Finite Group Databasebeta

← #11867768

Subgroups — #11867768

Order 1152.

1416 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596304C211yesmaximal, characteristic
2C2×#64160C211yesmaximal, characteristic, complemented
3C2×#64110C211yesmaximal, characteristic, complemented
4#10595997C211yesmaximal, characteristic, complemented
5C32×#416C211yesmaximal, characteristic, complemented
6C2×#63643C211yesmaximal, characteristic, complemented
7#10592694C211yesmaximal, characteristic, complemented
8#10592515C212yesmaximal, complemented
9#10592512C212yesmaximal, complemented
10#10592479C212yesmaximal, complemented
11#10592477C212yesmaximal, complemented
12#78841—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C2×#3828C2211yescharacteristic
15C22×#634C2211yescharacteristic
16C22×C32×C8C2211yescharacteristic
17C32×#117C2211yescharacteristic
18#64160—24no—
19#64155C2212yes—
20C2×#3756C2212yes—
21#64148—22no—
22#64147C2212yes—
23C2×#3754C2212yes—
24#64139—21no—
25C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
26#64133—21no—
27C2×#3748C2211yescharacteristic
28#64110—24no—
29C2×#3743C2212yes—
30C2×C8×C3⋊D3C2212yes—
31#64106C2212yes—
32#64106C2212yes—
33#64095—22no—
34#64084—21no—
35#64078—21no—
36C32×#103C2211yescharacteristic
37C32×#103C2211yescharacteristic
38C32×#102C2211yescharacteristic
39C32×#102C2211yescharacteristic
40#63643C2212yes—
41#63643C2212yes—
42#63635C2212yes—
43#63634C2212yes—
44#5833—34no—
45C2×#951—34no—
46C2×#901—34no—
47#5526—34no—
48C3×#416D314yescomplemented
49C2×#816—34no—
50#4995—34no—
51#4816—38no—
52#4813—38no—
53#4780—38no—
54#4778—38no—
55C32×#41—22no—
56C32×#41—22no—
57C2×C32×Q8C2311yescharacteristic
58C2×C32×D4—21no—
59C2×C32×D4C2311yescharacteristic
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C2×#638—22no—
63C2×#638D412yes—
64C22×C3⋊Dic3—21no—
65C22×C3⋊Dic3—21no—
66#3828—22no—
67#3828—22no—
68C2×C4⋊(C3⋊D3)D411yescharacteristic
69C2×C4×C3⋊D3C2312yes—
70C2×#634—22no—
71C2×#634—22no—
72C2×#634—21no—
73C2×#634D411yescharacteristic
74C2×#634C2312yes—
75C2×C32×C8—22no—
76C2×C32×C8C2311yescharacteristic
77C2×C32×C8C2311yescharacteristic
78C32×C4⋊C4C2311yescharacteristic
79C32×C4⋊C4C2311yescharacteristic
80C32×C22⋊C4—21no—
81C32×C42—21no—
82#3757—22no—
83#3756—24no—
84#3754—24no—
85#3752—22no—
86#3752—22no—
87#3752—22no—
88C4⋊(C3⋊Dic3)—21no—
89C4⋊(C3⋊Dic3)—22no—
90C4⋊(C3⋊Dic3)—22no—
91C4⋊(C3⋊Dic3)C2312yes—
92#3750—22no—
93#3748D414yescomplemented
94C2×C3⋊(C3⋊C8)C2312yes—
95#3743—24no—
96C8×C3⋊D3—24no—
97#3063—91noSylow
98C2×C3×#41C2×D314yes—
99C2×#235—34no—
100C22×Dic6—34no—
101C22×C3×C8C2×D314yes—
102C3×#117C2×D314yes—
103#951—616no—
104#946—38no—
105C2×#216—38no—
106#939—68no—
107#938—38no—
108C2×#214—38no—
109#930—64no—
110C2×C4⋊Dic3—34no—
111#924—64no—
112C2×#208—34no—
113#901—616no—
114C2×ΓL(1,𝔽25)—38no—
115C2×D3×C8—38no—
116#897—38no—
117#897—38no—
118#886—68no—
119#875—64no—
120#869—64no—
121C3×#103C2×D314yes—
122C3×#103C2×D314yes—
123C3×#102C2×D314yes—
124C3×#102C2×D314yes—
125#816—38no—
126#816—38no—
127#808—38no—
128#807—38no—
129C23×C32—21no—
130C23×C32C2×D411yescharacteristic
131C22×C3⋊D3C2×D412yes—
132C32×Q8—21no—
133C32×Q8—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—42no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D412yes—
143C2×C32×C4#4112yes—
144C2×C32×C4C2411yescharacteristic
145#638—24no—
146#638—82no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—42no—
150C2×C3⋊Dic3—81no—
151C2×C3⋊Dic3C2×D412yes—
152C4⋊(C3⋊D3)—22no—
153C4×C3⋊D3—22no—
154C4×C3⋊D3—22no—
155#634—22no—
156#634—22no—
157#634—22no—
158#634—42no—
159C32×C8—21no—
160C32×C8—21no—
161C3⋊(C3⋊C8)C2×D414yes—
162C2×#137—91no—
163C2×#131—91no—
164#537—91no—
165#536—91no—
166#470—92no—
167#468—92no—
168#467—92no—
169#465—92no—
170#428—91no—
171#416C3⋊D311yescharacteristic, complemented
172C2×#103—91no—
173C3×#41—28no—
174C3×#41—28no—
175C2×C3×Q8C22×D314yes—
176C2×C3×D4—24no—
177C2×C3×D4C22×D314yes—
178C22×C3×C4—24no—
179C22×C3×C4C22×D314yes—
180C2×#67—38no—
181C2×#67—68no—
182C22×Dic3—64no—
183C22×Dic3—64no—
184#235—68no—
185#235—68no—
186C2×D12—34no—
187C2×C4×D3—38no—
188C2×Dic6—34no—
189C2×Dic6—38no—
190C2×Dic6—68no—
191C2×Dic6—68no—
192C2×Dic6—64no—
193C2×C3×C8—28no—
194C2×C3×C8C22×D314yes—
195C2×C3×C8C22×D314yes—
196C3×C4⋊C4C22×D314yes—
197C3×C4⋊C4C22×D314yes—
198C3×C22⋊C4—24no—
199C3×C42—24no—
200#217—68no—
201#216—616no—
202#214—616no—
203(C2×D3)⋊C4—68no—
204(C2×D3)⋊C4—68no—
205(C2×D3)⋊C4—68no—
206C4⋊Dic3—38no—
207C4⋊Dic3—68no—
208C4⋊Dic3—68no—
209C4⋊Dic3—64no—
210Dic3⋊C4—68no—
211#208—316no—
212C2×C3⋊C8—38no—
213ΓL(1,𝔽25)—616no—
214D3×C8—616no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32#12112yes—
221C22×C32C22×D411yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4#12112yes—
229C32×C4C22×D411yescharacteristic
230C32×C4C2×#4111yescharacteristic
231C3⋊Dic3—22no—
232C3⋊Dic3—42no—
233C3⋊Dic3—42no—
234C2×#41—91no—
235C2×#41C2×C3⋊D311yescharacteristic
236C22×Q8—91no—
237#137—184no—
238C2×Q16—92no—
239C2×SD16—92no—
240#131—184no—
241C2×M16—91no—
242C2×M16—92no—
243C22×C8—92no—
244C22×C8C2×C3⋊D311yescharacteristic
245C4⋊Q8—181no—
246#124—181no—
247#123—182no—
248C22⋊Q8—92no—
249C22⋊Q8—92no—
250C22⋊Q8—92no—
251C22⋊Q8—181no—
252#121—92no—
253#121—181no—
254#121—182no—
255#117C2×C3⋊D311yescharacteristic
256C2×C4⋊C4—91no—
257#107—92no—
258#106—92no—
259#103—92no—
260#103—92no—
261#103C2×C3⋊D311yescharacteristic
262#103C2×C3⋊D311yescharacteristic
263#102C2×C3⋊D311yescharacteristic
264#102C2×C3⋊D311yescharacteristic
265C23×C3—24no—
266C23×C3D3×D414yes—
267C22×D3—38no—
268C3×Q8—24no—
269C3×Q8—24no—
270C3×D4—24no—
271C3×D4—24no—
272C3×D4—48no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—48no—
277C2×C3×C4#23518yes—
278C2×C3×C4D3×D414yes—
279C2×C3×C4D3×D418yes—
280C2×C3×C4C23×D314yes—
281#67—616no—
282#67—248no—
283C2×Dic3—38no—
284C2×Dic3—68no—
285C2×Dic3—68no—
286C2×Dic3—128no—
287C2×Dic3—244no—
288D12—68no—
289C4×D3—68no—
290C4×D3—68no—
291Dic6—68no—
292Dic6—68no—
293Dic6—68no—
294Dic6—128no—
295C3×C8—24no—
296C3×C8—24no—
297C3⋊C8—316no—
298C2×C32—22no—
299C2×C32—81no—
300C2×C32C2×#12111yescharacteristic
301C2×C32#57711yescharacteristic
302C2×C32#57811yescharacteristic
303C3⋊D3—42no—
304#41—22no—
305#41—22no—
306#41—182no—
307#41—182no—
308C2×Q8—91no—
309C2×Q8—92no—
310C2×Q8—181no—
311C2×Q8—182no—
312C2×Q8—182no—
313C2×Q8C22×C3⋊D311yescharacteristic
314C2×D4—21no—
315C2×D4—92no—
316C2×D4—91no—
317C2×D4—182no—
318C2×D4C22×C3⋊D311yescharacteristic
319C22×C4—21no—
320C22×C4—92no—
321C22×C4—181no—
322C22×C4—181no—
323C22×C4C22×C3⋊D311yescharacteristic
324Q16—184no—
325SD16—184no—
326M16—94no—
327M16—184no—
328C2×C8—22no—
329C2×C8—92no—
330C2×C8—184no—
331C2×C8C22×C3⋊D311yescharacteristic
332C2×C8C22×C3⋊D311yescharacteristic
333C4⋊C4—92no—
334C4⋊C4—182no—
335C4⋊C4—182no—
336C4⋊C4—182no—
337C4⋊C4—181no—
338C4⋊C4C22×C3⋊D311yescharacteristic
339C4⋊C4C22×C3⋊D311yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—182no—
342C22⋊C4—182no—
343C22⋊C4—182no—
344C22⋊C4—182no—
345C42—21no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3#88418yes—
352C22×C3C2×D3×D414yes—
353C2×D3—68no—
354C2×D3—68no—
355C2×D3—68no—
356C3×C4—44no—
357C3×C4—44no—
358C3×C4—44no—
359C3×C4#88418yes—
360C3×C4C2×#23514yes—
361C3×C4C2×D3×D414yes—
362Dic3—68no—
363Dic3—128no—
364Dic3—128no—
365C32#306311yescharacteristic, complemented, Sylow
366C23—21no—
367C23—92no—
368C23D4×C3⋊D311yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—182no—
372Q8—182no—
373Q8—182no—
374Q8—362no—
375D4—21no—
376D4—21no—
377D4—42no—
378D4—184no—
379D4—182no—
380D4—722no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—92no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4—362no—
391C2×C4—721no—
392C2×C4#382812yes—
393C2×C4D4×C3⋊D311yescharacteristic
394C2×C4D4×C3⋊D312yes—
395C2×C4C23×C3⋊D311yescharacteristic
396C8—21no—
397C8—21no—
398C8—94no—
399C2×C3—28no—
400C2×C3—84no—
401C2×C3C2×#88414yes—
402C2×C3#577014yes—
403C2×C3#577114yes—
404D3—128no—
405C22—21no—
406C22—21no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—182no—
411C22—182no—
412C22—182no—
413C22#6409312yes—
414C22C2×D4×C3⋊D311yescharacteristic
415C4—41no—
416C4—41no—
417C4—41no—
418C4—182no—
419C4—362no—
420C4—362no—
421C4#6409312yes—
422C4C2×#382811yescharacteristic
423C4C2×D4×C3⋊D311yescharacteristic
424C3#7884114yescomplemented
425C2—22no—
426C2—81no—
427C2—362no—
428C2C2×#6409311yescharacteristic
429C2#1059624111yescharacteristic
430C2#1059624211yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#416C2
socleC22×C32C22×D4
prime radical#10596304C2
hypercenter#416C3⋊D3
hypocenterC32#3063
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#64160C2semidirect
3. C2×#64110C2semidirect
4. #10595997C2semidirect
5. C32×#416C2semidirect
6. C2×#63643C2semidirect
7. #10592694C2semidirect
8. #10592515C2semidirect
9. #10592512C2semidirect
10. #10592479C2semidirect
11. #10592477C2semidirect
48. C3×#416D3semidirect
93. #3748D4semidirect
171. #416C3⋊D3semidirect
365. C32#3063semidirect
424. C3#78841semidirect

Maximal subgroups

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

subgroupindex
#105963042
C2×#641602
C2×#641102
#105959972
C32×#4162
C2×#636432
#105926942
#105925152
#105925122
#105924792
#105924772
#788413