Finite Group Databasebeta

← #11803812

Subgroups — #11803812

Order 1152.

1181 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64096C211yesmaximal, characteristic
2C2×C4×#627C211yesmaximal, characteristic
3C2×#63840C211yesmaximal, characteristic
4C3×#4978C212yesmaximal
5#10591335C212yesmaximal
6#10591332C212yesmaximal
7#10591273C211yesmaximal, characteristic
8#10591252C211yesmaximal, characteristic
9#10591247C211yesmaximal, characteristic
10#10591236C211yesmaximal, characteristic
11#10591192C212yesmaximal
12#74948—32nomaximal
13C22×#634C2211yescharacteristic
14C22×#627C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×#3750C2211yescharacteristic
17C2×#3750C2211yescharacteristic
18C2×C4×C3⋊Dic3C2211yescharacteristic
19C2×C4×C3⋊Dic3C2211yescharacteristic
20C2×C4×C3⋊Dic3C2211yescharacteristic
21#64096—24no—
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×C4×Dic3C2212yes—
24C2×#3724C2211yescharacteristic
25C2×Dic3⋊Dic3C2212yes—
26C2×Dic3⋊Dic3C2212yes—
27C2×Dic32C2211yescharacteristic
28C2×Dic32C2211yescharacteristic
29C4×#627—24no—
30#63840C418yescomplemented
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35#63579C2212yes—
36#63579C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39C2×#888—32no—
40C2×#887—32no—
41C2×#887—61no—
42C2×C4×Dic6—32no—
43C3×#387—32no—
44#4980—32no—
45#4978D312yes—
46#4933—32no—
47#4655—32no—
48#4654—32no—
49#4653—32no—
50#4652—32no—
51#4650—32no—
52#4648—32no—
53#4646—32no—
54#4646—32no—
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C22×C32×C4C2311yescharacteristic
58C22×C3⋊Dic3C2311yescharacteristic
59C22×C3⋊Dic3C2311yescharacteristic
60C22×C3⋊Dic3C2311yescharacteristic
61C22×C3⋊Dic3C2311yescharacteristic
62C2×#634—22no—
63C2×#634—22no—
64C2×#634C2×C414yes—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C2×#627—22no—
70C2×#627—22no—
71C2×#627C2×C414yes—
72C32×C4⋊C4—22no—
73#3750—22no—
74#3750—22no—
75C4×C3⋊Dic3—22no—
76C4×C3⋊Dic3—22no—
77C4×C3⋊Dic3D414yes—
78C3×Dic3⋊C4C2×C418yes—
79C3×C4×Dic3—24no—
80#3724—22no—
81Dic3⋊Dic3—24no—
82Dic3⋊Dic3C2×C418yes—
83Dic32—22no—
84Dic32C2×C414yes—
85#2327—91noSylow
86C22×Dic6—32no—
87C22×Dic6—32no—
88C22×Dic6—61no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—32no—
91C2×C3×C4⋊C4C2×D312yes—
92C2×C3×C42—32no—
93C2×C4⋊Dic3—32no—
94C2×C4⋊Dic3—32no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4—32no—
98C2×Dic3⋊C4—32no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—61no—
101C2×Dic3⋊C4—61no—
102C2×Dic3⋊C4C2×D312yes—
103C2×C4×Dic3—32no—
104C2×C4×Dic3—32no—
105C2×C4×Dic3—32no—
106C2×C4×Dic3—32no—
107C2×C4×Dic3—32no—
108C2×C4×Dic3—61no—
109C2×C4×Dic3—61no—
110C2×C4×Dic3—61no—
111C2×C4×Dic3C2×D312yes—
112#888—316no—
113#887—68no—
114#887—68no—
115C4×Dic6—68no—
116C3×#95—32no—
117C3×#95—32no—
118C3×#95—32no—
119C3×#95—32no—
120C3⋊#95—32no—
121C3⋊#95—32no—
122C3⋊#95—32no—
123C3⋊#95—32no—
124C3⋊#95—32no—
125C3⋊#95—32no—
126C3⋊#95—32no—
127C3⋊#95—32no—
128C3⋊#95C2×D312yes—
129C3⋊#95C2×D312yes—
130C3⋊#95C2×D312yes—
131C3⋊#95C2×D312yes—
132C23×C32C2411yescharacteristic
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4C22×C412yes—
139C2×C32×C4C2×D412yes—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3C22×C412yes—
146C2×C3⋊Dic3C22×C412yes—
147C2×C3⋊Dic3C2×D412yes—
148C2×C3⋊Dic3C2×D412yes—
149C2×C3⋊Dic3#4112yes—
150#634—28no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3C22×C414yes—
162C2×C3×Dic3C22×C414yes—
163#627—28no—
164C2×C4⋊Q8—91no—
165C2×C4×Q8—91no—
166C2×C4×Q8—91no—
167#391—91no—
168#391—91no—
169#387—32no—
170#387—92no—
171#387—92no—
172#387—92no—
173#382—91no—
174#376—91no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—32no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4C22×D312yes—
183C22×C3×C4C22×D312yes—
184C22×C3×C4C22×D312yes—
185C22×Dic3—32no—
186C22×Dic3—32no—
187C22×Dic3—32no—
188C22×Dic3—32no—
189C22×Dic3—61no—
190C22×Dic3—61no—
191C22×Dic3—61no—
192C22×Dic3—61no—
193C22×Dic3C22×D312yes—
194C22×Dic3C22×D312yes—
195C22×Dic3C22×D312yes—
196C22×Dic3C22×D312yes—
197C2×Dic6—38no—
198C2×Dic6—38no—
199C2×Dic6—64no—
200C2×Dic6—64no—
201C2×Dic6—64no—
202C2×Dic6—64no—
203C2×Dic6—64no—
204C2×Dic6—64no—
205C2×Dic6—64no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—38no—
209C3×C42—64no—
210C4⋊Dic3—38no—
211C4⋊Dic3—64no—
212Dic3⋊C4—38no—
213Dic3⋊C4—64no—
214Dic3⋊C4—64no—
215Dic3⋊C4—64no—
216Dic3⋊C4—64no—
217Dic3⋊C4—64no—
218Dic3⋊C4—64no—
219Dic3⋊C4C4×D318yes—
220C4×Dic3—24no—
221C4×Dic3—38no—
222C4×Dic3—38no—
223C4×Dic3—64no—
224C4×Dic3—64no—
225C4×Dic3—64no—
226C4×Dic3—64no—
227C4×Dic3—64no—
228C4×Dic3—64no—
229C22×C32C23×C411yescharacteristic
230C22×C32C22×D411yescharacteristic
231C22×C32C2×#4111yescharacteristic
232C22×C322−1+412yes—
233C22×C322−1+412yes—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—41no—
237C3⋊Dic3—22no—
238C3⋊Dic3—22no—
239C3⋊Dic3—41no—
240C3⋊Dic3C4×D414yes—
241C3×Dic3—24no—
242C3×Dic3—24no—
243C3×Dic3—42no—
244C3×Dic3—42no—
245C22×Q8—91no—
246C22×Q8—91no—
247C4⋊Q8—98no—
248C4×Q8—184no—
249C4×Q8—184no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—92no—
252C2×C4⋊C4—91no—
253C2×C4⋊C4—92no—
254C2×C4⋊C4—91no—
255C2×C4⋊C4—91no—
256C2×C4⋊C4D3211yescharacteristic
257C2×C42—32no—
258C2×C42—91no—
259C2×C42—91no—
260C2×C42—91no—
261C2×C42—91no—
262C2×C42—91no—
263#95—32no—
264#95—32no—
265#95—32no—
266#95—32no—
267#95—92no—
268#95—92no—
269#95—92no—
270#95—92no—
271C23×C3—21no—
272C23×C3C23×D312yes—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—34no—
286C2×C3×C4—34no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4C2×C4×D314yes—
300C2×C3×C4D3×D414yes—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—34no—
312C2×Dic3—34no—
313C2×Dic3—34no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—64no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—122no—
333C2×Dic3C2×C4×D314yes—
334C2×Dic3C2×C4×D314yes—
335Dic6—616no—
336Dic6—616no—
337Dic6—616no—
338C2×C32C2×C4×D411yescharacteristic
339C2×C32#51912yes—
340C2×C32#53611yescharacteristic
341C2×C32#53611yescharacteristic
342C2×C32#54112yes—
343C2×Q8—94no—
344C2×Q8—94no—
345C2×Q8—182no—
346C2×Q8—182no—
347C2×Q8—182no—
348C2×Q8—182no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4C2×D3211yescharacteristic
358C22×C4C2×D3211yescharacteristic
359C22×C4C2×D3211yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—38no—
362C4⋊C4—98no—
363C4⋊C4—182no—
364C4⋊C4—182no—
365C4⋊C4—184no—
366C4⋊C4—182no—
367C42—64no—
368C42—94no—
369C42—94no—
370C42—182no—
371C42—182no—
372C42—182no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3C22×C4×D312yes—
379C22×C3C2×D3×D412yes—
380C22×C3#100712yes—
381C22×C3D3×#4112yes—
382C22×C3#101012yes—
383C22×C3#101012yes—
384C22×C3#101012yes—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—64no—
392C3×C4—64no—
393C3×C4—122no—
394C3×C4—122no—
395Dic3—24no—
396Dic3—24no—
397Dic3—38no—
398Dic3—42no—
399Dic3—42no—
400Dic3—64no—
401Dic3—64no—
402Dic3—64no—
403Dic3—64no—
404Dic3—64no—
405Dic3—122no—
406Dic3—122no—
407C32#232711yescharacteristic, complemented, Sylow
408C23C22×D3211yescharacteristic
409Q8—188no—
410Q8—188no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—34no—
417C2×C4—34no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—181no—
434C2×C4—182no—
435C2×C4—181no—
436C2×C4—182no—
437C2×C4—181no—
438C2×C4C4×D3212yes—
439C2×C4#381112yes—
440C2×C3—22no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3#551612yes—
446C2×C3C4×D3×D412yes—
447C2×C3#557212yes—
448C2×C3#562812yes—
449C2×C3#564812yes—
450C2×C3#569612yes—
451C2×C3#570112yes—
452C22C2×C4×D3211yescharacteristic
453C22#6430611yescharacteristic
454C22#6430912yes—
455C22#6431512yes—
456C22C2×#381111yescharacteristic
457C4—22no—
458C4—22no—
459C4—41no—
460C4—64no—
461C4—64no—
462C4—94no—
463C4—122no—
464C4—122no—
465C4—182no—
466C4—182no—
467C4—361no—
468C3—21no—
469C3#7494812yescomplemented
470C2#1059435712yes—
471C2#1059447412yes—
472C2C4×#381111yescharacteristic
473C2#1059527111yescharacteristic
474C2#1059540511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, prime radical, cosocleC23×C32C24
hypercenterC2×C4⋊C4D32
hypocenterC32#2327
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
30. #63840C4semidirect
407. C32#2327semidirect
469. C3#74948semidirect

Maximal subgroups

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

subgroupindex
C2×#640962
C2×C4×#6272
C2×#638402
C3×#49782
#105913352
#105913322
#105912732
#105912522
#105912472
#105912362
#105911922
#749483