Finite Group Databasebeta

← #11867765

Subgroups — #11867765

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
1C32×#537C211yesmaximal, characteristic
2#10596296C211yesmaximal, characteristic, complemented
3C2×#64160C211yesmaximal, characteristic, complemented
4C2×#64157C211yesmaximal, characteristic, complemented
5#10592932C211yesmaximal, characteristic, complemented
6#10592751C212yesmaximal, complemented
7#10592750C212yesmaximal, complemented
8#10592741C212yesmaximal, complemented
9#10592740C212yesmaximal, complemented
10C2×#63637C211yesmaximal, characteristic, complemented
11#10592663C211yesmaximal, characteristic, complemented
12#83423—34nomaximal
13C2×C32×#41C2211yescharacteristic
14C2×C32×#41C2211yescharacteristic
15C22×#634C2211yescharacteristic
16C32×C22⋊Q8—21no—
17C32×C22⋊Q8C2212yes—
18C32×#121—21no—
19C32×#121—22no—
20C32×#121C2212yes—
21C2×C32×C4⋊C4C2211yescharacteristic
22#64160—24no—
23#64157—24no—
24#64153—21no—
25C2×#3756C2212yes—
26#64145—21no—
27#64144—22no—
28C2×#3754C2212yes—
29#64135C2211yescharacteristic
30#64133C2212yes—
31#64133C2212yes—
32C2×#3748C2212yes—
33C2×#3748C2211yescharacteristic
34C22×C3⋊(C3⋊C8)C2211yescharacteristic
35C22×C3⋊(C3⋊C8)C2212yes—
36#63662C2211yescharacteristic
37#63662C2211yescharacteristic
38#63659C2211yescharacteristic
39#63659C2211yescharacteristic
40#63637C2212yes—
41#63637C2212yes—
42#63635C2212yes—
43#63634C2212yes—
44C3×#537D314yes—
45#5825—34no—
46C2×#951—34no—
47C2×#948—34no—
48#5233—34no—
49#5052—38no—
50#5051—38no—
51#5042—38no—
52#5041—38no—
53C2×#810—34no—
54#4964—34no—
55C32×#41—22no—
56C32×#41—22no—
57C32×#41—22no—
58C32×#41—22no—
59C2×C32×Q8D411yescharacteristic
60C2×C32×Q8C2311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4—22no—
63C2×C32×D4D412yes—
64C2×C32×D4D411yescharacteristic
65C2×C32×D4C2311yescharacteristic
66C22×C32×C4—21no—
67C22×C32×C4—21no—
68C22×C32×C4C2312yes—
69C22×C32×C4C2311yescharacteristic
70C22×C3⋊Dic3—21no—
71C2×#634—22no—
72C2×#634—22no—
73C2×#634—21no—
74C2×#634C2312yes—
75C32×C4⋊C4—21no—
76C32×C4⋊C4—22no—
77C32×C4⋊C4C2312yes—
78C32×C22⋊C4—22no—
79C32×C22⋊C4—22no—
80#3757—22no—
81#3757—22no—
82#3757—21no—
83#3756—24no—
84#3754—24no—
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86C4⋊(C3⋊Dic3)C2311yescharacteristic
87#3750—22no—
88#3750—22no—
89C4×C3⋊Dic3—21no—
90#3748—24no—
91#3748D414yescomplemented
92C2×C3⋊(C3⋊C8)—22no—
93C2×C3⋊(C3⋊C8)—24no—
94C2×C3⋊(C3⋊C8)C2311yescharacteristic
95C2×C3⋊(C3⋊C8)C2311yescharacteristic
96C2×C3⋊(C3⋊C8)C2312yes—
97#3063—91noSylow
98C2×C3×#41C2×D314yes—
99C2×C3×#41C2×D314yes—
100C22×Dic6—34no—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8C2×D318yes—
103C3×#121—24no—
104C3×#121—28no—
105C3×#121C2×D318yes—
106C2×C3×C4⋊C4C2×D314yes—
107#951—616no—
108#948—616no—
109#944—64no—
110C2×#216—38no—
111#936—64no—
112#935—68no—
113C2×#214—38no—
114#926—34no—
115#924—38no—
116#924—38no—
117C2×#208—34no—
118C2×#208—38no—
119C22×C3⋊C8—34no—
120C22×C3⋊C8—38no—
121#835—34no—
122#835—34no—
123#832—34no—
124#832—34no—
125#810—38no—
126#810—38no—
127#808—38no—
128#807—38no—
129C23×C32—21no—
130C23×C32C2×D412yes—
131C23×C32C2×D411yescharacteristic
132C32×Q8—22no—
133C32×Q8—21no—
134C32×Q8—21no—
135C32×D4—24no—
136C32×D4—22no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—42no—
140C32×D4—82no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—42no—
146C2×C32×C4—81no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4#4112yes—
151C2×C32×C4C2411yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—42no—
156#634—22no—
157#634—22no—
158#634—42no—
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)C2×D414yes—
162C2×#137—91no—
163C2×#131—91no—
164#537C3⋊D311yescharacteristic
165#536—91no—
166#470—92no—
167#468—92no—
168#467—92no—
169#465—92no—
170#428—91no—
171#416—91no—
172C2×#103—91no—
173C3×#41—28no—
174C3×#41—28no—
175C3×#41—28no—
176C3×#41—28no—
177C2×C3×Q8#6714yes—
178C2×C3×Q8C22×D314yes—
179C2×C3×D4—24no—
180C2×C3×D4—28no—
181C2×C3×D4#6718yes—
182C2×C3×D4#6714yes—
183C2×C3×D4C22×D314yes—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4C22×D318yes—
187C22×C3×C4C22×D314yes—
188C22×Dic3—64no—
189C2×Dic6—38no—
190C2×Dic6—68no—
191C2×Dic6—68no—
192C2×Dic6—64no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4C22×D318yes—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198#217—68no—
199#217—68no—
200#217—64no—
201#216—616no—
202#214—616no—
203C4⋊Dic3—34no—
204C4⋊Dic3—34no—
205Dic3⋊C4—68no—
206Dic3⋊C4—68no—
207C4×Dic3—64no—
208#208—316no—
209#208—616no—
210C2×C3⋊C8—38no—
211C2×C3⋊C8—34no—
212C2×C3⋊C8—34no—
213C2×C3⋊C8—616no—
214C2×C3⋊C8—68no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32#12112yes—
224C22×C32C22×D411yescharacteristic
225C32×C4—22no—
226C32×C4—41no—
227C32×C4—42no—
228C32×C4#12112yes—
229C32×C4C22×D411yescharacteristic
230C32×C4C2×#4111yescharacteristic
231C3⋊Dic3—42no—
232C3⋊Dic3—41no—
233C3⋊Dic3—41no—
234C2×#41C2×C3⋊D311yescharacteristic
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—91no—
244C22×C8—92no—
245C4⋊Q8—181no—
246#124—181no—
247#123—182no—
248C22⋊Q8—21no—
249C22⋊Q8—92no—
250C22⋊Q8—92no—
251C22⋊Q8C2×C3⋊D312yes—
252#121—21no—
253#121—22no—
254#121C2×C3⋊D312yes—
255#117—91no—
256C2×C4⋊C4C2×C3⋊D311yescharacteristic
257#107—92no—
258#106—92no—
259#103—91no—
260#103—91no—
261#103—92no—
262#103—92no—
263#102—91no—
264#102—91no—
265C23×C3—24no—
266C23×C3C2×#6718yes—
267C23×C3C2×#6714yes—
268C3×Q8—28no—
269C3×Q8—24no—
270C3×Q8—24no—
271C3×D4—216no—
272C3×D4—28no—
273C3×D4—24no—
274C3×D4—24no—
275C3×D4—48no—
276C3×D4—88no—
277C2×C3×C4—28no—
278C2×C3×C4—28no—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—48no—
282C2×C3×C4—84no—
283C2×C3×C4D3×D418yes—
284C2×C3×C4#23718yes—
285C2×C3×C4C2×#6718yes—
286C2×C3×C4C2×#6714yes—
287C2×C3×C4C23×D314yes—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C2×Dic3—68no—
291C2×Dic3—128no—
292Dic6—68no—
293Dic6—68no—
294Dic6—128no—
295C3⋊C8—316no—
296C3⋊C8—64no—
297C3⋊C8—64no—
298C2×C32—22no—
299C2×C32—42no—
300C2×C32—81no—
301C2×C32C2×#12111yescharacteristic
302C2×C32#57711yescharacteristic
303C2×C32#57811yescharacteristic
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308C2×Q8—92no—
309C2×Q8—181no—
310C2×Q8—182no—
311C2×Q8—182no—
312C2×Q8#63811yescharacteristic
313C2×Q8C22×C3⋊D311yescharacteristic
314C2×D4—21no—
315C2×D4—22no—
316C2×D4#63812yes—
317C2×D4#63811yescharacteristic
318C2×D4C22×C3⋊D311yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—181no—
322C22×C4C22×C3⋊D312yes—
323C22×C4C22×C3⋊D311yescharacteristic
324Q16—184no—
325SD16—184no—
326M16—94no—
327M16—184no—
328C2×C8—92no—
329C2×C8—91no—
330C2×C8—91no—
331C2×C8—184no—
332C2×C8—182no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—91no—
336C4⋊C4—91no—
337C4⋊C4—182no—
338C4⋊C4—182no—
339C4⋊C4C22×C3⋊D312yes—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—181no—
343C22⋊C4—182no—
344C22⋊C4—182no—
345C42—181no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—28no—
349C22×C3—28no—
350C22×C3—28no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3#93918yes—
355C22×C3C22×#6714yes—
356C3×C4—28no—
357C3×C4—44no—
358C3×C4—48no—
359C3×C4#93918yes—
360C3×C4C2×D3×D414yes—
361C3×C4C2×#23714yes—
362Dic3—128no—
363Dic3—124no—
364Dic3—124no—
365C32#306311yescharacteristic, complemented, Sylow
366C23—21no—
367C23C2×#63812yes—
368C23C2×#63811yescharacteristic
369Q8—22no—
370Q8—21no—
371Q8—21no—
372Q8—182no—
373Q8—182no—
374Q8—362no—
375D4—24no—
376D4—21no—
377D4—21no—
378D4—22no—
379D4—42no—
380D4—82no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—182no—
390C2×C4—362no—
391C2×C4D4×C3⋊D312yes—
392C2×C4#383012yes—
393C2×C4C2×#63812yes—
394C2×C4C2×#63811yescharacteristic
395C2×C4C23×C3⋊D311yescharacteristic
396C8—94no—
397C8—181no—
398C8—181no—
399C2×C3—28no—
400C2×C3—48no—
401C2×C3—84no—
402C2×C3C2×#93914yes—
403C2×C3#583614yes—
404C2×C3#583814yes—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22#6414812yes—
414C22C22×#63811yescharacteristic
415C4—22no—
416C4—41no—
417C4—42no—
418C4—362no—
419C4—361no—
420C4—361no—
421C4#6414812yes—
422C4C2×D4×C3⋊D311yescharacteristic
423C4C2×#383011yescharacteristic
424C3#8342314yescomplemented
425C2—22no—
426C2—42no—
427C2—81no—
428C2C2×#6414811yescharacteristic
429C2#1059630711yescharacteristic
430C2#1059630911yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC32×#537C2
socleC22×C32C22×D4
hypercenter#537C3⋊D3
hypocenterC32#3063
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
2. #10596296C2semidirect
3. C2×#64160C2semidirect
4. C2×#64157C2semidirect
5. #10592932C2semidirect
6. #10592751C2semidirect
7. #10592750C2semidirect
8. #10592741C2semidirect
9. #10592740C2semidirect
10. C2×#63637C2semidirect
11. #10592663C2semidirect
91. #3748D4semidirect
365. C32#3063semidirect
424. C3#83423semidirect

Maximal subgroups

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

subgroupindex
C32×#5372
#105962962
C2×#641602
C2×#641572
#105929322
#105927512
#105927502
#105927412
#105927402
C2×#636372
#105926632
#834233