Finite Group Databasebeta

← #11775154

Subgroups — #11775154

Order 1152.

699 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 94 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#10596067C211yesmaximal, characteristic
2#10591252C211yesmaximal, characteristic, complemented
3#10591066C211yesmaximal, characteristic, complemented
4C3×#4537C212yesmaximal, complemented
5#10589151C212yesmaximal, complemented
6#68278—32nomaximal
7C2×#3830—21no—
8C32×#121C2211yescharacteristic
9#64149C411yescharacteristic, complemented
10#64148—21no—
11D4×C3⋊Dic3C411yescharacteristic, complemented
12#64139C2211yescharacteristic
13C2×C4×C3⋊Dic3C2211yescharacteristic
14#64102C411yescharacteristic, complemented
15#64097C411yescharacteristic, complemented
16#64094—21no—
17#64090—21no—
18C2×Dic32—21no—
19C3×C3⋊#95C2212yes—
20C3×#830C2212yes—
21#63579—21no—
22#63572—21no—
23#63555—21no—
24#5596—32no—
25#5596—61no—
26#4719—32no—
27#4648—32no—
28C3×#327—32no—
29#4537D312yescomplemented
30#4470—32no—
31#4451—32no—
32C2×C32×D4—21no—
33C2×C32×D4C2×C411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C2×#638—21no—
36C2×#638—21no—
37C2×#638—21no—
38C22×C3⋊Dic3—21no—
39C22×C3⋊Dic3D411yescharacteristic
40C22×C3⋊Dic3D411yescharacteristic
41#3830—44no—
42C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×#634—21no—
45C22×C3×Dic3—22no—
46C32×C4⋊C4C2×C411yescharacteristic
47C32×C22⋊C4—21no—
48#3757—21no—
49#3752—21no—
50#3752—21no—
51C4⋊(C3⋊Dic3)C2×C411yescharacteristic
52#3750—21no—
53C4×C3⋊Dic3C2×C411yescharacteristic
54C4×C3⋊Dic3C2×C411yescharacteristic
55C4×C3⋊Dic3D411yescharacteristic
56C4×C3⋊Dic3D411yescharacteristic
57C2×C3×C3⋊C8—22no—
58Dic32—24no—
59#1505—91noSylow
60C2×#237—62no—
61C2×#237—62no—
62C3×#121—21no—
63C3×#121C2×D312yes—
64#940—32no—
65#940—61no—
66#939—62no—
67#939—62no—
68D4×Dic3—32no—
69D4×Dic3—61no—
70#930—32no—
71#930—61no—
72C2×C4×Dic3—32no—
73C2×C4×Dic3—61no—
74C2×C4×Dic3—62no—
75#893—32no—
76#893—61no—
77#888—32no—
78#888—61no—
79#885—62no—
80#885—62no—
81#67⋊C4—62no—
82#67⋊C4—62no—
83C3×C22⋊C8—32no—
84C3×#95—32no—
85C3⋊#95—62no—
86C3⋊#95C2×D312yes—
87#830C2×D312yes—
88#815—62no—
89Dic3⋊C8—62no—
90C23×C32—21no—
91C23×C32C2×D411yescharacteristic
92C22×C3⋊D3—21no—
93C32×D4—41no—
94C32×D4—42no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4C22×C411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99#638—42no—
100#638—42no—
101#638—42no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—42no—
106C2×C3⋊Dic3C22⋊C411yescharacteristic
107C2×C3⋊Dic3C22⋊C411yescharacteristic
108C2×C3⋊Dic3C22⋊C411yescharacteristic
109C2×C3⋊Dic3C22⋊C411yescharacteristic
110C4⋊(C3⋊D3)—41no—
111C4×C3⋊D3—42no—
112#634—42no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—42no—
116C3×C3⋊C8—42no—
117#532—91no—
118#432—91no—
119#376—91no—
120#327—32no—
121#327—92no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4C4×D312yes—
126C22×C3×C4—21no—
127C22×C3×C4—62no—
128C22×C3×C4C22×D312yes—
129C2×#67—62no—
130C2×#67—62no—
131C2×#67—62no—
132C2×#67—62no—
133C2×#67—62no—
134C2×#67—62no—
135C22×Dic3—22no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—61no—
139C22×Dic3—61no—
140C22×Dic3—62no—
141C22×Dic3—62no—
142#237—128no—
143#237—124no—
144#237—124no—
145C2×D12—32no—
146C2×D12—61no—
147C2×C4×D3—62no—
148C2×C4×D3—62no—
149C2×Dic6—62no—
150C2×Dic6—62no—
151C2×C3×C8—62no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4C4×D312yes—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—22no—
156#217—62no—
157#217—62no—
158(C2×D3)⋊C4—62no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—62no—
162C4⋊Dic3—32no—
163C4⋊Dic3—61no—
164Dic3⋊C4—62no—
165Dic3⋊C4—62no—
166C4×Dic3—32no—
167C4×Dic3—32no—
168C4×Dic3—32no—
169C4×Dic3—32no—
170C4×Dic3—61no—
171C4×Dic3—61no—
172C4×Dic3—61no—
173C4×Dic3—61no—
174C4×Dic3—68no—
175C2×C3⋊C8—22no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32C2×C22⋊C411yescharacteristic
182C22×C32AGL(1,ℤ/8)12yes—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—41no—
189C3⋊Dic3—21no—
190C3⋊Dic3—21no—
191C3⋊Dic3—21no—
192C3⋊Dic3—41no—
193C3⋊Dic3#10412yes—
194C3×Dic3—44no—
195C2×#41—181no—
196C4⋊Q8—91no—
197#127—91no—
198#124—181no—
199#121—91no—
200#121—181no—
201#121D3211yescharacteristic
202C4×D4—91no—
203C4×D4—91no—
204C4×D4—181no—
205C2×C42—91no—
206C2×C42—181no—
207C4⋊C8—181no—
208C22⋊C8—32no—
209#97—181no—
210#95—32no—
211#95—181no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—21no—
215C23×C3D3×D412yes—
216C22×D3—62no—
217C22×D3—62no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—44no—
221C3×D4—42no—
222C3×D4—42no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—64no—
230C2×C3×C4—122no—
231C2×C3×C4—122no—
232C2×C3×C4C2×C4×D312yes—
233C2×C3×C4D3×D412yes—
234#67—124no—
235#67—124no—
236#67—124no—
237#67—124no—
238#67—124no—
239#67—122no—
240#67—122no—
241C2×Dic3—24no—
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—32no—
245C2×Dic3—32no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—64no—
253C2×Dic3—64no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—122no—
259C2×Dic3—124no—
260C2×Dic3—124no—
261D12—122no—
262D12—122no—
263C4×D3—124no—
264C4×D3—124no—
265Dic6—124no—
266Dic6—124no—
267C3×C8—122no—
268C3⋊C8—42no—
269C2×C32—22no—
270C2×C32—81no—
271C2×C32#41011yescharacteristic
272C2×C32#41811yescharacteristic
273C2×C32C2×#10411yescharacteristic
274C3⋊D3—81no—
275#41—364no—
276C2×Q8—181no—
277C2×D4—21no—
278C2×D4—91no—
279C2×D4—181no—
280C2×D4—181no—
281C2×D4—181no—
282C2×D4#62411yescharacteristic
283C22×C4—62no—
284C22×C4—91no—
285C22×C4—91no—
286C22×C4—181no—
287C22×C4—181no—
288C22×C4C2×D3211yescharacteristic
289C2×C8—62no—
290C4⋊C4—91no—
291C4⋊C4—181no—
292C4⋊C4#62411yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22⋊C4—181no—
297C42—91no—
298C42—91no—
299C42—91no—
300C42—91no—
301C42—184no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3D3×C22⋊C412yes—
315C22×C3#91112yes—
316C22×C3#91412yes—
317C2×D3—122no—
318C2×D3—122no—
319C2×D3—122no—
320C2×D3—122no—
321C2×D3—122no—
322C2×D3—122no—
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—124no—
330Dic3—34no—
331Dic3—44no—
332Dic3—62no—
333Dic3—62no—
334Dic3—62no—
335Dic3—62no—
336Dic3—62no—
337Dic3—62no—
338Dic3—121no—
339Dic3—122no—
340Dic3—122no—
341C32#150511yescharacteristic, complemented, Sylow
342C23—21no—
343C23—181no—
344C23#381111yescharacteristic
345Q8—362no—
346D4—41no—
347D4—42no—
348D4—361no—
349D4—362no—
350D4—362no—
351D4—362no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—64no—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—91no—
358C2×C4—91no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—181no—
364C2×C4—362no—
365C2×C4—362no—
366C2×C4C2×#62411yescharacteristic
367C2×C4#381111yescharacteristic
368C8—122no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—42no—
375C2×C3—82no—
376C2×C3—82no—
377C2×C3#474012yes—
378C2×C3#477012yes—
379C2×C3D3×#10412yes—
380D3—242no—
381D3—242no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—361no—
388C22—361no—
389C22—361no—
390C22#6393712yes—
391C22#6397411yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—92no—
396C4—124no—
397C4—181no—
398C4—181no—
399C4—181no—
400C4—361no—
401C3—21no—
402C3#6827812yescomplemented
403C2—22no—
404C2—81no—
405C2—721no—
406C2#1059142611yescharacteristic
407C2#1059152911yescharacteristic
408C2#1059167111yescharacteristic
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#121C22
socleC22×C32C2×C22⋊C4
prime radical#10596067C2
cosocleC22×C32×C4C23
hypercenter#121D32
hypocenterC32#1505
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10591252C2semidirect
3. #10591066C2semidirect
4. C3×#4537C2semidirect
5. #10589151C2semidirect
9. #64149C4semidirect
11. D4×C3⋊Dic3C4semidirect
14. #64102C4semidirect
15. #64097C4semidirect
29. #4537D3semidirect
341. C32#1505semidirect
402. C3#68278semidirect

Maximal subgroups

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

subgroupindex
#105960672
#105912522
#105910662
C3×#45372
#105891512
#682783