Finite Group Databasebeta

← #123×#157 #11802156

Subgroups — #123×#157 #11802156

Order 1152.

1030 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#157C211yesmaximal, characteristic
2C22×C4×#157C211yesmaximal, characteristic, complemented
3C3⋊D3×#123C211yesmaximal, characteristic
4#10593826C211yesmaximal, characteristic
5#10593811C211yesmaximal, characteristic, complemented
6#10593758C212yesmaximal, complemented
7C4⋊C4×#157C212yesmaximal, complemented
8#10593747C212yesmaximal, complemented
9#10593746C211yesmaximal, characteristic, complemented
10C22⋊C4×#157C212yesmaximal, complemented
11C22⋊C4×#157C211yesmaximal, characteristic, complemented
12C23×#157C2211yescharacteristic
13C23×#157C2211yescharacteristic
14C23×#157C2211yescharacteristic
15C2×D4×C3⋊D3C2211yescharacteristic
16C22×C4×C3⋊D3C2211yescharacteristic
17C2×C22⋊#157C2211yescharacteristic
18C2×C22⋊#157C2211yescharacteristic
19C2×C22⋊#157C2211yescharacteristic
20C2×C22⋊#157C2212yes—
21D4×#157—24no—
22C2×C4⋊#157C2211yescharacteristic
23C2×C4⋊#157C2212yes—
24C2×C4×#157—24no—
25C2×C4×#157C2212yes—
26C2×C4×#157C2212yes—
27C2×C4×#157C2211yescharacteristic
28C2×C4×#157C2212yes—
29C32×#123C411yescharacteristic, complemented
30#64144C411yescharacteristic, complemented
31#64138C411yescharacteristic, complemented
32#64103C412yescomplemented
33C4⋊C4×C3⋊D3C2212yes—
34#64095C411yescharacteristic, complemented
35#64092C412yescomplemented
36C22⋊C4×C3⋊D3C2211yescharacteristic
37C22⋊C4×C3⋊D3C2212yes—
38#63781C2212yes—
39#63781C2212yes—
40#63781C2212yes—
41#63781C2212yes—
42#63781C2212yes—
43D3×#123—62no—
44C23×C3⋊D3C2311yescharacteristic
45C23×C3⋊D3C2311yescharacteristic
46C22×#157—22no—
47C22×#157—24no—
48C22×#157—24no—
49C22×#157—22no—
50C22×#157—24no—
51C22×#157D414yes—
52C22×#157C2311yescharacteristic
53C22×#157C2312yes—
54C22×#157C2311yescharacteristic
55C22×#157C2311yescharacteristic
56C22×#157C2312yes—
57C22×#157C2311yescharacteristic
58C2×C32×D4C2×C411yescharacteristic
59C22×C32×C4C2×C411yescharacteristic, complemented
60C2×#638C2×C411yescharacteristic
61C2×#638C2×C411yescharacteristic
62C22×C3⋊Dic3C2×C411yescharacteristic, complemented
63D4×C3⋊D3—22no—
64D4×C3⋊D3—22no—
65C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
66C2×C4×C3⋊D3—22no—
67C2×C4×C3⋊D3—22no—
68C2×C4×C3⋊D3C2×C412yescomplemented
69C2×C4×C3⋊D3C2312yes—
70C2×C4×C3⋊D3C2312yes—
71C2×C4×C3⋊D3C2311yescharacteristic
72C22⋊#157—22no—
73C22⋊#157—22no—
74C22⋊#157—22no—
75C22⋊#157—24no—
76C4⋊#157—22no—
77C4⋊#157—24no—
78C4×#157—24no—
79C4×#157—24no—
80C4×#157—22no—
81C4×#157—24no—
82C32×C4⋊C4C2×C412yescomplemented
83C32×C22⋊C4C2×C412yescomplemented
84C32×C22⋊C4C2×C411yescharacteristic, complemented
85#3757C2×C412yescomplemented
86#3757C2×C411yescharacteristic, complemented
87#3752C2×C412yescomplemented
88#3752C2×C412yescomplemented
89#3752C2×C412yescomplemented
90C4⋊(C3⋊Dic3)C2×C412yescomplemented
91#3750C2×C412yescomplemented
92#3750C2×C412yescomplemented
93C4×#123—91nomaximal, Sylow
94C2×D3×D4—62no—
95C22×C4×D3—62no—
96C3×#123—22no—
97#935—62no—
98#929—62no—
99#894—64no—
100D3×C4⋊C4—64no—
101#886—62no—
102#883—64no—
103D3×C22⋊C4—62no—
104D3×C22⋊C4—64no—
105C23×C32C22×C411yescharacteristic
106C23×C32C22×C411yescharacteristic
107C22×C3⋊D3—22no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3—21no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—21no—
113C22×C3⋊D3C22×C411yescharacteristic
114C22×C3⋊D3C22×C411yescharacteristic
115C22×C3⋊D3C2×D412yes—
116C22×C3⋊D3#4112yes—
117C22×C3⋊D3C2411yescharacteristic
118C2×#157—28no—
119C2×#157—22no—
120C2×#157—22no—
121C2×#157—22no—
122C2×#157—28no—
123C2×#157—21no—
124C2×#157—21no—
125C2×#157—22no—
126C2×#157—22no—
127C2×#157—22no—
128C2×#157—22no—
129C2×#157—22no—
130C2×#157—44no—
131C2×#157C2×D412yes—
132C2×#157C2×D412yes—
133C2×#157#4114yes—
134C2×#157#4114yes—
135C32×D4—22no—
136C2×C32×C4—22no—
137C2×C32×C4C22×C411yescharacteristic
138C2×C32×C4C22×C412yes—
139C2×C32×C4C22×C412yes—
140#638—22no—
141#638—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3C22×C411yescharacteristic
144C2×C3⋊Dic3C22×C412yes—
145C2×C3⋊Dic3C22×C412yes—
146C4⋊(C3⋊D3)—22no—
147C4×C3⋊D3—24no—
148C4×C3⋊D3—22no—
149C4×C3⋊D3—21no—
150C4×C3⋊D3—22no—
151C4×C3⋊D3—22no—
152C4×C3⋊D3—22no—
153C4×C3⋊D3—21no—
154C2×#123—91no—
155C2×C4×D4—91no—
156C22×C42—91no—
157#388—92no—
158#387—92no—
159#386—91no—
160#385—91no—
161#381—91no—
162C4×C4⋊C4—92no—
163C4×C22⋊C4—92no—
164C4×C22⋊C4—91no—
165C23×D3—62no—
166C23×D3—62no—
167C2×C3×D4—22no—
168C22×C3×C4—22no—
169C2×#67—62no—
170C2×#67—62no—
171C22×Dic3—62no—
172D3×D4—124no—
173D3×D4—124no—
174C2×D12—62no—
175C2×C4×D3—64no—
176C2×C4×D3—64no—
177C2×C4×D3—64no—
178C2×C4×D3—62no—
179C2×C4×D3—124no—
180C2×C4×D3—124no—
181C3×C4⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—22no—
184#217—64no—
185#217—62no—
186(C2×D3)⋊C4—64no—
187(C2×D3)⋊C4—64no—
188(C2×D3)⋊C4—64no—
189C4⋊Dic3—64no—
190Dic3⋊C4—64no—
191Dic3⋊C4—64no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32C4×D412yes—
196C22×C32C23×C411yescharacteristic
197C2×C3⋊D3—22no—
198C2×C3⋊D3—22no—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—24no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—22no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—42no—
206C2×C3⋊D3C4×D412yes—
207C2×C3⋊D3C22×D411yescharacteristic
208C2×C3⋊D3C2×#4112yes—
209C2×C3⋊D3C2×#4112yes—
210C2×C3⋊D3C2×#4111yescharacteristic
211#157—22no—
212#157—24no—
213#157—41no—
214#157—41no—
215#157—42no—
216#157#12314yesdirect product factor
217C32×C4—22no—
218C32×C4—21no—
219C32×C4—22no—
220C3⋊Dic3—22no—
221C3⋊Dic3—21no—
222C3⋊Dic3—22no—
223C22×D4—91no—
224C23×C4—91no—
225C23×C4—91no—
226C23×C4—91no—
227C23×C4—91no—
228#123—91no—
229#123—91no—
230#123—92no—
231#123—92no—
232#123—91no—
233#123#15711yescharacteristic, direct product factor
234C4×D4—184no—
235C2×C4⋊C4—92no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—92no—
238C2×C22⋊C4—91no—
239C2×C22⋊C4—92no—
240C2×C22⋊C4—91no—
241C2×C22⋊C4—91no—
242C2×C22⋊C4—91no—
243C2×C22⋊C4—92no—
244C2×C42—91no—
245C2×C42—92no—
246C2×C42—92no—
247C2×C42—92no—
248C2×C42—184no—
249#95—92no—
250#95—92no—
251#95—92no—
252#95—92no—
253#95—92no—
254C23×C3—22no—
255C23×C3—22no—
256C22×D3—62no—
257C22×D3—62no—
258C22×D3—64no—
259C22×D3—64no—
260C22×D3—62no—
261C22×D3—124no—
262C22×D3—124no—
263C22×D3—122no—
264C22×D3—122no—
265C22×D3—124no—
266C22×D3—124no—
267C3×D4—44no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—44no—
272#67—124no—
273#67—124no—
274C2×Dic3—62no—
275C2×Dic3—64no—
276C2×Dic3—64no—
277C2×Dic3—124no—
278D12—124no—
279C4×D3—124no—
280C4×D3—124no—
281C4×D3—124no—
282C4×D3—124no—
283C4×D3—128no—
284C4×D3—122no—
285C4×D3—122no—
286C2×C32—22no—
287C2×C32—41no—
288C2×C32C2×C4×D411yescharacteristic
289C2×C32C4×#4112yes—
290C3⋊D3—22no—
291C3⋊D3—41no—
292C3⋊D3C2×#12311yescharacteristic
293C3⋊D3#52511yescharacteristic
294C3⋊D3#52912yes—
295C24—91no—
296C24—91no—
297C2×D4—91no—
298C2×D4—91no—
299C2×D4—91no—
300C2×D4—182no—
301C2×D4—182no—
302C2×D4C2×#15711yescharacteristic
303C22×C4—91no—
304C22×C4—94no—
305C22×C4—91no—
306C22×C4—92no—
307C22×C4—91no—
308C22×C4—92no—
309C22×C4—92no—
310C22×C4—91no—
311C22×C4—92no—
312C22×C4—92no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—182no—
316C22×C4—182no—
317C22×C4—184no—
318C22×C4—184no—
319C22×C4—182no—
320C22×C4—184no—
321C22×C4—182no—
322C22×C4C2×#15711yescharacteristic, complemented
323C4⋊C4—92no—
324C4⋊C4—92no—
325C4⋊C4—92no—
326C4⋊C4—182no—
327C4⋊C4—184no—
328C4⋊C4C2×#15712yescomplemented
329C22⋊C4—91no—
330C22⋊C4—92no—
331C22⋊C4—92no—
332C22⋊C4—92no—
333C22⋊C4—92no—
334C22⋊C4—182no—
335C22⋊C4—184no—
336C22⋊C4—182no—
337C22⋊C4—182no—
338C22⋊C4C2×#15711yescharacteristic, complemented
339C22⋊C4C2×#15712yescomplemented
340C42—184no—
341C42—184no—
342C42—182no—
343C42—184no—
344C22×C3—24no—
345C22×C3—22no—
346C22×C3—44no—
347C22×C3—42no—
348C22×C3—44no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—128no—
357C2×D3—124no—
358C2×D3—122no—
359C2×D3—124no—
360C2×D3—242no—
361C2×D3—242no—
362C2×D3—244no—
363C3×C4—44no—
364C3×C4—42no—
365C3×C4—44no—
366Dic3—124no—
367Dic3—122no—
368Dic3—124no—
369C32C4×#12311yescharacteristic, complemented, Sylow
370C23—91no—
371C23—92no—
372C23—92no—
373C23—91no—
374C23—91no—
375C23—182no—
376C23—182no—
377C23—181no—
378C23—182no—
379C23—182no—
380C23—181no—
381C23C22×#15711yescharacteristic
382C23C22×#15711yescharacteristic
383D4—22no—
384D4—182no—
385D4—182no—
386D4—182no—
387C2×C4—22no—
388C2×C4—92no—
389C2×C4—94no—
390C2×C4—92no—
391C2×C4—94no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—91no—
395C2×C4—182no—
396C2×C4—181no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4—182no—
400C2×C4—184no—
401C2×C4—181no—
402C2×C4—188no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—188no—
407C2×C4—182no—
408C2×C4—181no—
409C2×C4—182no—
410C2×C4—181no—
411C2×C4—182no—
412C2×C4—182no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—364no—
416C2×C4C22×#15712yes—
417C2×C4C22×#15712yes—
418C2×C4C22×#15711yescharacteristic
419C2×C3—22no—
420C2×C3—24no—
421C2×C3—44no—
422C2×C3—82no—
423D3—62no—
424D3—62no—
425D3—64no—
426D3—124no—
427D3—242no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—91no—
432C22—91no—
433C22—92no—
434C22—92no—
435C22—92no—
436C22—182no—
437C22—182no—
438C22—184no—
439C22—182no—
440C22—181no—
441C22—182no—
442C22—361no—
443C22—361no—
444C22—362no—
445C22D4×#15712yes—
446C22C23×#15711yescharacteristic
447C4—22no—
448C4—21no—
449C4—22no—
450C4—94no—
451C4—182no—
452C4—184no—
453C4—182no—
454C4—181no—
455C4—182no—
456C4—361no—
457C4—361no—
458C4—362no—
459C3—22no—
460C2—22no—
461C2—41no—
462C2—91no—
463C2—91no—
464C2—92no—
465C2—182no—
466C2—361no—
467C2C2×D4×#15711yescharacteristic
468C2#41×#15712yes—
subgroupG/H
center, FrattiniC22C23×#157
commutator, socleC22×C32C23×C4
FittingC32×#123C4
prime radicalC2×D4×C3⋊D3C22
cosocleC22×C3⋊D3C24
hypercenter#123#157
hypocenterC32C4×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×#157C2semidirect
5. #10593811C2semidirect
6. #10593758C2semidirect
7. C4⋊C4×#157C2semidirect
8. #10593747C2semidirect
9. #10593746C2semidirect
10. C22⋊C4×#157C2semidirect
11. C22⋊C4×#157C2semidirect
29. C32×#123C4semidirect
30. #64144C4semidirect
31. #64138C4semidirect
32. #64103C4semidirect
34. #64095C4semidirect
35. #64092C4semidirect
59. C22×C32×C4C2×C4semidirect
62. C22×C3⋊Dic3C2×C4semidirect
68. C2×C4×C3⋊D3C2×C4semidirect
82. C32×C4⋊C4C2×C4semidirect
83. C32×C22⋊C4C2×C4semidirect
84. C32×C22⋊C4C2×C4semidirect
85. #3757C2×C4semidirect
86. #3757C2×C4semidirect
87. #3752C2×C4semidirect
88. #3752C2×C4semidirect
89. #3752C2×C4semidirect
90. C4⋊(C3⋊Dic3)C2×C4semidirect
91. #3750C2×C4semidirect
92. #3750C2×C4semidirect
216. #157#123direct (N × H)
233. #123#157direct (N × H)
322. C22×C4C2×#157semidirect
328. C4⋊C4C2×#157semidirect
338. C22⋊C4C2×#157semidirect
339. C22⋊C4C2×#157semidirect
369. C32C4×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#1572
C22×C4×#1572
C3⋊D3×#1232
#105938262
#105938112
#105937582
C4⋊C4×#1572
#105937472
#105937462
C22⋊C4×#1572
C4×#1239