Finite Group Databasebeta

← #11808201

Subgroups — #11808201

Order 1152.

2832 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3830C211yesmaximal, characteristic, complemented
2C2×D4×C3⋊Dic3C212yesmaximal, complemented
3C2×#64090C211yesmaximal, characteristic, complemented
4C32×#392C211yesmaximal, characteristic, complemented
5#10592922C212yesmaximal, complemented
6#10592921C211yesmaximal, characteristic, complemented
7#10592660C212yesmaximal, complemented
8#10592646C211yesmaximal, characteristic, complemented
9C22⋊C4×C3⋊Dic3C212yesmaximal, complemented
10#10592365C211yesmaximal, characteristic, complemented
11#10592355C211yesmaximal, characteristic, complemented
12#76665—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C22×C32×D4C2212yes—
15C22×#638C2211yescharacteristic
16C22×#638C2212yes—
17C23×C3⋊Dic3C2211yescharacteristic
18C23×C3⋊Dic3C2212yes—
19C2×#3830—28no—
20C2×#3830—24no—
21C22×C4×C3⋊D3C2211yescharacteristic
22C22×#634C2211yescharacteristic
23C2×C32×C22⋊C4C2211yescharacteristic
24C2×C32×C22⋊C4C2212yes—
25C2×#3757C2212yes—
26C2×#3757C2212yes—
27C2×#3757C2212yes—
28D4×C3⋊Dic3—28no—
29C2×#3752C2212yes—
30C2×#3752C2211yescharacteristic
31C2×C4⋊(C3⋊Dic3)C2212yes—
32C2×#3750C2211yescharacteristic
33C2×C4×C3⋊Dic3C2212yes—
34C2×C4×C3⋊Dic3C2211yescharacteristic
35#64090—24no—
36C32×#95C2211yescharacteristic
37#63658C2212yes—
38#63658C2211yescharacteristic
39#63658C2212yes—
40C22×#237—34no—
41C2×D4×Dic3—38no—
42C2×#67⋊C4—34no—
43C3×#392D314yescomplemented
44#5223—38no—
45#5222—34no—
46#4961—38no—
47#4947—34no—
48Dic3×C22⋊C4—38no—
49#4666—34no—
50#4656—34no—
51C24×C32C2312yes—
52C24×C32C2311yescharacteristic
53C23×C3⋊D3C2311yescharacteristic
54C2×C32×D4—24no—
55C2×C32×D4—24no—
56C2×C32×D4—24no—
57C2×C32×D4—22no—
58C2×C32×D4—24no—
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2312yes—
62C2×#638—24no—
63C2×#638—22no—
64C2×#638—22no—
65C2×#638—24no—
66C2×#638—24no—
67C2×#638—22no—
68C22×C3⋊Dic3—24no—
69C22×C3⋊Dic3—24no—
70C22×C3⋊Dic3—22no—
71C22×C3⋊Dic3—24no—
72C22×C3⋊Dic3D418yescomplemented
73C22×C3⋊Dic3C2311yescharacteristic
74C22×C3⋊Dic3C2312yes—
75C22×C3⋊Dic3C2312yes—
76C22×C3⋊Dic3C2311yescharacteristic
77C22×C3⋊Dic3C2311yescharacteristic
78#3830—232no—
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3D414yescomplemented
81C2×#634—22no—
82C2×#634—24no—
83C32×C22⋊C4—22no—
84C32×C22⋊C4—24no—
85#3757—24no—
86#3757—24no—
87#3757—24no—
88#3752—22no—
89#3752—24no—
90C4⋊(C3⋊Dic3)—24no—
91#3750—22no—
92C4×C3⋊Dic3—22no—
93C4×C3⋊Dic3—24no—
94#2388—91noSylow
95C22×C3×D4C2×D314yes—
96C22×C3×D4C2×D318yes—
97C22×#67—38no—
98C22×#67—34no—
99C23×Dic3—38no—
100C23×Dic3—34no—
101C2×#237—616no—
102C2×#237—632no—
103C22×C4×D3—34no—
104C22×Dic6—34no—
105C2×C3×C22⋊C4C2×D318yes—
106C2×C3×C22⋊C4C2×D314yes—
107C2×#217—38no—
108C2×#217—38no—
109C2×#217—38no—
110D4×Dic3—632no—
111C2×(C2×D3)⋊C4—38no—
112C2×(C2×D3)⋊C4—34no—
113C2×C4⋊Dic3—38no—
114C2×Dic3⋊C4—34no—
115C2×C4×Dic3—38no—
116C2×C4×Dic3—34no—
117#67⋊C4—616no—
118C3×#95C2×D314yes—
119C3⋊#95—38no—
120C3⋊#95—34no—
121C3⋊#95—38no—
122C23×C32—22no—
123C23×C32—21no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—22no—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32C2×D414yes—
133C23×C32#4112yescomplemented
134C23×C32C2411yescharacteristic
135C22×C3⋊D3—22no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3C2×D412yes—
140C32×D4—28no—
141C32×D4—216no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—22no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4C2×D412yes—
148C2×C32×C4#4114yescomplemented
149#638—216no—
150#638—28no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—216no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—44no—
162C2×C3⋊Dic3C2×D414yes—
163C2×C3⋊Dic3C2×D414yes—
164C2×C3⋊Dic3C2×D412yes—
165C2×C3⋊Dic3#4112yescomplemented
166C4×C3⋊D3—28no—
167#634—28no—
168C22×#41—91no—
169C2×C4×D4—92no—
170C2×C4×D4—91no—
171#395—91no—
172#394—91no—
173#394—92no—
174#393—91no—
175#392—92no—
176#392C3⋊D311yescharacteristic, complemented
177C4×C22⋊C4—92no—
178C4×C22⋊C4—91no—
179C24×C3C22×D318yes—
180C24×C3C22×D314yes—
181C23×D3—34no—
182C2×C3×D4—216no—
183C2×C3×D4—216no—
184C2×C3×D4—216no—
185C2×C3×D4—28no—
186C2×C3×D4—216no—
187C22×C3×C4C22×D314yes—
188C22×C3×C4C22×D314yes—
189C22×C3×C4C22×D318yes—
190C2×#67—616no—
191C2×#67—68no—
192C2×#67—68no—
193C2×#67—616no—
194C2×#67—616no—
195C2×#67—68no—
196C22×Dic3—34no—
197C22×Dic3—332no—
198C22×Dic3—34no—
199C22×Dic3—38no—
200C22×Dic3—38no—
201C22×Dic3—34no—
202C22×Dic3—616no—
203C22×Dic3—616no—
204C22×Dic3—68no—
205C22×Dic3—616no—
206#237—6128no—
207C2×C4×D3—316no—
208C2×C4×D3—616no—
209C2×Dic6—616no—
210C2×Dic6—68no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—216no—
213#217—616no—
214#217—616no—
215#217—616no—
216(C2×D3)⋊C4—68no—
217(C2×D3)⋊C4—616no—
218C4⋊Dic3—616no—
219Dic3⋊C4—68no—
220C4×Dic3—68no—
221C4×Dic3—616no—
222C22×C32—24no—
223C22×C32—22no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—41no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32C22×D411yescharacteristic
235C22×C32C22×D412yes—
236C22×C32C2×#4111yescharacteristic
237C22×C32C2×#4112yes—
238C22×C32C2×#4111yescharacteristic
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—22no—
242C2×C3⋊D3—42no—
243C2×C3⋊D3—42no—
244C32×C4—22no—
245C32×C4—24no—
246C32×C4—41no—
247C3⋊Dic3—24no—
248C3⋊Dic3—22no—
249C3⋊Dic3—41no—
250C3⋊Dic3—42no—
251C3⋊Dic3#12014yes—
252C2×#41—188no—
253C2×#41—184no—
254C22×Q8—91no—
255C22×D4—91no—
256C22×D4—92no—
257C22×D4C2×C3⋊D312yes—
258C22×D4C2×C3⋊D311yescharacteristic
259C23×C4—92no—
260C23×C4—91no—
261C23×C4—91no—
262C4×D4—184no—
263C4×D4—188no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—92no—
266C2×C22⋊C4—92no—
267C2×C22⋊C4—92no—
268C2×C22⋊C4—92no—
269C2×C22⋊C4—91no—
270C2×C22⋊C4—92no—
271C2×C22⋊C4C2×C3⋊D311yescharacteristic
272C2×C22⋊C4C2×C3⋊D312yes—
273C2×C42—91no—
274C2×C42—92no—
275#95—92no—
276#95—92no—
277#95—91no—
278#95C2×C3⋊D311yescharacteristic
279C23×C3—28no—
280C23×C3—24no—
281C23×C3—28no—
282C23×C3—28no—
283C23×C3—28no—
284C23×C3—28no—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3—28no—
288C23×C3—24no—
289C23×C3D3×D4116yes—
290C23×C3#23718yescomplemented
291C23×C3C23×D314yes—
292C22×D3—38no—
293C22×D3—68no—
294C22×D3—68no—
295C22×D3—64no—
296C22×D3—64no—
297C3×D4—232no—
298C3×D4—264no—
299C2×C3×C4—28no—
300C2×C3×C4—28no—
301C2×C3×C4—28no—
302C2×C3×C4—24no—
303C2×C3×C4—28no—
304C2×C3×C4D3×D418yes—
305C2×C3×C4#237116yescomplemented
306#67—664no—
307#67—632no—
308C2×Dic3—38no—
309C2×Dic3—316no—
310C2×Dic3—316no—
311C2×Dic3—38no—
312C2×Dic3—664no—
313C2×Dic3—64no—
314C2×Dic3—68no—
315C2×Dic3—68no—
316C2×Dic3—64no—
317C2×Dic3—68no—
318C2×Dic3—64no—
319C2×Dic3—68no—
320C2×Dic3—68no—
321C2×Dic3—68no—
322C2×Dic3—1216no—
323C4×D3—632no—
324Dic6—632no—
325C2×C32—44no—
326C2×C32—42no—
327C2×C32C2×#12011yescharacteristic
328C2×C32#52512yes—
329C2×C32#52511yescharacteristic
330C2×C32#53211yescharacteristic
331C2×C32#53212yes—
332C3⋊D3—42no—
333C24—91no—
334C24C22×C3⋊D312yes—
335C24C22×C3⋊D311yescharacteristic
336#41—1832no—
337C2×Q8—184no—
338C2×Q8—182no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—184no—
345C2×D4—182no—
346C2×D4—182no—
347C2×D4—184no—
348C2×D4—184no—
349C2×D4—182no—
350C22×C4—91no—
351C22×C4—91no—
352C22×C4—92no—
353C22×C4—98no—
354C22×C4—94no—
355C22×C4—92no—
356C22×C4—91no—
357C22×C4—184no—
358C22×C4—184no—
359C22×C4—184no—
360C22×C4—182no—
361C22×C4—184no—
362C22×C4C22×C3⋊D311yescharacteristic
363C22×C4C22×C3⋊D311yescharacteristic
364C22×C4C22×C3⋊D312yes—
365C4⋊C4—184no—
366C4⋊C4—182no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—182no—
370C22⋊C4—184no—
371C22⋊C4—184no—
372C22⋊C4—184no—
373C22⋊C4—184no—
374C42—184no—
375C42—182no—
376C22×C3—216no—
377C22×C3—216no—
378C22×C3—216no—
379C22×C3—216no—
380C22×C3—28no—
381C22×C3—44no—
382C22×C3—48no—
383C22×C3—48no—
384C22×C3—44no—
385C22×C3—48no—
386C22×C3—48no—
387C22×C3—48no—
388C22×C3C2×D3×D414yes—
389C22×C3C2×D3×D418yes—
390C22×C3C2×#23718yes—
391C22×C3C2×#23714yes—
392C22×C3D3×#4114yes—
393C2×D3—68no—
394C2×D3—68no—
395C2×D3—68no—
396C2×D3—128no—
397C2×D3—128no—
398C3×C4—216no—
399C3×C4—28no—
400C3×C4—44no—
401Dic3—316no—
402Dic3—616no—
403Dic3—68no—
404Dic3—124no—
405Dic3—128no—
406C32#238811yescharacteristic, complemented, Sylow
407C23—21no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—21no—
417C23—92no—
418C23—182no—
419C23—182no—
420C23—181no—
421C23—181no—
422C23D4×C3⋊D314yes—
423C23#383012yescomplemented
424C23C23×C3⋊D311yescharacteristic
425Q8—188no—
426D4—28no—
427D4—216no—
428D4—1816no—
429D4—188no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—92no—
436C2×C4—94no—
437C2×C4—94no—
438C2×C4—92no—
439C2×C4—188no—
440C2×C4—1816no—
441C2×C4—181no—
442C2×C4—182no—
443C2×C4—182no—
444C2×C4—181no—
445C2×C4—182no—
446C2×C4—181no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—364no—
451C2×C4D4×C3⋊D312yes—
452C2×C4#383014yescomplemented
453C2×C3—416no—
454C2×C3—48no—
455C2×C3#558618yes—
456C2×C3D3×#12014yes—
457C2×C3#559618yes—
458C2×C3#560614yes—
459C2×C3#571614yes—
460D3—128no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—22no—
466C22—41no—
467C22—42no—
468C22—42no—
469C22—41no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—182no—
474C22—182no—
475C22—182no—
476C22—362no—
477C22—362no—
478C22C2×D4×C3⋊D311yescharacteristic
479C22C2×D4×C3⋊D312yes—
480C22C2×#383012yes—
481C22C2×#383011yescharacteristic
482C22#41×C3⋊D311yescharacteristic
483C4—24no—
484C4—22no—
485C4—41no—
486C4—94no—
487C4—184no—
488C4—182no—
489C4—361no—
490C4—362no—
491C3#7666514yescomplemented
492C2—44no—
493C2—42no—
494C2—362no—
495C2#1059605712yes—
496C2C3⋊D3×#12011yescharacteristic
497C2#1059606712yes—
498C2#1059607711yescharacteristic
499C2#1059618711yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
FittingC32×#392C2
hypercenter#392C3⋊D3
hypocenterC32#2388
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#3830C2semidirect
2. C2×D4×C3⋊Dic3C2semidirect
3. C2×#64090C2semidirect
4. C32×#392C2semidirect
5. #10592922C2semidirect
6. #10592921C2semidirect
7. #10592660C2semidirect
8. #10592646C2semidirect
9. C22⋊C4×C3⋊Dic3C2semidirect
10. #10592365C2semidirect
11. #10592355C2semidirect
43. C3×#392D3semidirect
72. C22×C3⋊Dic3D4semidirect
80. C2×C4×C3⋊D3D4semidirect
133. C23×C32#41semidirect
148. C2×C32×C4#41semidirect
165. C2×C3⋊Dic3#41semidirect
176. #392C3⋊D3semidirect
290. C23×C3#237semidirect
305. C2×C3×C4#237semidirect
406. C32#2388semidirect
423. C23#3830semidirect
452. C2×C4#3830semidirect
491. C3#76665semidirect

Maximal subgroups

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

subgroupindex
C22×#38302
C2×D4×C3⋊Dic32
C2×#640902
C32×#3922
#105929222
#105929212
#105926602
#105926462
C22⋊C4×C3⋊Dic32
#105923652
#105923552
#766653