Finite Group Databasebeta

← #11802826

Subgroups — #11802826

Order 1152.

1141 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63883C211yesmaximal, characteristic
2#10592686C211yesmaximal, characteristic
3C3×#4978C212yesmaximal, complemented
4#10591371C212yesmaximal
5#10591355C211yesmaximal, characteristic
6#10591331C212yesmaximal, complemented
7#10591272C212yesmaximal, complemented
8#10591250C212yesmaximal, complemented
9#10591233C212yesmaximal
10#75296—32nomaximal
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×#624C2211yescharacteristic
13C22×#624C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×#3752C2212yes—
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×Dic3⋊C4C2212yes—
18C2×C3×C4×Dic3C2212yes—
19C2×#3724C2211yescharacteristic
20C2×#3724C2211yescharacteristic
21C2×Dic3⋊Dic3C2212yes—
22C2×#3722C2212yes—
23C2×Dic32C2212yes—
24#63883C418yescomplemented
25#63658C2212yes—
26C3×C3⋊#95C2212yes—
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30#63579C2212yes—
31#63579C2212yes—
32#63579C2212yes—
33C2×D3×C4⋊C4—32no—
34C3×#387—32no—
35#4987—32no—
36#4987—61no—
37#4980—32no—
38#4978D312yescomplemented
39C4×(C2×D3)⋊C4—32no—
40#4933—32no—
41#4667—32no—
42#4666—32no—
43#4664—32no—
44#4664—32no—
45#4655—32no—
46#4654—32no—
47#4653—32no—
48#4652—32no—
49C23×C3⋊D3C2311yescharacteristic
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3⋊Dic3C2312yes—
53C22×C3⋊Dic3C2311yescharacteristic
54C2×C4×C3⋊D3—41no—
55C2×C4×C3⋊D3—41no—
56C2×C4×C3⋊D3C2×C414yes—
57C22×C3×Dic3C2312yes—
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2312yes—
61C2×#624—22no—
62C2×#624—22no—
63C2×#624—41no—
64C2×#624—41no—
65C2×#624C2×C414yes—
66C2×#624C2×C414yes—
67C32×C4⋊C4—41no—
68#3752—24no—
69C4⋊(C3⋊Dic3)—41no—
70C3×Dic3⋊C4C2×C418yes—
71C3×C4×Dic3—42no—
72#3724—41no—
73#3724—41no—
74Dic3⋊Dic3C2×C418yes—
75#3722—24no—
76Dic32—42no—
77#2313—91noSylow
78C22×C4×D3—32no—
79C22×C4×D3—32no—
80C22×C4×D3—32no—
81C22×C4×D3—61no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—32no—
84C2×C3×C4⋊C4C2×D312yes—
85C2×C3×C42—32no—
86C2×(C2×D3)⋊C4—32no—
87C2×(C2×D3)⋊C4—32no—
88C2×(C2×D3)⋊C4—32no—
89C2×(C2×D3)⋊C4—32no—
90C2×(C2×D3)⋊C4—62no—
91C2×C4⋊Dic3—32no—
92C2×C4⋊Dic3—32no—
93C2×C4⋊Dic3—61no—
94C2×Dic3⋊C4—32no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—32no—
97C2×Dic3⋊C4C2×D312yes—
98C2×C4×Dic3—32no—
99C2×C4×Dic3—32no—
100C2×C4×Dic3C2×D312yes—
101D3×C4⋊C4—316no—
102C3×#95—32no—
103C3×#95—32no—
104C3×#95—32no—
105C3×#95—32no—
106C3⋊#95—32no—
107C3⋊#95—32no—
108C3⋊#95—32no—
109C3⋊#95—32no—
110C3⋊#95—32no—
111C3⋊#95—32no—
112C3⋊#95—32no—
113C3⋊#95—32no—
114C3⋊#95—62no—
115C3⋊#95C2×D312yes—
116C3⋊#95C2×D312yes—
117C3⋊#95C2×D312yes—
118C3⋊#95C2×D312yes—
119C23×C32C2411yescharacteristic
120C22×C3⋊D3—21no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3C22×C412yes—
125C22×C3⋊D3#4112yes—
126C22×C3⋊D3#4112yes—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4C22×C412yes—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3C22×C412yes—
139C4×C3⋊D3—44no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3C22×C414yes—
151C2×C3×Dic3C22×C414yes—
152#624—28no—
153#624—44no—
154C22×C4⋊C4—91no—
155#387—32no—
156#387—92no—
157#387—92no—
158#387—92no—
159#381—92no—
160#380—91no—
161#380—91no—
162C4×C22⋊C4—92no—
163C23×D3—32no—
164C23×D3—61no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—32no—
168C22×C3×C4—32no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4C22×D312yes—
172C22×C3×C4C22×D312yes—
173C22×C3×C4C22×D312yes—
174C22×Dic3—32no—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—62no—
179C22×Dic3C22×D312yes—
180C22×Dic3C22×D312yes—
181C22×Dic3C22×D312yes—
182C22×Dic3C22×D312yes—
183C2×C4×D3—38no—
184C2×C4×D3—38no—
185C2×C4×D3—38no—
186C2×C4×D3—64no—
187C2×C4×D3—64no—
188C2×C4×D3—64no—
189C2×C4×D3—122no—
190C2×C4×D3—122no—
191C2×C4×D3—122no—
192C2×C4×D3—122no—
193C2×C4×D3—122no—
194C2×C4×D3—122no—
195C3×C4⋊C4—38no—
196C3×C4⋊C4—42no—
197C3×C4⋊C4—42no—
198C3×C42—122no—
199(C2×D3)⋊C4—64no—
200(C2×D3)⋊C4—64no—
201(C2×D3)⋊C4—64no—
202(C2×D3)⋊C4—64no—
203(C2×D3)⋊C4—124no—
204C4⋊Dic3—38no—
205C4⋊Dic3—122no—
206C4⋊Dic3—122no—
207Dic3⋊C4—38no—
208Dic3⋊C4—122no—
209Dic3⋊C4—122no—
210Dic3⋊C4C4×D318yes—
211C4×Dic3—42no—
212C4×Dic3—122no—
213C4×Dic3—122no—
214C22×C32C23×C411yescharacteristic
215C22×C32C2×#4111yescharacteristic
216C22×C32C2×#4111yescharacteristic
217C22×C32H(5,𝔽2)11yescharacteristic
218C22×C32H(5,𝔽2)11yescharacteristic
219C22×C322−1+411yescharacteristic
220C22×C322−1+411yescharacteristic
221C2×C3⋊D3—22no—
222C2×C3⋊D3—22no—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3⋊D3—22no—
227C2×C3⋊D3#11714yes—
228C32×C4—22no—
229C32×C4—42no—
230C3⋊Dic3—22no—
231C3⋊Dic3—42no—
232C3×Dic3—24no—
233C3×Dic3—24no—
234C3×Dic3—42no—
235C3×Dic3—42no—
236C23×C4—91no—
237C23×C4—91no—
238C23×C4—91no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—98no—
241C2×C4⋊C4—92no—
242C2×C4⋊C4—91no—
243C2×C4⋊C4—91no—
244C2×C4⋊C4—91no—
245C2×C4⋊C4D3211yescharacteristic
246C2×C22⋊C4—92no—
247C2×C22⋊C4—92no—
248C2×C42—32no—
249C2×C42—92no—
250#95—32no—
251#95—32no—
252#95—32no—
253#95—32no—
254#95—92no—
255#95—92no—
256#95—92no—
257#95—92no—
258C23×C3—21no—
259C23×C3C23×D312yes—
260C22×D3—34no—
261C22×D3—34no—
262C22×D3—34no—
263C22×D3—62no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—121no—
271C22×D3—121no—
272C22×D3—121no—
273C22×D3—121no—
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—42no—
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×C4—62no—
300C2×C3×C4C2×C4×D314yes—
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—62no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—121no—
322C2×Dic3—121no—
323C2×Dic3—122no—
324C2×Dic3—122no—
325C2×Dic3—122no—
326C2×Dic3C2×C4×D314yes—
327C2×Dic3C2×C4×D314yes—
328C4×D3—616no—
329C4×D3—128no—
330C4×D3—128no—
331C4×D3—128no—
332C2×C32C2×#11711yescharacteristic
333C2×C32#51811yescharacteristic
334C2×C32#51911yescharacteristic
335C2×C32#53911yescharacteristic
336C2×C32#53911yescharacteristic
337C2×C32#53911yescharacteristic
338C2×C32#53911yescharacteristic
339C3⋊D3—24no—
340C24—91no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—94no—
346C22×C4—94no—
347C22×C4—91no—
348C22×C4—94no—
349C22×C4—92no—
350C22×C4—182no—
351C22×C4—182no—
352C22×C4—361no—
353C22×C4—361no—
354C22×C4—361no—
355C22×C4—361no—
356C22×C4C2×D3212yes—
357C22×C4C2×D3211yescharacteristic
358C4⋊C4—38no—
359C4⋊C4—41no—
360C4⋊C4—98no—
361C4⋊C4—361no—
362C4⋊C4—361no—
363C4⋊C4—361no—
364C22⋊C4—184no—
365C22⋊C4—184no—
366C42—122no—
367C42—362no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3C22×C4×D312yes—
376C22×C3#100412yes—
377C22×C3#100712yes—
378C22×C3D3×#4112yes—
379C22×C3D3×#4112yes—
380C22×C3#100912yes—
381C22×C3#101012yes—
382C2×D3—38no—
383C2×D3—64no—
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—64no—
387C2×D3—64no—
388C2×D3—64no—
389C2×D3—64no—
390C2×D3—122no—
391C2×D3—122no—
392C2×D3—122no—
393C2×D3—122no—
394C2×D3—122no—
395C2×D3—122no—
396C3×C4—24no—
397C3×C4—24no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—64no—
401C3×C4—64no—
402C3×C4—82no—
403C3×C4—122no—
404C3×C4—122no—
405Dic3—24no—
406Dic3—24no—
407Dic3—42no—
408Dic3—42no—
409Dic3—64no—
410Dic3—122no—
411Dic3—122no—
412Dic3—122no—
413Dic3—242no—
414C32#231311yescharacteristic, complemented, Sylow
415C23—92no—
416C23—92no—
417C23—92no—
418C23—181no—
419C23—181no—
420C23—181no—
421C23—181no—
422C23C22×D3211yescharacteristic
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—34no—
429C2×C4—34no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—92no—
441C2×C4—188no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4—182no—
447C2×C4—364no—
448C2×C4—364no—
449C2×C4C4×D3212yes—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3D3×#11712yes—
458C2×C3#554512yes—
459C2×C3#557212yes—
460C2×C3#563812yes—
461C2×C3#566012yes—
462C2×C3#569212yes—
463C2×C3#570712yes—
464D3—68no—
465D3—124no—
466C22—94no—
467C22—182no—
468C22—182no—
469C22—182no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22C2×C4×D3211yescharacteristic
474C22#6430811yescharacteristic
475C22#6430911yescharacteristic
476C22#6431211yescharacteristic
477C22#6431611yescharacteristic
478C22#6431911yescharacteristic
479C22#6433011yescharacteristic
480C4—22no—
481C4—42no—
482C4—64no—
483C4—64no—
484C4—122no—
485C4—122no—
486C4—182no—
487C4—362no—
488C3—21no—
489C3#7529612yescomplemented
490C2—184no—
491C2#1059434711yescharacteristic
492C2#1059448311yescharacteristic
493C2#1059449111yescharacteristic
494C2#1059460211yescharacteristic
495C2#1059505811yescharacteristic
496C2#1059520911yescharacteristic
497C2#1059541511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32#2313
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4978C2semidirect
6. #10591331C2semidirect
7. #10591272C2semidirect
8. #10591250C2semidirect
24. #63883C4semidirect
38. #4978D3semidirect
414. C32#2313semidirect
489. C3#75296semidirect

Maximal subgroups

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

subgroupindex
C2×#638832
#105926862
C3×#49782
#105913712
#105913552
#105913312
#105912722
#105912502
#105912332
#752963