Finite Group Databasebeta

← #11824543

Subgroups — #11824543

Order 1152.

2172 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#121C211yesmaximal, characteristic, complemented
2C2×#64148C212yesmaximal, complemented
3C2×C4×#638C211yesmaximal, characteristic, complemented
4#10592922C211yesmaximal, characteristic, complemented
5#10592921C211yesmaximal, characteristic, complemented
6#10592919C212yesmaximal, complemented
7C2×#63658C211yesmaximal, characteristic, complemented
8#10592691C211yesmaximal, characteristic, complemented
9#10592682C211yesmaximal, characteristic, complemented
10#10592655C212yesmaximal, complemented
11#10592643C212yesmaximal, complemented
12#82671—34nomaximal
13C22×C32×D4C2212yes—
14C22×C32×D4C2211yescharacteristic
15C23×C32×C4C2211yescharacteristic
16C22×#638C2212yes—
17C22×#638C2211yescharacteristic
18C23×C3⋊Dic3C2212yes—
19C22×C4×C3⋊D3C2211yescharacteristic
20C32×#121—24no—
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2212yes—
23C2×#3757C2212yes—
24C2×#3757C2212yes—
25C2×#3757C2211yescharacteristic
26C2×#3757C2212yes—
27#64148—28no—
28C4×#638—24no—
29C2×#3752C2212yes—
30C2×#3752C2211yescharacteristic
31C2×#3750C2212yes—
32C2×#3750C2211yescharacteristic
33C2×C4×C3⋊Dic3C2211yescharacteristic
34#63658C2212yes—
35#63658C2212yes—
36#63658C2211yescharacteristic
37#63658C2211yescharacteristic
38#63658C2212yes—
39C2×C3×#121D314yescomplemented
40C2×#939—38no—
41C2×C4×#67—34no—
42#5223—34no—
43#5222—34no—
44#5220—38no—
45C2×C3⋊#95—34no—
46#4992—34no—
47#4983—34no—
48#4956—38no—
49#4944—38no—
50C24×C32C2312yes—
51C24×C32C2311yescharacteristic
52C23×C3⋊D3C2311yescharacteristic
53C2×C32×D4—24no—
54C2×C32×D4—24no—
55C2×C32×D4—24no—
56C2×C32×D4—22no—
57C2×C32×D4—42no—
58C22×C32×C4—42no—
59C22×C32×C4D414yescomplemented
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C2×#638—22no—
64C2×#638—22no—
65C2×#638—24no—
66C2×#638—24no—
67C2×#638—42no—
68C2×#638D414yescomplemented
69C22×C3⋊Dic3—24no—
70C22×C3⋊Dic3—24no—
71C22×C3⋊Dic3—42no—
72C22×C3⋊Dic3C2311yescharacteristic
73C22×C3⋊Dic3C2312yes—
74C22×C3⋊Dic3C2311yescharacteristic
75C22×C3⋊Dic3C2311yescharacteristic
76C22×C3⋊Dic3C2312yes—
77C2×C4×C3⋊D3—22no—
78C2×C4×C3⋊D3—42no—
79C32×C4⋊C4—22no—
80C32×C22⋊C4—24no—
81#3757—22no—
82#3757—24no—
83#3757—24no—
84#3757—42no—
85#3752—24no—
86#3752—22no—
87#3750—22no—
88#3750—24no—
89C4×C3⋊Dic3—22no—
90#2543—91noSylow
91C22×C3×D4C2×D318yes—
92C22×C3×D4C2×D314yes—
93C23×C3×C4C2×D314yes—
94C22×#67—34no—
95C22×#67—38no—
96C23×Dic3—38no—
97C22×C4×D3—34no—
98C3×#121—216no—
99C2×C3×C4⋊C4C2×D314yes—
100C2×C3×C22⋊C4C2×D318yes—
101C2×#217—38no—
102C2×#217—38no—
103C2×#217—34no—
104C2×#217—38no—
105#939—632no—
106C4×#67—616no—
107C2×(C2×D3)⋊C4—38no—
108C2×(C2×D3)⋊C4—34no—
109C2×Dic3⋊C4—38no—
110C2×Dic3⋊C4—34no—
111C2×C4×Dic3—34no—
112C3⋊#95—38no—
113C3⋊#95—38no—
114C3⋊#95—34no—
115C3⋊#95—34no—
116C3⋊#95—38no—
117C23×C32—22no—
118C23×C32—22no—
119C23×C32—22no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32C2×D412yes—
127C23×C32C2×D412yes—
128C23×C32#4112yes—
129C23×C32C2411yescharacteristic
130C22×C3⋊D3—21no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3—22no—
134C22×C3⋊D3C2×D412yes—
135C32×D4—44no—
136C32×D4—48no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—44no—
143C2×C32×C4C2×D412yes—
144C2×C32×C4C2×D412yes—
145#638—28no—
146#638—48no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—48no—
159C2×C3⋊Dic3C2×D412yes—
160C2×C3⋊Dic3#4114yes—
161C4×C3⋊D3—44no—
162C2×#121—92no—
163C2×#121C3⋊D311yescharacteristic, complemented
164C2×C4×D4—91no—
165#398—91no—
166#397—91no—
167#394—91no—
168#392—91no—
169#388—92no—
170#386—92no—
171#386—92no—
172C2×#95—91no—
173C24×C3C22×D318yes—
174C24×C3C22×D314yes—
175C23×D3—34no—
176C2×C3×D4—216no—
177C2×C3×D4—216no—
178C2×C3×D4—216no—
179C2×C3×D4—28no—
180C2×C3×D4—48no—
181C22×C3×C4—48no—
182C22×C3×C4#67116yescomplemented
183C22×C3×C4C22×D318yes—
184C22×C3×C4C22×D314yes—
185C22×C3×C4C22×D314yes—
186C2×#67—316no—
187C2×#67—616no—
188C2×#67—68no—
189C2×#67—68no—
190C2×#67—616no—
191C2×#67—128no—
192C22×Dic3—38no—
193C22×Dic3—34no—
194C22×Dic3—38no—
195C22×Dic3—34no—
196C22×Dic3—34no—
197C22×Dic3—616no—
198C22×Dic3—616no—
199C22×Dic3—128no—
200C2×C4×D3—68no—
201C2×C4×D3—128no—
202C3×C4⋊C4—28no—
203C3×C22⋊C4—216no—
204#217—616no—
205#217—616no—
206#217—68no—
207#217—128no—
208(C2×D3)⋊C4—616no—
209(C2×D3)⋊C4—68no—
210Dic3⋊C4—616no—
211Dic3⋊C4—68no—
212C4×Dic3—68no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—24no—
216C22×C32—24no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—42no—
223C22×C32—42no—
224C22×C32#12114yes—
225C22×C32C22×D411yescharacteristic
226C22×C32C22×D411yescharacteristic
227C22×C32C2×#4111yescharacteristic
228C22×C32C2×#4112yes—
229C22×C32H(5,𝔽2)12yes—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—24no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—42no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—42no—
238C3⋊Dic3—22no—
239C3⋊Dic3—24no—
240C3⋊Dic3—41no—
241C3⋊Dic3—41no—
242C3⋊Dic3—42no—
243C22×D4—91no—
244C22×D4—92no—
245C22×D4C2×C3⋊D312yes—
246C22×D4C2×C3⋊D311yescharacteristic
247C23×C4—92no—
248C23×C4—91no—
249C23×C4C2×C3⋊D311yescharacteristic
250#121—24no—
251#121—188no—
252C4×D4—184no—
253C2×C4⋊C4—91no—
254C2×C4⋊C4—92no—
255C2×C4⋊C4C2×C3⋊D311yescharacteristic
256C2×C22⋊C4—91no—
257C2×C22⋊C4—92no—
258C2×C22⋊C4—92no—
259C2×C22⋊C4—91no—
260C2×C22⋊C4—92no—
261C2×C22⋊C4—92no—
262C2×C22⋊C4C2×C3⋊D312yes—
263C2×C42—91no—
264#95—92no—
265#95—91no—
266#95—91no—
267#95—92no—
268#95—92no—
269C23×C3—28no—
270C23×C3—28no—
271C23×C3—28no—
272C23×C3—28no—
273C23×C3—28no—
274C23×C3—28no—
275C23×C3—28no—
276C23×C3—28no—
277C23×C3—28no—
278C23×C3D3×D418yes—
279C23×C3#23718yes—
280C23×C3C2×#6718yes—
281C23×C3C23×D314yes—
282C22×D3—38no—
283C22×D3—64no—
284C22×D3—68no—
285C22×D3—68no—
286C22×D3—64no—
287C3×D4—416no—
288C3×D4—432no—
289C2×C3×C4—28no—
290C2×C3×C4—28no—
291C2×C3×C4—28no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—416no—
295C2×C3×C4C2×#6718yes—
296C2×C3×C4C2×#6718yes—
297#67—632no—
298#67—1232no—
299C2×Dic3—38no—
300C2×Dic3—316no—
301C2×Dic3—68no—
302C2×Dic3—64no—
303C2×Dic3—68no—
304C2×Dic3—64no—
305C2×Dic3—68no—
306C2×Dic3—68no—
307C2×Dic3—68no—
308C2×Dic3—68no—
309C2×Dic3—68no—
310C2×Dic3—64no—
311C2×Dic3—64no—
312C2×Dic3—1232no—
313C4×D3—1216no—
314C2×C32—24no—
315C2×C32—82no—
316C2×C32C2×#12111yescharacteristic
317C2×C32#53211yescharacteristic
318C2×C32#53411yescharacteristic
319C2×C32#54612yes—
320C2×C32#55312yes—
321C3⋊D3—42no—
322C24—91no—
323C24C22×C3⋊D312yes—
324C24C22×C3⋊D311yescharacteristic
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—42no—
330C2×D4—94no—
331C2×D4—184no—
332C2×D4—182no—
333C2×D4—182no—
334C2×D4—184no—
335C2×D4—362no—
336C22×C4—42no—
337C22×C4—92no—
338C22×C4—91no—
339C22×C4—92no—
340C22×C4—91no—
341C22×C4—91no—
342C22×C4—184no—
343C22×C4—184no—
344C22×C4—182no—
345C22×C4—362no—
346C22×C4—362no—
347C22×C4#63814yescomplemented
348C22×C4C22×C3⋊D312yes—
349C22×C4C22×C3⋊D311yescharacteristic
350C22×C4C22×C3⋊D311yescharacteristic
351C4⋊C4—22no—
352C4⋊C4—184no—
353C4⋊C4—182no—
354C22⋊C4—24no—
355C22⋊C4—184no—
356C22⋊C4—184no—
357C22⋊C4—184no—
358C22⋊C4—182no—
359C22⋊C4—182no—
360C22⋊C4—362no—
361C42—182no—
362C22×C3—28no—
363C22×C3—28no—
364C22×C3—216no—
365C22×C3—216no—
366C22×C3—48no—
367C22×C3—48no—
368C22×C3—48no—
369C22×C3—48no—
370C22×C3—48no—
371C22×C3—48no—
372C22×C3—48no—
373C22×C3#938116yes—
374C22×C3C2×D3×D414yes—
375C22×C3C2×#23714yes—
376C22×C3#100418yes—
377C22×C3D3×#4118yes—
378C22×C3C22×#6714yes—
379C2×D3—68no—
380C2×D3—616no—
381C2×D3—124no—
382C2×D3—124no—
383C2×D3—128no—
384C3×C4—28no—
385C3×C4—28no—
386C3×C4—48no—
387Dic3—68no—
388Dic3—616no—
389Dic3—124no—
390Dic3—124no—
391Dic3—128no—
392C32#254311yescharacteristic, complemented, Sylow
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—92no—
403C23—181no—
404C23—182no—
405C23—182no—
406C23—181no—
407C23D4×C3⋊D312yes—
408C23#383012yes—
409C23C2×#63812yes—
410C23C23×C3⋊D311yescharacteristic
411D4—44no—
412D4—48no—
413D4—188no—
414D4—368no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—44no—
421C2×C4—92no—
422C2×C4—94no—
423C2×C4—182no—
424C2×C4—181no—
425C2×C4—182no—
426C2×C4—181no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4—364no—
435C2×C4—368no—
436C2×C4C2×#63812yes—
437C2×C4C2×#63812yes—
438C2×C3—216no—
439C2×C3—88no—
440C2×C3#560118yes—
441C2×C3#561018yes—
442C2×C3C2×#93814yes—
443C2×C3#580414yes—
444C2×C3#583214yes—
445D3—128no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—182no—
458C22—184no—
459C22—361no—
460C22—361no—
461C22—362no—
462C22#6414714yes—
463C22C2×D4×C3⋊D311yescharacteristic
464C22C2×#383011yescharacteristic
465C22#6436112yes—
466C22#41×C3⋊D312yes—
467C22C22×#63811yescharacteristic
468C4—22no—
469C4—22no—
470C4—42no—
471C4—182no—
472C4—184no—
473C4—361no—
474C4—361no—
475C4—362no—
476C3#8267114yescomplemented
477C2—24no—
478C2—82no—
479C2—362no—
480C2#1059607212yes—
481C2#1059608112yes—
482C2C2×#6414711yescharacteristic
483C2#1059627511yescharacteristic
484C2#1059630311yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
FittingC2×C32×#121C2
hypercenterC2×#121C3⋊D3
hypocenterC32#2543
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C32×#121C2semidirect
2. C2×#64148C2semidirect
3. C2×C4×#638C2semidirect
4. #10592922C2semidirect
5. #10592921C2semidirect
6. #10592919C2semidirect
7. C2×#63658C2semidirect
8. #10592691C2semidirect
9. #10592682C2semidirect
10. #10592655C2semidirect
11. #10592643C2semidirect
39. C2×C3×#121D3semidirect
59. C22×C32×C4D4semidirect
68. C2×#638D4semidirect
163. C2×#121C3⋊D3semidirect
182. C22×C3×C4#67semidirect
347. C22×C4#638semidirect
392. C32#2543semidirect
476. C3#82671semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C2×#641482
C2×C4×#6382
#105929222
#105929212
#105929192
C2×#636582
#105926912
#105926822
#105926552
#105926432
#826713