Finite Group Databasebeta

← #11922991

Subgroups — #11922991

Order 1152.

664 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#10596575C211yesmaximal, characteristic
2C3×A4×M16C211yesmaximal, characteristic, complemented
3C2×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593634C212yesmaximal, complemented
5C8×C3⋊S4C212yesmaximal, complemented
6#85442—33nomaximal
7#83757—31nomaximal
8C2×C3×C4×A4C2211yescharacteristic
9#64269C412yes—
10C4⋊(C3⋊S4)C411yescharacteristic
11C4×C3⋊S4C2212yes—
12#64259C411yescharacteristic
13#64116—41nomaximal
14C3×C8×A4C2212yes—
15C3⋊(A4⋊C8)C2211yescharacteristic
16C3⋊(A4⋊C8)C2211yescharacteristic
17#5912—33no—
18C22×C3×M16D311yescharacteristic, complemented
19#5791—31no—
20C2×#907—31no—
21A4×M16D313yescomplemented
22C2×A4⋊C8—33no—
23#5400—36no—
24C8×S4—36no—
25C2×#830—31no—
26#5129—31no—
27#5128—32no—
28#5127—32no—
29#5122—31no—
30#5119—31no—
31#5111—32no—
32C8×#67—32no—
33C22×C3×A4C2×C411yescharacteristic
34C2×C3⋊S4C2×C412yes—
35#3828—41no—
36C3×C4×A4C2×C411yescharacteristic
37C3×C4×A4C2311yescharacteristic
38C3⋊SL(2,ℤ/4)C2×C412yes—
39C32×M16—41no—
40C2×C3⋊(C3⋊C8)—41no—
41#3743—42no—
42C8×C3⋊D3—42no—
43#2933—91noSylow
44C23×C3×C4C2×D311yescharacteristic
45C2×#235—31no—
46C2×C4×A4C2×D313yes—
47GL(2,ℤ/4)—36no—
48C4⋊S4—33no—
49C4×S4—36no—
50A4⋊Q8—33no—
51C2×C3×M16—31no—
52C2×C3×M16—32no—
53C2×C3×M16—31no—
54C22×C3×C8C2×D312yes—
55#953—32no—
56#930—31no—
57#930—31no—
58#929—32no—
59C4×#67—32no—
60C4×#67—32no—
61#926—31no—
62#924—31no—
63#924—31no—
64C22×C3⋊C8—31no—
65C22×C3⋊C8—31no—
66#907—61no—
67#907—62no—
68#907—61no—
69#907—123no—
70C2×ΓL(1,𝔽25)—32no—
71C2×D3×C8—32no—
72C8×A4C2×D316yes—
73A4⋊C8—33no—
74A4⋊C8—33no—
75#830—31no—
76#830—31no—
77#830—31no—
78#830—31no—
79(C2×D3)⋊C8—32no—
80(C2×D3)⋊C8—32no—
81#815—32no—
82Dic3⋊C8—32no—
83Dic3⋊C8—32no—
84C8×Dic3—32no—
85C2×C3×A4—21no—
86C2×C3×A4C22×C411yescharacteristic
87C3⋊S4—22no—
88C2×C32×C4—41no—
89#638—42no—
90C4⋊(C3⋊D3)—41no—
91C4×C3⋊D3—42no—
92#634—41no—
93C32×C8—42no—
94C3⋊(C3⋊C8)—41no—
95C3⋊(C3⋊C8)—41no—
96C2×#131—91no—
97C22×M16C3⋊D311yescharacteristic, complemented
98#525—91no—
99#436—92no—
100#435—92no—
101#435—92no—
102C8×D4—92no—
103#424—91no—
104#408—91no—
105C2×C22⋊C8—91no—
106#405—91no—
107C24×C3C4×D311yescharacteristic
108C22×A4C4×D313yes—
109C2×S4—36no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4C4×D311yescharacteristic
115C22×C3×C4C22×D311yescharacteristic
116C2×#67—32no—
117C2×#67—32no—
118C2×#67—32no—
119#235—61no—
120#235—61no—
121#235—61no—
122#235—61no—
123#235—123no—
124C2×D12—31no—
125C2×C4×D3—32no—
126C2×Dic6—31no—
127C4×A4C4×D313yes—
128C4×A4C22×D313yes—
129SL(2,ℤ/4)—36no—
130C3×M16—31no—
131C3×M16—32no—
132C3×M16—43no—
133C3×M16—61no—
134C3×M16S411yescharacteristic, complemented
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137#217—32no—
138#217—32no—
139#217—32no—
140(C2×D3)⋊C4—32no—
141(C2×D3)⋊C4—32no—
142C4⋊Dic3—31no—
143C4⋊Dic3—31no—
144Dic3⋊C4—32no—
145Dic3⋊C4—32no—
146C4×Dic3—32no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—123no—
156ΓL(1,𝔽25)—62no—
157ΓL(1,𝔽25)—62no—
158ΓL(1,𝔽25)—126no—
159D3×C8—62no—
160D3×C8—62no—
161D3×C8—126no—
162C22×C32—41no—
163C2×C3⋊D3—42no—
164C3×A4#13111yescharacteristic, complemented
165C32×C4—41no—
166C32×C4—41no—
167C3⋊Dic3—42no—
168C2×#41—91no—
169C23×C4C2×C3⋊D311yescharacteristic
170#131—181no—
171#131—182no—
172#131—181no—
173C2×M16—31no—
174C2×M16—31no—
175C2×M16—32no—
176C2×M16—92no—
177C22×C8—92no—
178C22×C8—91no—
179C22×C8—91no—
180C22×C8C2×C3⋊D312yes—
181#123—92no—
182C22⋊Q8—91no—
183C22⋊Q8—91no—
184#121—91no—
185#121—91no—
186#120—92no—
187C4×D4—92no—
188C4×D4—92no—
189#117—91no—
190C4⋊C8—92no—
191C4⋊C8—92no—
192C22⋊C8—92no—
193C22⋊C8—92no—
194C22⋊C8—91no—
195C22⋊C8—91no—
196C22⋊C8—91no—
197C22⋊C8—91no—
198#97—92no—
199C4×C8—92no—
200C23×C3—21no—
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—61no—
204C23×C3C2×C4×D311yescharacteristic
205C22×D3—32no—
206C2×A4—23no—
207C2×A4C2×C4×D313yes—
208S4—66no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—43no—
217C2×C3×C4—61no—
218C2×C3×C4C2×S411yescharacteristic
219#67—62no—
220#67—62no—
221#67—62no—
222#67—62no—
223#67—126no—
224#67—122no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—32no—
228D12—61no—
229D12—61no—
230D12—123no—
231C4×D3—62no—
232C4×D3—62no—
233C4×D3—126no—
234Dic6—61no—
235Dic6—61no—
236Dic6—123no—
237C3×C8—32no—
238C3×C8—46no—
239C3×C8C2×S412yes—
240C3⋊C8—61no—
241C3⋊C8—61no—
242C3⋊C8—61no—
243C3⋊C8—61no—
244C3⋊C8—123no—
245C3⋊C8—123no—
246C2×C32—41no—
247C2×C32—81no—
248C3⋊D3—82no—
249C24C4×C3⋊D311yescharacteristic
250#41—181no—
251#41—181no—
252#41—181no—
253#41—181no—
254C2×Q8—91no—
255C2×D4—91no—
256C2×D4—92no—
257C2×D4—92no—
258C2×D4—92no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—92no—
264C22×C4C4×C3⋊D311yescharacteristic
265C22×C4C22×C3⋊D311yescharacteristic
266M16—31no—
267M16—32no—
268M16—61no—
269M16—182no—
270M16—182no—
271M16C3⋊S411yescharacteristic, complemented
272C2×C8—32no—
273C2×C8—32no—
274C2×C8—91no—
275C2×C8—91no—
276C2×C8—91no—
277C2×C8—91no—
278C2×C8—182no—
279C2×C8—181no—
280C2×C8—181no—
281C2×C8—181no—
282C2×C8—181no—
283C2×C8—182no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—92no—
287C4⋊C4—92no—
288C22⋊C4—92no—
289C22⋊C4—92no—
290C22⋊C4—92no—
291C22⋊C4—92no—
292C22⋊C4—92no—
293C42—92no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—43no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3#90711yescharacteristic, complemented
303C22×C3C4×S411yescharacteristic
304C2×D3—62no—
305C2×D3—62no—
306C2×D3—62no—
307C2×D3—126no—
308A4#90713yescomplemented
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—43no—
312C3×C4—43no—
313C3×C4C4×S411yescharacteristic
314C3×C4C22×S411yescharacteristic
315Dic3—62no—
316Dic3—62no—
317Dic3—62no—
318Dic3—126no—
319C32—41noSylow
320C23—21no—
321C23—31no—
322C23—31no—
323C23—61no—
324C23—92no—
325C23C2×C4×C3⋊D311yescharacteristic
326Q8—181no—
327Q8—181no—
328D4—181no—
329D4—182no—
330D4—182no—
331D4—182no—
332D4—182no—
333D4—181no—
334D4—362no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—92no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—182no—
347C2×C4—182no—
348C2×C4C2×C3⋊S411yescharacteristic
349C8—32no—
350C8—181no—
351C8—181no—
352C8—181no—
353C8—181no—
354C8C2×C3⋊S412yes—
355C2×C3—21no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—43no—
359C2×C3—61no—
360C2×C3—83no—
361C2×C3C2×C4×S411yescharacteristic
362D3—122no—
363D3—246no—
364C22—31no—
365C22—31no—
366C22—31no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22—182no—
372C22—182no—
373C22—182no—
374C22#6411611yescharacteristic, complemented
375C22C4×C3⋊S411yescharacteristic
376C4—31no—
377C4—31no—
378C4—182no—
379C4—182no—
380C4—182no—
381C4C4×C3⋊S411yescharacteristic
382C4C22×C3⋊S411yescharacteristic
383C3—43no—
384C3#8544211yescharacteristic, complemented
385C2—21no—
386C2—31no—
387C2—31no—
388C2—61no—
389C2—362no—
390C2C2×C4×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC4C22×C3⋊S4
commutatorC2×C3×A4C22×C4
FittingC22×C3×M16D3
socleC23×C3C2×C4×D3
prime radical#10596575C2
cosocleC3×C4×A4C23
hypercenterM16C3⋊S4
hypocenterC3×A4#131
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×A4×M16C2semidirect
3. C2×C3⋊(A4⋊C8)C2semidirect
4. #10593634C2semidirect
5. C8×C3⋊S4C2semidirect
18. C22×C3×M16D3semidirect
21. A4×M16D3semidirect
97. C22×M16C3⋊D3semidirect
134. C3×M16S4semidirect
164. C3×A4#131semidirect
271. M16C3⋊S4semidirect
302. C22×C3#907semidirect
308. A4#907semidirect
374. C22#64116semidirect
384. C3#85442semidirect

Maximal subgroups

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

subgroupindex
#105965752
C3×A4×M162
C2×C3⋊(A4⋊C8)2
#105936342
C8×C3⋊S42
#854423
#837573
#641164