×Dic6—62no—
138C2×Dic6—62no—
139C2×Dic6—61no—
140C2×Dic6—123no—
141C4×A4—26no—
142C4×A4D1216yes—
143SL(2,ℤ/4)—66no—
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—62no—
151#217—61no—
152#217—62no—
153#217—61no—
154(C2×D3)⋊C4—62no—
155(C2×D3)⋊C4—62no—
156(C2×D3)⋊C4—62no—
157(C2×D3)⋊C4—126no—
158(C2×D3)⋊C4—122no—
159C4⋊Dic3—62no—
160C4⋊Dic3—62no—
161C4⋊Dic3—62no—
162C4⋊Dic3—126no—
163C4⋊Dic3—122no—
164Dic3⋊C4—64no—
165Dic3⋊C4—62no—
166C4×Dic3—62no—
167C22×C32—41no—
168C2×C3⋊D3—42no—
169C2×C3⋊D3—81no—
170C2×C3⋊D3—81no—
171C3×A4C22⋊Q811yescharacteristic, complemented
172C32×C4—42no—
173C32×C4—82no—
174C3⋊Dic3—82no—
175C3⋊Dic3—81no—
176C22×Q8—91no—
177C22×D4—91no—
178C23×C4—91no—
179C23×C4C2×C3⋊D312yes—
180C23×C4C2×C3⋊D311yescharacteristic
181C22⋊Q8—181no—
182C22⋊Q8—182no—
183C22⋊Q8—182no—
184C22⋊Q8—182no—
185C22⋊Q8—181no—
186C4×D4—182no—
187C4×D4—182no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4—91no—
193C2×C4⋊C4—92no—
194C2×C4⋊C4—92no—
195C2×C4⋊C4—92no—
196C2×C22⋊C4—91no—
197C2×C22⋊C4—92no—
198C2×C22⋊C4—91no—
199C2×C22⋊C4—92no—
200C2×C22⋊C4—92no—
201C2×C42—91no—
202#95—92no—
203#95—92no—
204#95—92no—
205#95—92no—
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—31no—
210C23×C3C2×D1211yescharacteristic
211C23×C3#23711yescharacteristic
212C23×C3D3×Q811yescharacteristic
213C22×D3—32no—
214C22×D3—61no—
215C22×D3—61no—
216C22×D3—61no—
217C22×D3—61no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—123no—
221C2×A4C2×D1213yes—
222C2×A4#23713yes—
223C2×A4D3×Q813yes—
224S4—66no—
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—62no—
240#67—62no—
241#67—62no—
242#67—62no—
243C2×Dic3—32no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—62no—
258C2×Dic3—123no—
259C2×Dic3—126no—
260C4×D3—122no—
261C4×D3—122no—
262C4×D3—246no—
263Dic6—122no—
264Dic6—122no—
265Dic6—246no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C3⋊D3—82no—
270C24—91no—
271C24C22×C3⋊D311yescharacteristic
272C2×Q8—181no—
273C2×Q8—182no—
274C2×Q8—182no—
275C2×Q8—181no—
276C2×D4—92no—
277C2×D4—92no—
278C2×D4—181no—
279C2×D4—181no—
280C2×D4—181no—
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—91no—
290C22×C4—91no—
291C22×C4—92no—
292C22×C4—92no—
293C22×C4—91no—
294C22×C4—182no—
295C22×C4—181no—
296C22×C4—182no—
297C22×C4—181no—
298C22×C4C4⋊(C3⋊D3)12yes—
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—61no—
302C4⋊C4—182no—
303C4⋊C4—184no—
304C4⋊C4—182no—
305C4⋊C4—182no—
306C4⋊C4—182no—
307C4⋊C4—362no—
308C4⋊C4C3⋊S411yescharacteristic, complemented
309C22⋊C4—182no—
310C22⋊C4—182no—
311C22⋊C4—181no—
312C22⋊C4—182no—
313C22⋊C4—181no—
314C22⋊C4—182no—
315C22⋊C4—182no—
316C22⋊C4—362no—
317C42—182no—
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#89711yescharacteristic, complemented
330C22×C3C22×S411yescharacteristic
331C2×D3—62no—
332C2×D3—62no—
333C2×D3—62no—
334C2×D3—126no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—243no—
340C2×D3—243no—
341A4#89713yescomplemented
342C3×C4—22no—
343C3×C4—32no—
344C3×C4—46no—
345C3×C4—62no—
346C3×C4—86no—
347C3×C4C4⋊S412yes—
348Dic3—62no—
349Dic3—121no—
350Dic3—122no—
351Dic3—122no—
352Dic3—121no—
353Dic3—246no—
354Dic3—243no—
355C32—41noSylow
356C23—31no—
357C23—31no—
358C23—31no—
359C23—31no—
360C23—92no—
361C23—181no—
362C23—181no—
363C23—181no—
364C23—181no—
365C23—181no—
366C23—181no—
367C23C2×C4⋊(C3⋊D3)11yescharacteristic
368C23#383011yescharacteristic
369C23Q8×C3⋊D311yescharacteristic
370Q8—362no—
371Q8—362no—
372D4—182no—
373D4—182no—
374D4—182no—
375D4—182no—
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—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—182no—
391C2×C4—181no—
392C2×C4—182no—
393C2×C4—182no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—182no—
397C2×C4—181no—
398C2×C4—182no—
399C2×C4—182no—
400C2×C4—181no—
401C2×C4—362no—
402C2×C4—362no—
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×C3#591411yescharacteristic
414C2×C3Q8×S411yescharacteristic
415D3—122no—
416D3—246no—
417C22—31no—
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—61no—
427C22—182no—
428C22—182no—
429C22—182no—
430C22—361no—
431C22—361no—
432C22—361no—
433C22—361no—
434C22#6410611yescharacteristic, complemented
435C22C22×C3⋊S411yescharacteristic
436C4—22no—
437C4—32no—
438C4—62no—
439C4—182no—
440C4—361no—
441C4—362no—
442C4—361no—
443C4—362no—
444C4C4⋊(C3⋊S4)12yes—
445C3—43no—
446C3#8531211yescharacteristic, complemented
447C2—31no—
448C2—31no—
449C2—31no—
450C2—31no—
451C2—362no—
452C2C2×C4⋊(C3⋊S4)11yescharacteristic
453C2#1059657711yescharacteristic
454C2Q8×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC22×C3×A4C23
FittingC22×C3×C4⋊C4D3
socleC24×C3C22×D3
prime radicalC22×C3⋊S4C22
hypercenterC4⋊C4C3⋊S4
hypocenterC3×A4C22⋊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
2. C2×#64259C2semidirect
3. C3×A4×C4⋊C4C2semidirect
5. C4⋊(C3⋊SL(2,ℤ/4))C2semidirect
18. C22×C3×C4⋊C4D3semidirect
22. A4×C4⋊C4D3semidirect
35. C2×C3⋊S4Q8semidirect
98. C22×C4⋊C4C3⋊D3semidirect
149. C3×C4⋊C4S4semidirect
171. C3×A4C22⋊Q8semidirect
308. C4⋊C4C3⋊S4semidirect
329. C22×C3#897semidirect
341. A4#897semidirect
434. C22#64106semidirect
446. C3#85312semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊S42
C2×#642592
C3×A4×C4⋊C42
#105936472
C4⋊(C3⋊SL(2,ℤ/4))2
#853123
#821943
#641064