Finite Group Databasebeta

← #11805269

Subgroups — #11805269

Order 1152.

1305 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#4980C212yesmaximal, complemented
3#10591355C211yesmaximal, characteristic
4#10591355C211yesmaximal, characteristic
5#10591325C211yesmaximal, characteristic
6C4×#3722C211yesmaximal, characteristic
7#10591271C212yesmaximal, complemented
8#10591264C212yesmaximal, complemented
9#10591260C211yesmaximal, characteristic
10#10591260C211yesmaximal, characteristic
11#10591243C212yesmaximal, complemented
12#74962—32nomaximal
13C22×C4×C3⋊D3C2211yescharacteristic
14C22×C4×C3⋊D3C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×#3750C2211yescharacteristic
19C2×#3750C2211yescharacteristic
20C4⋊C4×C3⋊D3C418yescomplemented
21C2×C3×C4⋊Dic3C2212yes—
22C2×C3×C4×Dic3C2212yes—
23C2×#3724C2212yes—
24C2×#3724C2212yes—
25C2×#3722C2211yescharacteristic
26C2×#3722C2211yescharacteristic
27C2×#3722C2211yescharacteristic
28C2×#3722C2211yescharacteristic
29C2×Dic32C2212yes—
30C2×Dic32C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35#63579C2212yes—
36#63579C2212yes—
37C2×D3×C4⋊C4—32no—
38C2×D3×C4⋊C4—61no—
39C3×#387—32no—
40#4987—32no—
41#4980D312yescomplemented
42#4976—32no—
43C4×(C2×D3)⋊C4—32no—
44C4×Dic3⋊C4—32no—
45#4667—32no—
46#4667—32no—
47#4666—32no—
48#4666—32no—
49#4654—32no—
50#4653—32no—
51#4649—32no—
52#4647—32no—
53C23×C3⋊D3C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C22×C3⋊Dic3C2311yescharacteristic
58C22×C3⋊Dic3C2311yescharacteristic
59C22×C3⋊Dic3C2311yescharacteristic
60C2×C4×C3⋊D3—24no—
61C2×C4×C3⋊D3—42no—
62C2×C4×C3⋊D3—42no—
63C2×C4×C3⋊D3C2×C414yes—
64C2×C4×C3⋊D3C2×C414yes—
65C2×C4×C3⋊D3C2×C414yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C32×C4⋊C4C2×C414yes—
71C4⋊(C3⋊Dic3)C2×C414yes—
72#3750C2×C414yes—
73#3750C2×C414yes—
74C3×C4⋊Dic3—42no—
75C3×C4×Dic3—42no—
76#3724—24no—
77#3724—42no—
78#3722—22no—
79#3722—22no—
80#3722—22no—
81#3722Q814yes—
82Dic32—24no—
83Dic32—42no—
84#2345—91noSylow
85C22×C4×D3—32no—
86C22×C4×D3—32no—
87C22×C4×D3—32no—
88C22×C4×D3—61no—
89C22×C4×D3—61no—
90C22×C4×D3—61no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—32no—
93C2×C3×C4⋊C4C2×D312yes—
94C2×C3×C42—32no—
95C2×(C2×D3)⋊C4—32no—
96C2×(C2×D3)⋊C4—32no—
97C2×(C2×D3)⋊C4—32no—
98C2×(C2×D3)⋊C4—32no—
99C2×C4⋊Dic3—32no—
100C2×C4⋊Dic3—61no—
101C2×C4⋊Dic3C2×D312yes—
102C2×Dic3⋊C4—32no—
103C2×Dic3⋊C4—32no—
104C2×Dic3⋊C4—32no—
105C2×Dic3⋊C4—32no—
106C2×Dic3⋊C4—32no—
107C2×Dic3⋊C4—32no—
108C2×Dic3⋊C4—61no—
109C2×Dic3⋊C4—61no—
110C2×C4×Dic3—32no—
111C2×C4×Dic3—32no—
112C2×C4×Dic3—32no—
113C2×C4×Dic3—32no—
114C2×C4×Dic3C2×D312yes—
115D3×C4⋊C4—316no—
116D3×C4⋊C4—68no—
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⋊#95C2×D312yes—
126C3⋊#95C2×D312yes—
127C3⋊#95C2×D312yes—
128C3⋊#95C2×D312yes—
129C23×C32C2411yescharacteristic
130C22×C3⋊D3—22no—
131C22×C3⋊D3—22no—
132C22×C3⋊D3C22×C412yes—
133C22×C3⋊D3C2×Q812yes—
134C22×C3⋊D3#4112yes—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4C22×C412yes—
139C2×C32×C4C22×C412yes—
140C2×C32×C4C22×C412yes—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3C22×C412yes—
144C2×C3⋊Dic3C22×C412yes—
145C2×C3⋊Dic3C22×C412yes—
146C2×C3⋊Dic3#4114yes—
147C4×C3⋊D3—28no—
148C4×C3⋊D3—44no—
149C4×C3⋊D3—44no—
150C2×C3×Dic3—22no—
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×Dic3C2×Q814yes—
162C22×C4⋊C4—91no—
163#389—92no—
164#387—32no—
165#387—92no—
166#380—91no—
167#380—91no—
168#380—91no—
169C4×C4⋊C4—92no—
170C4×C22⋊C4—91no—
171C4×C22⋊C4—91no—
172C4×C22⋊C4—91no—
173C23×D3—32no—
174C23×D3—61no—
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—61no—
189C22×Dic3—61no—
190C22×Dic3—61no—
191C22×Dic3C22×D312yes—
192C22×Dic3C22×D312yes—
193C22×Dic3C22×D312yes—
194C22×Dic3C22×D312yes—
195C2×C4×D3—38no—
196C2×C4×D3—38no—
197C2×C4×D3—38no—
198C2×C4×D3—64no—
199C2×C4×D3—64no—
200C2×C4×D3—64no—
201C2×C4×D3—64no—
202C2×C4×D3—64no—
203C2×C4×D3—68no—
204C2×C4×D3—122no—
205C2×C4×D3—122no—
206C2×C4×D3—122no—
207C2×C4×D3—122no—
208C2×C4×D3—124no—
209C2×C4×D3—124no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—122no—
212C3×C4⋊C4C4×D318yes—
213C3×C42—122no—
214(C2×D3)⋊C4—38no—
215(C2×D3)⋊C4—64no—
216(C2×D3)⋊C4—64no—
217(C2×D3)⋊C4—64no—
218C4⋊Dic3—38no—
219C4⋊Dic3—42no—
220C4⋊Dic3—64no—
221Dic3⋊C4—38no—
222Dic3⋊C4—38no—
223Dic3⋊C4—64no—
224Dic3⋊C4—64no—
225Dic3⋊C4—64no—
226Dic3⋊C4—64no—
227Dic3⋊C4—122no—
228Dic3⋊C4—122no—
229C4×Dic3—42no—
230C4×Dic3—64no—
231C4×Dic3—64no—
232C4×Dic3—122no—
233C4×Dic3—122no—
234C22×C32C23×C411yescharacteristic
235C22×C32C22×Q811yescharacteristic
236C22×C32C2×#4112yes—
237C22×C32C2×#4111yescharacteristic
238C22×C32H(5,𝔽2)12yes—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—24no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3C4×Q814yes—
244C32×C4—22no—
245C32×C4—22no—
246C32×C4—22no—
247C3⋊Dic3—22no—
248C3⋊Dic3—22no—
249C3⋊Dic3#11714yes—
250C3×Dic3—24no—
251C3×Dic3—42no—
252C3×Dic3—42no—
253C3×Dic3—42no—
254C23×C4—91no—
255C23×C4—91no—
256C23×C4—91no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—91no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—98no—
262C2×C4⋊C4—92no—
263C2×C4⋊C4—92no—
264C2×C4⋊C4D3211yescharacteristic
265C2×C22⋊C4—91no—
266C2×C22⋊C4—91no—
267C2×C22⋊C4—91no—
268C2×C22⋊C4—91no—
269C2×C42—32no—
270C2×C42—92no—
271C2×C42—92no—
272#95—32no—
273#95—32no—
274#95—32no—
275#95—32no—
276#95—92no—
277#95—92no—
278C23×C3—21no—
279C23×C3C23×D312yes—
280C22×D3—34no—
281C22×D3—34no—
282C22×D3—34no—
283C22×D3—62no—
284C22×D3—62no—
285C22×D3—62no—
286C22×D3—64no—
287C22×D3—64no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—62no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—34no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4C2×C4×D314yes—
317C2×C3×C4C2×C4×D314yes—
318C2×C3×C4C2×C4×D314yes—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—34no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—34no—
334C2×Dic3—34no—
335C2×Dic3—62no—
336C2×Dic3—64no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—122no—
344C2×Dic3—122no—
345C2×Dic3D3×Q814yes—
346C4×D3—616no—
347C4×D3—616no—
348C4×D3—128no—
349C4×D3—128no—
350C4×D3—128no—
351C4×D3—128no—
352C2×C32C2×#11711yescharacteristic
353C2×C32C2×C4×Q811yescharacteristic
354C2×C32#51811yescharacteristic
355C2×C32#55312yes—
356C2×C32#55412yes—
357C3⋊D3—24no—
358C24—91no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4—94no—
367C22×C4—94no—
368C22×C4—94no—
369C22×C4—184no—
370C22×C4—362no—
371C22×C4—362no—
372C22×C4C2×D3211yescharacteristic
373C22×C4C2×D3211yescharacteristic
374C22×C4C2×D3211yescharacteristic
375C4⋊C4—94no—
376C4⋊C4—94no—
377C4⋊C4—94no—
378C4⋊C4—122no—
379C4⋊C4—184no—
380C4⋊C4—362no—
381C4⋊C4#62414yes—
382C22⋊C4—94no—
383C22⋊C4—182no—
384C22⋊C4—182no—
385C22⋊C4—182no—
386C42—122no—
387C42—184no—
388C42—362no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3C22×C4×D312yes—
395C22×C3#100412yes—
396C22×C3C2×D3×Q812yes—
397C22×C3D3×#4112yes—
398C22×C3D3×#4112yes—
399C22×C3D3×#4112yes—
400C22×C3#100912yes—
401C2×D3—38no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—64no—
405C2×D3—68no—
406C2×D3—68no—
407C2×D3—64no—
408C2×D3—64no—
409C2×D3—64no—
410C2×D3—64no—
411C2×D3—64no—
412C2×D3—64no—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—64no—
420C3×C4—122no—
421C3×C4—122no—
422C3×C4—122no—
423Dic3—24no—
424Dic3—38no—
425Dic3—42no—
426Dic3—42no—
427Dic3—42no—
428Dic3—64no—
429Dic3—64no—
430Dic3—64no—
431Dic3—122no—
432Dic3—122no—
433C32#234511yescharacteristic, complemented, Sylow
434C23—92no—
435C23—92no—
436C23—92no—
437C23—182no—
438C23—182no—
439C23C22×D3211yescharacteristic
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—34no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—94no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—188no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4—364no—
463C2×C4—364no—
464C2×C4C2×#62412yes—
465C2×C4C2×#62412yes—
466C2×C4C2×#62412yes—
467C2×C3—21no—
468C2×C3—22no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—22no—
472C2×C3D3×#11712yes—
473C2×C3#554512yes—
474C2×C3C4×D3×Q812yes—
475C2×C3#563612yes—
476C2×C3#566112yes—
477C2×C3#569012yes—
478C2×C3#570912yes—
479D3—68no—
480D3—68no—
481C22—94no—
482C22—184no—
483C22—182no—
484C22—182no—
485C22—184no—
486C22#6430612yes—
487C22#6431412yes—
488C22D32×Q811yescharacteristic
489C22#6431911yescharacteristic
490C22C22×#62411yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—64no—
495C4—94no—
496C4—122no—
497C4—122no—
498C4—122no—
499C4—182no—
500C4—182no—
501C3—21no—
502C3#7496212yescomplemented
503C2—184no—
504C2#1059439412yes—
505C2#1059447312yes—
506C2#1059496911yescharacteristic
507C2#1059522311yescharacteristic
508C2Q8×#62411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32#2345
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4980C2semidirect
7. #10591271C2semidirect
8. #10591264C2semidirect
11. #10591243C2semidirect
20. C4⋊C4×C3⋊D3C4semidirect
41. #4980D3semidirect
433. C32#2345semidirect
502. C3#74962semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×C3⋊D32
C3×#49802
#105913552
#105913252
C4×#37222
#105912712
#105912642
#105912602
#105912432
#749623