Finite Group Databasebeta

← #11858880

Subgroups — #11858880

Order 1152.

847 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63855C211yesmaximal, characteristic
2#10592817C211yesmaximal, characteristic, complemented
3C3×#5126C211yesmaximal, characteristic
4C3×Dic3×M16C211yesmaximal, characteristic, complemented
5#10591120C211yesmaximal, characteristic, complemented
6#10591069C211yesmaximal, characteristic
7Dic3×#208C211yesmaximal, characteristic, complemented
8#10590697C214yesmaximal, complemented
9#10590696C214yesmaximal, complemented
10#78620—31nomaximal
11#78518—31nomaximal
12C22×C3×C4×D3C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×C4×C3⋊Dic3C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C2×Dic3⋊Dic3C412yes—
20C2×(C2×D3)⋊Dic3C412yes—
21#63855C412yes—
22#63855C412yes—
23#63855C2212yes—
24#63855C2212yes—
25#63641C2214yescomplemented
26C3×(C2×D3)⋊C8C2214yes—
27C3×#815C2212yescomplemented
28C3×C8×Dic3C2212yescomplemented
29#63572C2214yescomplemented
30(C2×D3)⋊(C3⋊C8)C2214yes—
31#63554C2212yescomplemented
32Dic3×C3⋊C8C2212yescomplemented
33C2×#926—31no—
34C2×#872—31no—
35C3×#407—31no—
36C3×C4×M16—31no—
37#5207—31no—
38#5126—31no—
39#5126D311yescharacteristic
40#5118—31no—
41#5118—31no—
42#5118—31no—
43#5118—31no—
44#5118—61no—
45Dic3×M16—31no—
46Dic3×M16D311yescharacteristic, complemented
47C4×#208—31no—
48#4720—34no—
49#4720—34no—
50#4709—34no—
51#4690—34no—
52C23×C3×D3C2×C411yescharacteristic
53C22×C32×C4C2311yescharacteristic
54C22×C3⋊Dic3C2×C412yes—
55C22×C3×Dic3C2×C411yescharacteristic
56C22×C3×Dic3C2×C412yes—
57C2×C3×C4×D3—22no—
58C2×C3×C4×D3—22no—
59C2×C3×C4×D3C2×C412yes—
60C2×C3×C4×D3C2312yes—
61C32×M16—22no—
62C2×C32×C8C2312yes—
63C4×C3⋊Dic3C2×C411yescharacteristic
64C4×C3⋊Dic3C2×C411yescharacteristic
65C4×C3⋊Dic3C2312yes—
66#3748—22no—
67C2×C3⋊(C3⋊C8)C2312yes—
68C3×C4×Dic3C2×C412yes—
69C3×C4×Dic3C2311yescharacteristic
70C3×C4×Dic3C2311yescharacteristic
71C3×#208—22no—
72C3×#208—22no—
73C2×C3×C3⋊C8C2312yes—
74C2×C3×C3⋊C8C2312yes—
75Dic3⋊Dic3C2×C412yes—
76Dic3⋊Dic3C2×C412yes—
77Dic3⋊Dic3C2×C412yes—
78Dic3⋊Dic3C2×C412yes—
79(C2×D3)⋊Dic3C2×C412yes—
80(C2×D3)⋊Dic3C2×C412yes—
81(C2×D3)⋊Dic3C2×C412yes—
82(C2×D3)⋊Dic3C2×C412yes—
83#2921—91noSylow
84C23×C3×C4—31no—
85C22×C4×D3C2×D311yescharacteristic
86C2×C3×M16—21no—
87C2×C3×M16—31no—
88C2×C3×M16—31no—
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C2×D311yescharacteristic
91C2×C3×C42—31no—
92C2×#217—32no—
93C2×(C2×D3)⋊C4—32no—
94#926—32no—
95#926—32no—
96#926—32no—
97#926—32no—
98C2×C4⋊Dic3—32no—
99C2×Dic3⋊C4—32no—
100C2×C4×Dic3—31no—
101C2×C4×Dic3—31no—
102C2×C4×Dic3—61no—
103C2×C4×Dic3C2×D311yescharacteristic
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—61no—
107C2×#208C2×D311yescharacteristic
108C2×#208C2×D311yescharacteristic
109#872—32no—
110#872—32no—
111#872—32no—
112#872—32no—
113C3×C22⋊C8—34no—
114C3×#97—32no—
115C3×C4×C8—32no—
116#830—34no—
117(C2×D3)⋊C8—34no—
118(C2×D3)⋊C8C2×D314yes—
119#815—32no—
120#815C2×D312yescomplemented
121Dic3⋊C8—34no—
122Dic3⋊C8—34no—
123Dic3⋊C8—34no—
124Dic3⋊C8—34no—
125Dic3⋊C8—64no—
126C8×Dic3—32no—
127C8×Dic3C2×D312yescomplemented
128#803—32no—
129C4×C3⋊C8—32no—
130C23×C32C22×C411yescharacteristic
131C22×C3×D3—22no—
132C22×C3×D3—42no—
133C22×C3×D3C22×C412yes—
134C2×C32×C4C22×C411yescharacteristic
135C2×C32×C4#4112yes—
136C2×C32×C4#4112yes—
137C2×C32×C4C2411yescharacteristic
138C2×C3⋊Dic3—42no—
139C2×C3⋊Dic3C22×C412yes—
140C2×C3⋊Dic3C22×C412yes—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3C22×C412yes—
144C2×C3×Dic3C22×C412yes—
145C2×C3×Dic3C22×C412yes—
146C3×C4×D3—24no—
147C3×C4×D3—22no—
148C3×C4×D3—22no—
149C32×C8—22no—
150C3⋊(C3⋊C8)—22no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—22no—
153C2×#117—91no—
154#433—94no—
155#433—94no—
156#423—91no—
157#423—91no—
158#407—31no—
159#407—91no—
160C4×M16—31no—
161C4×M16—91no—
162C24×C3—31no—
163C23×D3C4×D311yescharacteristic
164C22×C3×C4—21no—
165C22×C3×C4—31no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—32no—
169C22×C3×C4—62no—
170C22×C3×C4—62no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C22×D311yescharacteristic
173C22×Dic3—32no—
174C22×Dic3—32no—
175C22×Dic3—62no—
176C22×Dic3C4×D311yescharacteristic
177C22×Dic3C4×D312yes—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3C4×D312yes—
181C2×C4×D3C22×D312yes—
182C3×M16—22no—
183C3×M16—22no—
184C3×M16—42no—
185C3×M16—62no—
186C3×M16—62no—
187C2×C3×C8—22no—
188C2×C3×C8—32no—
189C2×C3×C8—32no—
190C2×C3×C8C22×D312yes—
191C2×C3×C8C22×D312yes—
192C3×C42—32no—
193C3×C42—31no—
194C3×C42—31no—
195#217—32no—
196#217—32no—
197#217—32no—
198#217—32no—
199(C2×D3)⋊C4—32no—
200(C2×D3)⋊C4—32no—
201(C2×D3)⋊C4—32no—
202(C2×D3)⋊C4—32no—
203C4⋊Dic3—32no—
204C4⋊Dic3—32no—
205C4⋊Dic3—32no—
206C4⋊Dic3—32no—
207Dic3⋊C4—32no—
208Dic3⋊C4—32no—
209Dic3⋊C4—32no—
210Dic3⋊C4—32no—
211C4×Dic3—31no—
212C4×Dic3—31no—
213C4×Dic3—31no—
214C4×Dic3—31no—
215C4×Dic3—32no—
216C4×Dic3—32no—
217C4×Dic3—62no—
218C4×Dic3—61no—
219C4×Dic3—61no—
220C4×Dic3C4×D312yes—
221C4×Dic3C22×D311yescharacteristic
222C4×Dic3C22×D311yescharacteristic
223#208—22no—
224#208—22no—
225#208—62no—
226#208—62no—
227#208—122no—
228C2×C3⋊C8—32no—
229C2×C3⋊C8—32no—
230C2×C3⋊C8—62no—
231C2×C3⋊C8C22×D312yes—
232C2×C3⋊C8C22×D312yes—
233C22×C32—22no—
234C22×C32#11712yes—
235C22×C32C23×C411yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—24no—
238C2×C3×D3—44no—
239C32×C4#11712yes—
240C32×C4C2×#4111yescharacteristic
241C32×C4C2×#4111yescharacteristic
242C3⋊Dic3—22no—
243C3⋊Dic3—22no—
244C3×Dic3—22no—
245C3×Dic3—22no—
246C3×Dic3—22no—
247C23×C4—31no—
248C2×M16—31no—
249C2×M16—31no—
250C2×M16—91no—
251C2×M16D3211yescharacteristic
252#117—92no—
253#117—92no—
254#117—92no—
255#117—92no—
256C2×C4⋊C4—92no—
257C2×C22⋊C4—92no—
258C2×C42—31no—
259C2×C42—91no—
260C4⋊C8—94no—
261C4⋊C8—94no—
262C22⋊C8—34no—
263C22⋊C8—94no—
264#97—32no—
265#97—92no—
266C4×C8—32no—
267C4×C8—92no—
268C23×C3—21no—
269C23×C3—32no—
270C23×C3—62no—
271C23×C3—122no—
272C23×C3C2×C4×D311yescharacteristic
273C23×C3C2×C4×D311yescharacteristic
274C22×D3—22no—
275C22×D3—42no—
276C22×D3C2×C4×D312yes—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—62no—
285C2×C3×C4—64no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—122no—
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4C2×C4×D311yescharacteristic
291C2×C3×C4#23512yes—
292C2×C3×C4#23512yes—
293C2×C3×C4#23712yes—
294C2×C3×C4#23712yes—
295C2×C3×C4C23×D311yescharacteristic
296C2×C3×C4C23×D311yescharacteristic
297C2×Dic3—22no—
298C2×Dic3—32no—
299C2×Dic3—32no—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—42no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—122no—
306C2×Dic3—122no—
307C2×Dic3—124no—
308C2×Dic3C2×C4×D312yes—
309C2×Dic3C2×C4×D312yes—
310C2×Dic3C2×C4×D312yes—
311C4×D3—24no—
312C4×D3—22no—
313C4×D3—22no—
314C3×C8—22no—
315C3×C8—22no—
316C3×C8—42no—
317C3×C8—62no—
318C3×C8—62no—
319C3⋊C8—22no—
320C3⋊C8—22no—
321C3⋊C8—62no—
322C3⋊C8—62no—
323C3⋊C8—122no—
324C2×C32—22no—
325C2×C32C2×#11711yescharacteristic
326C2×C32#56812yes—
327C3×D3—42no—
328C24—31no—
329C22×C4—31no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—92no—
336C22×C4C2×D3211yescharacteristic
337M16—22no—
338M16—62no—
339M16—62no—
340M16—182no—
341C2×C8—32no—
342C2×C8—32no—
343C2×C8—92no—
344C2×C8C2×D3212yes—
345C4⋊C4—92no—
346C4⋊C4—92no—
347C4⋊C4—92no—
348C4⋊C4—92no—
349C22⋊C4—92no—
350C22⋊C4—92no—
351C22⋊C4—92no—
352C22⋊C4—92no—
353C42—31no—
354C42—31no—
355C42—32no—
356C42—92no—
357C42—91no—
358C42—91no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—42no—
364C22×C3—62no—
365C22×C3—64no—
366C22×C3—124no—
367C22×C3#87212yes—
368C22×C3#87712yes—
369C22×C3C22×C4×D311yescharacteristic
370C22×C3C22×C4×D311yescharacteristic
371C2×D3—22no—
372C2×D3—24no—
373C2×D3—44no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—22no—
377C3×C4—62no—
378C3×C4—62no—
379C3×C4—62no—
380C3×C4#87212yes—
381C3×C4#87712yes—
382C3×C4C2×#23511yescharacteristic
383C3×C4C2×#23511yescharacteristic
384C3×C4C2×#23711yescharacteristic
385C3×C4C2×#23711yescharacteristic
386Dic3—22no—
387Dic3—22no—
388Dic3—22no—
389Dic3—64no—
390Dic3—62no—
391Dic3—62no—
392Dic3—62no—
393Dic3—62no—
394Dic3—122no—
395C32#292111yescharacteristic, complemented, Sylow
396C23—32no—
397C23—62no—
398C23—122no—
399C23C4×D3211yescharacteristic
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—64no—
407C2×C4—92no—
408C2×C4—92no—
409C2×C4—122no—
410C2×C4—362no—
411C2×C4#379512yes—
412C2×C4C4×D3211yescharacteristic
413C2×C4#380412yes—
414C2×C4C22×D3211yescharacteristic
415C8—22no—
416C8—62no—
417C8—62no—
418C8—182no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—42no—
424C2×C3—122no—
425C2×C3C2×#87211yescharacteristic
426C2×C3C2×#87711yescharacteristic
427C2×C3#574512yes—
428C2×C3#574512yes—
429D3—42no—
430C22—22no—
431C22—62no—
432C22—64no—
433C22—124no—
434C22#6387712yes—
435C22C2×C4×D3211yescharacteristic
436C4—62no—
437C4—62no—
438C4—62no—
439C4—182no—
440C4—182no—
441C4#6387712yes—
442C4C2×#379511yescharacteristic
443C4C2×#380411yescharacteristic
444C3—21no—
445C3#7851811yescharacteristic, complemented
446C3#7862011yescharacteristic, complemented
447C2—22no—
448C2—122no—
449C2#1059472212yes—
450C2C2×#6387711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC2×C32×M16C22
prime radicalC23×C3×D3C2×C4
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2921
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592817C2semidirect
4. C3×Dic3×M16C2semidirect
5. #10591120C2semidirect
7. Dic3×#208C2semidirect
8. #10590697C2semidirect
9. #10590696C2semidirect
25. #63641C22semidirect
27. C3×#815C22semidirect
28. C3×C8×Dic3C22semidirect
29. #63572C22semidirect
31. #63554C22semidirect
32. Dic3×C3⋊C8C22semidirect
46. Dic3×M16D3semidirect
120. #815C2×D3semidirect
127. C8×Dic3C2×D3semidirect
395. C32#2921semidirect
445. C3#78518semidirect
446. C3#78620semidirect

Maximal subgroups

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

subgroupindex
C2×#638552
#105928172
C3×#51262
C3×Dic3×M162
#105911202
#105910692
Dic3×#2082
#105906972
#105906962
#786203
#785183