Finite Group Databasebeta

← #11824528

Subgroups — #11824528

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×#64104C211yesmaximal, characteristic, complemented
2C2×#64093C212yesmaximal, complemented
3C2×#64090C211yesmaximal, characteristic, complemented
4C32×#397C211yesmaximal, characteristic, complemented
5#10592682C211yesmaximal, characteristic, complemented
6#10592659C211yesmaximal, characteristic, complemented
7#10592657C212yesmaximal, complemented
8#10592371C211yesmaximal, characteristic, complemented
9#10592368C212yesmaximal, complemented
10#10592364C212yesmaximal, complemented
11C3⋊D3×#95C211yesmaximal, characteristic, complemented
12#77682—34nomaximal
13C22×#638C2212yes—
14C22×#638C2211yescharacteristic
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C4⋊(C3⋊D3)C2212yes—
17C22×C4⋊(C3⋊D3)C2211yescharacteristic
18C22×C4×C3⋊D3C2212yes—
19C22×C4×C3⋊D3C2211yescharacteristic
20C2×C32×C4⋊C4C2211yescharacteristic
21C2×C32×C22⋊C4C2211yescharacteristic
22C2×C32×C22⋊C4C2212yes—
23C2×#3752C2211yescharacteristic
24C2×#3752C2212yes—
25C2×#3752C2212yes—
26C2×#3752C2212yes—
27C2×#3752C2212yes—
28C2×#3750C2212yes—
29C2×#3750C2211yescharacteristic
30C2×C4×C3⋊Dic3C2211yescharacteristic
31#64104—24no—
32#64093—28no—
33#64090—24no—
34C32×#95C2211yescharacteristic
35C32×#95C2212yes—
36#63658C2212yes—
37#63658C2211yescharacteristic
38#63658C2212yes—
39C2×#895—34no—
40C2×#884—38no—
41C2×#67⋊C4—34no—
42C3×#397D314yescomplemented
43#4983—34no—
44#4960—34no—
45#4958—38no—
46#4672—34no—
47#4669—38no—
48#4665—38no—
49D3×#95—34no—
50C24×C32C2311yescharacteristic
51C23×C3⋊D3C2312yes—
52C23×C3⋊D3C2311yescharacteristic
53C22×C32×C4C2312yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C2×#638—22no—
58C2×#638—22no—
59C2×#638—24no—
60C2×#638—24no—
61C2×#638—42no—
62C2×#638D414yescomplemented
63C22×C3⋊Dic3—22no—
64C22×C3⋊Dic3—42no—
65C22×C3⋊Dic3C2311yescharacteristic
66C22×C3⋊Dic3C2311yescharacteristic
67C22×C3⋊Dic3C2311yescharacteristic
68C22×C3⋊Dic3C2312yes—
69C2×C4⋊(C3⋊D3)—24no—
70C2×C4⋊(C3⋊D3)—24no—
71C2×C4⋊(C3⋊D3)—24no—
72C2×C4⋊(C3⋊D3)—22no—
73C2×C4⋊(C3⋊D3)—42no—
74C2×C4×C3⋊D3—24no—
75C2×C4×C3⋊D3—24no—
76C2×C4×C3⋊D3—42no—
77C2×C4×C3⋊D3—42no—
78C2×C4×C3⋊D3D414yescomplemented
79C32×C4⋊C4—22no—
80C32×C22⋊C4—24no—
81C32×C22⋊C4—22no—
82#3752—22no—
83#3752—24no—
84#3752—24no—
85#3752—24no—
86#3752—42no—
87#3750—22no—
88#3750—24no—
89C4×C3⋊Dic3—22no—
90#2543—91noSylow
91C22×#67—34no—
92C22×#67—38no—
93C23×Dic3—34no—
94C22×D12—38no—
95C22×D12—34no—
96C22×C4×D3—34no—
97C22×C4×D3—38no—
98C2×C3×C4⋊C4C2×D314yes—
99C2×C3×C22⋊C4C2×D318yes—
100C2×C3×C22⋊C4C2×D314yes—
101C2×(C2×D3)⋊C4—34no—
102C2×(C2×D3)⋊C4—38no—
103C2×(C2×D3)⋊C4—38no—
104C2×(C2×D3)⋊C4—38no—
105C2×(C2×D3)⋊C4—38no—
106C2×Dic3⋊C4—38no—
107C2×Dic3⋊C4—34no—
108C2×C4×Dic3—34no—
109#895—616no—
110#884—632no—
111#67⋊C4—616no—
112C3×#95C2×D314yes—
113C3×#95C2×D318yes—
114C3⋊#95—38no—
115C3⋊#95—34no—
116C3⋊#95—38no—
117C23×C32—21no—
118C23×C32—21no—
119C23×C32—22no—
120C23×C32—22no—
121C23×C32C2×D412yes—
122C23×C32C2411yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3—22no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3—22no—
132C22×C3⋊D3C2×D412yes—
133C22×C3⋊D3C2×D412yes—
134C22×C3⋊D3#4112yes—
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
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—22no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—44no—
155C2×C3⋊Dic3C2×D412yes—
156C2×C3⋊Dic3C2×D412yes—
157C2×C3⋊Dic3#4114yes—
158C4⋊(C3⋊D3)—44no—
159C4⋊(C3⋊D3)—48no—
160C4×C3⋊D3—44no—
161C4×C3⋊D3—48no—
162C2×#121—91no—
163C2×#121—92no—
164C2×C4×D4—91no—
165#398—91no—
166#397C3⋊D311yescharacteristic, complemented
167#394—91no—
168#392—91no—
169#388—92no—
170#386—92no—
171#386—92no—
172C2×#95—91no—
173C24×C3C22×D314yes—
174C23×D3—34no—
175C23×D3—38no—
176C22×C3×C4C22×D318yes—
177C22×C3×C4C22×D318yes—
178C22×C3×C4C22×D314yes—
179C22×C3×C4C22×D314yes—
180C2×#67—316no—
181C2×#67—616no—
182C2×#67—68no—
183C2×#67—68no—
184C2×#67—616no—
185C2×#67—128no—
186C22×Dic3—38no—
187C22×Dic3—34no—
188C22×Dic3—34no—
189C22×Dic3—34no—
190C22×Dic3—68no—
191C22×Dic3—128no—
192C2×D12—616no—
193C2×D12—616no—
194C2×D12—616no—
195C2×D12—68no—
196C2×D12—128no—
197C2×C4×D3—316no—
198C2×C4×D3—616no—
199C2×C4×D3—616no—
200C2×C4×D3—128no—
201C2×C4×D3—128no—
202C3×C4⋊C4—28no—
203C3×C22⋊C4—216no—
204C3×C22⋊C4—28no—
205(C2×D3)⋊C4—616no—
206(C2×D3)⋊C4—616no—
207(C2×D3)⋊C4—616no—
208(C2×D3)⋊C4—68no—
209(C2×D3)⋊C4—128no—
210Dic3⋊C4—616no—
211Dic3⋊C4—68no—
212C4×Dic3—68no—
213C22×C32—22no—
214C22×C32—24no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32C22×D411yescharacteristic
219C22×C32C22×D411yescharacteristic
220C22×C32C2×#4111yescharacteristic
221C22×C32C2×#4112yes—
222C22×C32H(5,𝔽2)12yes—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—24no—
226C2×C3⋊D3—24no—
227C2×C3⋊D3—42no—
228C2×C3⋊D3—42no—
229C2×C3⋊D3—42no—
230C2×C3⋊D3—42no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3—42no—
233C2×C3⋊D3—42no—
234C2×C3⋊D3#12114yes—
235C32×C4—22no—
236C32×C4—42no—
237C32×C4—41no—
238C32×C4—42no—
239C3⋊Dic3—22no—
240C3⋊Dic3—22no—
241C3⋊Dic3—24no—
242C3⋊Dic3—41no—
243C22×D4—91no—
244C22×D4—92no—
245C22×D4—92no—
246C22×D4—91no—
247C23×C4—91no—
248C23×C4—91no—
249C23×C4—92no—
250#121—188no—
251#121—184no—
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—92no—
260C2×C22⋊C4—92no—
261C2×C22⋊C4C2×C3⋊D311yescharacteristic
262C2×C22⋊C4C2×C3⋊D312yes—
263C2×C42—91no—
264#95—92no—
265#95—91no—
266#95—92no—
267#95C2×C3⋊D311yescharacteristic
268#95C2×C3⋊D312yes—
269C23×C3—24no—
270C23×C3—28no—
271C23×C3—24no—
272C23×C3—28no—
273C23×C3D3×D418yes—
274C23×C3C23×D314yes—
275C22×D3—38no—
276C22×D3—38no—
277C22×D3—38no—
278C22×D3—68no—
279C22×D3—68no—
280C22×D3—68no—
281C22×D3—68no—
282C22×D3—68no—
283C22×D3—68no—
284C22×D3—68no—
285C22×D3—68no—
286C22×D3—68no—
287C2×C3×C4—28no—
288C2×C3×C4—24no—
289C2×C3×C4—28no—
290C2×C3×C4—28no—
291C2×C3×C4—28no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—28no—
295C2×C3×C4—28no—
296C2×C3×C4D3×D418yes—
297#67—632no—
298#67—1232no—
299C2×Dic3—38no—
300C2×Dic3—38no—
301C2×Dic3—316no—
302C2×Dic3—64no—
303C2×Dic3—68no—
304C2×Dic3—64no—
305C2×Dic3—68no—
306C2×Dic3—68no—
307C2×Dic3—64no—
308C2×Dic3—68no—
309C2×Dic3—1216no—
310D12—1216no—
311D12—1232no—
312C4×D3—1216no—
313C4×D3—1232no—
314C2×C32—42no—
315C2×C32C2×#12111yescharacteristic
316C2×C32#53211yescharacteristic
317C2×C32#53411yescharacteristic
318C2×C32#54612yes—
319C2×C32#55312yes—
320C3⋊D3—24no—
321C3⋊D3—82no—
322C24—91no—
323C24—92no—
324C24C22×C3⋊D311yescharacteristic
325C2×D4—94no—
326C2×D4—184no—
327C2×D4—184no—
328C2×D4—184no—
329C2×D4—184no—
330C2×D4—182no—
331C2×D4—182no—
332C2×D4—184no—
333C2×D4—182no—
334C2×D4—362no—
335C2×D4—362no—
336C22×C4—92no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4—94no—
340C22×C4—91no—
341C22×C4—184no—
342C22×C4—184no—
343C22×C4—182no—
344C22×C4—362no—
345C22×C4—362no—
346C22×C4—362no—
347C22×C4C22×C3⋊D312yes—
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—22no—
356C22⋊C4—184no—
357C22⋊C4—184no—
358C22⋊C4—184no—
359C22⋊C4—182no—
360C22⋊C4—362no—
361C42—182no—
362C22×C3—216no—
363C22×C3—28no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—48no—
367C22×C3C2×D3×D414yes—
368C22×C3C2×D3×D414yes—
369C22×C3D3×#4118yes—
370C22×C3D3×#4114yes—
371C22×C3#100918yes—
372C2×D3—316no—
373C2×D3—68no—
374C2×D3—616no—
375C2×D3—68no—
376C2×D3—616no—
377C2×D3—128no—
378C2×D3—128no—
379C2×D3—128no—
380C2×D3—128no—
381C2×D3—128no—
382C2×D3—128no—
383C2×D3—128no—
384C3×C4—28no—
385C3×C4—48no—
386C3×C4—44no—
387C3×C4—48no—
388Dic3—68no—
389Dic3—616no—
390Dic3—68no—
391Dic3—124no—
392C32#254311yescharacteristic, complemented, Sylow
393C23—21no—
394C23—22no—
395C23—21no—
396C23—22no—
397C23—92no—
398C23—92no—
399C23—92no—
400C23—182no—
401C23—182no—
402C23—182no—
403C23—182no—
404C23—182no—
405C23—182no—
406C23—182no—
407C23—182no—
408C23—182no—
409C23D4×C3⋊D312yes—
410C23C23×C3⋊D311yescharacteristic
411D4—188no—
412D4—364no—
413D4—368no—
414D4—368no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—94no—
427C2×C4—181no—
428C2×C4—182no—
429C2×C4—181no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—181no—
433C2×C4—182no—
434C2×C4—364no—
435C2×C4—368no—
436C2×C4—364no—
437C2×C4D4×C3⋊D312yes—
438C2×C3—48no—
439C2×C3D3×#12114yes—
440C2×C3#565314yes—
441C2×C3#565418yes—
442C2×C3#566814yes—
443C2×C3#570918yes—
444D3—616no—
445D3—248no—
446C22—24no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—42no—
451C22—94no—
452C22—182no—
453C22—184no—
454C22—182no—
455C22—184no—
456C22—362no—
457C22—362no—
458C22—362no—
459C22—362no—
460C22—362no—
461C22—362no—
462C22—362no—
463C22C2×D4×C3⋊D311yescharacteristic
464C22C2×D4×C3⋊D311yescharacteristic
465C22#41×C3⋊D312yes—
466C22#41×C3⋊D311yescharacteristic
467C22#6436612yes—
468C4—22no—
469C4—42no—
470C4—41no—
471C4—42no—
472C4—182no—
473C4—184no—
474C4—182no—
475C4—361no—
476C3#7768214yescomplemented
477C2—42no—
478C2—184no—
479C2—722no—
480C2C3⋊D3×#12111yescharacteristic
481C2#1059612411yescharacteristic
482C2#1059612512yes—
483C2#1059613911yescharacteristic
484C2#1059618012yes—
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
FittingC32×#397C2
hypercenter#397C3⋊D3
hypocenterC32#2543
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#64104C2semidirect
2. C2×#64093C2semidirect
3. C2×#64090C2semidirect
4. C32×#397C2semidirect
5. #10592682C2semidirect
6. #10592659C2semidirect
7. #10592657C2semidirect
8. #10592371C2semidirect
9. #10592368C2semidirect
10. #10592364C2semidirect
11. C3⋊D3×#95C2semidirect
42. C3×#397D3semidirect
62. C2×#638D4semidirect
78. C2×C4×C3⋊D3D4semidirect
166. #397C3⋊D3semidirect
392. C32#2543semidirect
476. C3#77682semidirect

Maximal subgroups

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

subgroupindex
C2×#641042
C2×#640932
C2×#640902
C32×#3972
#105926822
#105926592
#105926572
#105923712
#105923682
#105923642
C3⋊D3×#952
#776823