Finite Group Databasebeta

← #11787154

Subgroups — #11787154

Order 1152.

1077 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C22×#3710C211yesmaximal, characteristic, complemented
3C2×#63557C211yesmaximal, characteristic, complemented
4C2×#63557C212yesmaximal, complemented
5C3×#4524C212yesmaximal, complemented
6#68668—32nomaximal
7C22×C4⋊(C3⋊D3)C411yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9C22×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C42C2211yescharacteristic
11C2×#3752C411yescharacteristic
12C2×#3752C411yescharacteristic
13C2×C4⋊(C3⋊Dic3)C411yescharacteristic
14C4×C4⋊(C3⋊D3)—22no—
15C4×C4⋊(C3⋊D3)—22no—
16C22×C3×C3⋊C8C2212yes—
17C22×C3×C3⋊C8C2212yes—
18C2×#3710—22no—
19C2×#3710—24no—
20#63557—22no—
21#63557—24no—
22C22×D3×C8—32no—
23C2×C4×D12—32no—
24C2×C4×D12—61no—
25C2×(C2×D3)⋊C8—32no—
26C2×(C2×D3)⋊C8—32no—
27C2×(C2×D3)⋊C8—32no—
28C3×#336—32no—
29#4524D312yescomplemented
30C23×C3⋊D3C2×C411yescharacteristic
31C23×C3⋊D3C2×C411yescharacteristic
32C22×C32×C4C2×C411yescharacteristic
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3⋊Dic3C2×C411yescharacteristic
36C22×C3⋊Dic3C2×C411yescharacteristic
37C2×C4⋊(C3⋊D3)—22no—
38C2×C4⋊(C3⋊D3)—22no—
39C2×C4⋊(C3⋊D3)C814yescomplemented
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—22no—
42C2×C4×C3⋊D3—21no—
43C2×C4×C3⋊D3—21no—
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3D412yes—
47C2×C4×C3⋊D3D412yes—
48C32×C42—21no—
49C32×C42—21no—
50#3752—22no—
51#3752—22no—
52C4⋊(C3⋊Dic3)—22no—
53C2×C3×C3⋊C8—22no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—22no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8D414yescomplemented
59#3710—28no—
60#1869—91noSylow
61C22×D12—32no—
62C22×D12—61no—
63C22×C4×D3—32no—
64C22×C4×D3—32no—
65C22×C4×D3—61no—
66C22×C4×D3—61no—
67C22×C3×C8—32no—
68C22×C3×C8—32no—
69C2×C3×C42—21no—
70C2×C3×C42C2×D312yes—
71C2×(C2×D3)⋊C4—32no—
72C2×(C2×D3)⋊C4—32no—
73C2×(C2×D3)⋊C4—61no—
74C2×(C2×D3)⋊C4—61no—
75C2×C4⋊Dic3—32no—
76C2×C4⋊Dic3—61no—
77C22×C3⋊C8C2×D312yes—
78C22×C3⋊C8C2×D312yes—
79C2×D3×C8—64no—
80C2×D3×C8—64no—
81C2×D3×C8—64no—
82C4×D12—64no—
83C4×D12—64no—
84C4×D12—64no—
85C4×D12—64no—
86(C2×D3)⋊C8—64no—
87(C2×D3)⋊C8—64no—
88(C2×D3)⋊C8—64no—
89C23×C32C22×C411yescharacteristic
90C22×C3⋊D3—21no—
91C22×C3⋊D3—21no—
92C22×C3⋊D3—22no—
93C22×C3⋊D3—22no—
94C22×C3⋊D3—22no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3C22⋊C412yes—
97C22×C3⋊D3C2×C812yes—
98C22×C3⋊D3C2×C812yescomplemented
99C22×C3⋊D3M1612yes—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4C2×C812yescomplemented
105C2×C32×C4C2×D412yes—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4#4112yes—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3C22⋊C412yes—
113C4⋊(C3⋊D3)—28no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3—24no—
116C4×C3⋊D3—22no—
117C4×C3⋊D3—41no—
118C4×C3⋊D3—42no—
119C4×C3⋊D3—41no—
120C3×C3⋊C8—24no—
121C3×C3⋊C8—42no—
122C23×C8—91no—
123C2×C4×D4—91no—
124C2×C22⋊C8—92no—
125C2×C22⋊C8—91no—
126#336—32no—
127C23×D3—32no—
128C23×D3—32no—
129C23×D3—61no—
130C23×D3—61no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C4×D312yes—
135C22×C3×C4C4×D312yes—
136C22×C3×C4C22×D312yes—
137C22×Dic3—32no—
138C22×Dic3—32no—
139C22×Dic3—61no—
140C22×Dic3—61no—
141C2×D12—38no—
142C2×D12—64no—
143C2×D12—64no—
144C2×D12—64no—
145C2×D12—64no—
146C2×D12—64no—
147C2×C4×D3—34no—
148C2×C4×D3—34no—
149C2×C4×D3—62no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×C4×D3—62no—
153C2×C4×D3—62no—
154C2×C4×D3—62no—
155C2×C4×D3—62no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—62no—
161C2×C4×D3—62no—
162C2×C4×D3—62no—
163C2×C4×D3—122no—
164C2×C4×D3—122no—
165C2×C4×D3—122no—
166C2×C4×D3—122no—
167C2×C3×C8—34no—
168C2×C3×C8—62no—
169C2×C3×C8—62no—
170C2×C3×C8—62no—
171C2×C3×C8—62no—
172C2×C3×C8—62no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—22no—
177(C2×D3)⋊C4—64no—
178(C2×D3)⋊C4—64no—
179(C2×D3)⋊C4—64no—
180(C2×D3)⋊C4—64no—
181C4⋊Dic3—64no—
182C4⋊Dic3—64no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8D1214yescomplemented
189D3×C8—616no—
190C22×C32C2×C22⋊C411yescharacteristic
191C22×C32C4×D412yes—
192C22×C32C22×C811yescharacteristic
193C22×C32C2×M1611yescharacteristic
194C22×C32#13112yes—
195C2×C3⋊D3—22no—
196C2×C3⋊D3—22no—
197C2×C3⋊D3—22no—
198C2×C3⋊D3—24no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—24no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—42no—
204C2×C3⋊D3C22⋊C814yescomplemented
205C32×C4—22no—
206C32×C4—41no—
207C32×C4#12011yescharacteristic
208C32×C4#12112yes—
209C32×C4#12311yescharacteristic
210C3⋊Dic3—22no—
211C3⋊Dic3—41no—
212C22×D4—91no—
213C23×C4—91no—
214C23×C4—91no—
215C22×C8—32no—
216C22×C8—32no—
217C22×C8—184no—
218C22×C8—182no—
219C4×D4—182no—
220C4×D4—182no—
221C2×C4⋊C4—91no—
222C2×C22⋊C4—91no—
223C2×C22⋊C4—91no—
224C2×C42D3211yescharacteristic
225C22⋊C8—184no—
226C22⋊C8—182no—
227C23×C3—21no—
228C23×C3C2×C4×D312yes—
229C22×D3—34no—
230C22×D3—34no—
231C22×D3—34no—
232C22×D3—34no—
233C22×D3—62no—
234C22×D3—62no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—122no—
250C22×D3—122no—
251C22×D3—122no—
252C22×D3—122no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—42no—
267C2×C3×C4D3×C814yescomplemented
268C2×C3×C4C2×D1212yes—
269C2×C3×C4#23512yes—
270C2×C3×C4D3×D412yes—
271C2×C3×C4D3×D412yes—
272C2×C3×C4D3×D412yes—
273C2×C3×C4#23912yes—
274C2×Dic3—34no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—122no—
283C2×Dic3—122no—
284D12—616no—
285D12—616no—
286C4×D3—64no—
287C4×D3—64no—
288C4×D3—64no—
289C4×D3—64no—
290C4×D3—122no—
291C4×D3—124no—
292C4×D3—122no—
293C4×D3—122no—
294C4×D3—124no—
295C4×D3—122no—
296C4×D3—122no—
297C4×D3—122no—
298C4×D3—122no—
299C4×D3—122no—
300C3×C8—64no—
301C3×C8—122no—
302C3⋊C8—24no—
303C3⋊C8—42no—
304C2×C32#38611yescharacteristic
305C2×C32C2×C22⋊C811yescharacteristic
306C2×C32#40811yescharacteristic
307C2×C32C8×D412yes—
308C2×C32#43512yes—
309C3⋊D3—24no—
310C3⋊D3—42no—
311C24—91no—
312C24—91no—
313C2×D4—94no—
314C2×D4—182no—
315C2×D4—182no—
316C22×C4—91no—
317C22×C4—92no—
318C22×C4—92no—
319C22×C4—91no—
320C22×C4—182no—
321C22×C4—182no—
322C22×C4—182no—
323C22×C4—182no—
324C22×C4—181no—
325C22×C4—181no—
326C22×C4#62411yescharacteristic
327C22×C4#62411yescharacteristic
328C22×C4C2×D3211yescharacteristic
329C2×C8—34no—
330C2×C8—62no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—188no—
336C4⋊C4—182no—
337C22⋊C4—182no—
338C22⋊C4—182no—
339C42—21no—
340C42—21no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3C4×D1212yes—
347C22×C3D3×C22⋊C412yes—
348C22×C3#89312yes—
349C22×C3C2×D3×C812yes—
350C22×C3#90112yes—
351C22×C3D3×M1612yes—
352C22×C3#90712yes—
353C2×D3—38no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—64no—
357C2×D3—64no—
358C2×D3—64no—
359C2×D3—64no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—64no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—122no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—122no—
369C2×D3—124no—
370C2×D3—122no—
371C2×D3—124no—
372C2×D3—122no—
373C2×D3—122no—
374C2×D3—122no—
375C2×D3—122no—
376C2×D3—122no—
377C3×C4—24no—
378C3×C4—24no—
379C3×C4—21no—
380C3×C4—22no—
381C3×C4—21no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4#88212yes—
385C3×C4#88412yes—
386C3×C4#89412yes—
387C3×C4#89512yes—
388Dic3—64no—
389Dic3—122no—
390Dic3—122no—
391Dic3—122no—
392C32#186911yescharacteristic, complemented, Sylow
393C23—92no—
394C23—92no—
395C23—92no—
396C23—92no—
397C23—182no—
398C23—182no—
399C23—182no—
400C23—181no—
401C23—182no—
402C23—181no—
403C23C2×#62411yescharacteristic
404D4—188no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—92no—
410C2×C4—182no—
411C2×C4—182no—
412C2×C4—184no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—181no—
416C2×C4—361no—
417C2×C4—362no—
418C2×C4—361no—
419C2×C4#371012yescomplemented
420C2×C4#379912yes—
421C2×C4D3×D1212yes—
422C2×C4#381111yescharacteristic
423C2×C4#381111yescharacteristic
424C8—64no—
425C8—122no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3#466512yes—
432C2×C3C8×D1212yes—
433C2×C3#470612yes—
434C2×C3D3×C22⋊C812yes—
435C2×C3#472712yes—
436C2×C3#483412yes—
437C2×C3#483612yes—
438D3—68no—
439D3—124no—
440D3—124no—
441D3—124no—
442C22—94no—
443C22—184no—
444C22—184no—
445C22—182no—
446C22—182no—
447C22—184no—
448C22—182no—
449C22—362no—
450C22—361no—
451C22—361no—
452C22C2×#371011yescharacteristic
453C22#6381712yes—
454C22#6381811yescharacteristic
455C22#6389412yes—
456C22#6397411yescharacteristic
457C4—22no—
458C4—41no—
459C4—182no—
460C4—361no—
461C4#6389711yescharacteristic
462C4#6391012yes—
463C4#6392311yescharacteristic
464C3—21no—
465C3#6866812yescomplemented
466C2—184no—
467C2—362no—
468C2#1059032712yes—
469C2#1059032812yes—
470C2#1059115211yescharacteristic
471C2#1059115311yescharacteristic
472C2#1059138011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutatorC22×C32C22×C8
FittingC2×C32×C42C22
socleC23×C32C22×C4
prime radicalC22×C4⋊(C3⋊D3)C4
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1869
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C22×#3710C2semidirect
3. C2×#63557C2semidirect
4. C2×#63557C2semidirect
5. C3×#4524C2semidirect
29. #4524D3semidirect
39. C2×C4⋊(C3⋊D3)C8semidirect
58. C2×C3×C3⋊C8D4semidirect
98. C22×C3⋊D3C2×C8semidirect
104. C2×C32×C4C2×C8semidirect
188. C2×C3⋊C8D12semidirect
204. C2×C3⋊D3C22⋊C8semidirect
267. C2×C3×C4D3×C8semidirect
392. C32#1869semidirect
419. C2×C4#3710semidirect
465. C3#68668semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊(C3⋊D3)2
C22×#37102
C2×#635572
C3×#45242
#686683