Finite Group Databasebeta

← #11775612

Subgroups — #11775612

Order 1152.

677 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#924C211yesmaximal, characteristic
2C2×#63657C211yesmaximal, characteristic
3C3×#5212C211yesmaximal, characteristic
4#10589153C214yesmaximal, complemented
5#71690—31nomaximal
6#69510—31nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×C3×Dic6C411yescharacteristic, complemented
9C22×C3⋊(C3⋊C8)—21no—
10C2×C3×#217—21no—
11C2×C3×#217—21no—
12C2×C3×C4⋊Dic3—21no—
13C2×C3×C4⋊Dic3C411yescharacteristic, complemented
14C3×#924—21no—
15C3×#924—21no—
16C3×#924C412yescomplemented
17C3×#924C2212yes—
18C2×C3×Dic3⋊C4—21no—
19#63657—21no—
20#63657C2212yes—
21C3×C3⋊#95C2212yes—
22C2×C3×C22⋊Q8—31no—
23C2×#924D311yescharacteristic
24C3×#380—31no—
25#5212D311yescharacteristic
26C2×#830—31no—
27C2×#830—31no—
28C2×#830—61no—
29#4538—34no—
30#4452—34no—
31C24×C32D411yescharacteristic
32C22×C32×C4—42no—
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4D411yescharacteristic
37C22×C32×C4D411yescharacteristic
38C22×C32×C4C2311yescharacteristic
39C22×C3×Dic3—21no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—22no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—42no—
44C2×C3×Dic6C2×C412yes—
45C2×C3⋊(C3⋊C8)—22no—
46C2×C3⋊(C3⋊C8)—42no—
47C3×#217—22no—
48C3×#217—22no—
49C3×#217—42no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—42no—
52C3×C4⋊Dic3C2×C412yes—
53C3×Dic3⋊C4—22no—
54C3×Dic3⋊C4—22no—
55#1520—91noSylow
56C22×C3×Q8—31no—
57C23×C3×C4—21no—
58C23×C3×C4C2×D311yescharacteristic
59C23×C3×C4C2×D311yescharacteristic
60C22×Dic6Dic311yescharacteristic, complemented
61C3×C22⋊Q8—32no—
62C3×C22⋊Q8—32no—
63C3×C22⋊Q8—61no—
64C3×C22⋊Q8—61no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—61no—
67C2×C3×C4⋊C4—61no—
68C2×C3×C22⋊C4—61no—
69C2×C3×C22⋊C4—61no—
70C2×#217—21no—
71C2×#217—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3Dic311yescharacteristic, complemented
74#924—21no—
75#924—21no—
76#924Dic312yescomplemented
77#924C2×D312yes—
78C2×Dic3⋊C4—21no—
79C22×C3⋊C8—62no—
80C22×C3⋊C8—61no—
81C22×C3⋊C8—61no—
82C3×#95—32no—
83C3⋊#95C2×D312yes—
84#830—32no—
85#830—32no—
86#830—62no—
87#830—62no—
88#830—61no—
89#830—61no—
90C23×C32—41no—
91C23×C32—41no—
92C23×C32C22⋊C411yescharacteristic
93C23×C32C22⋊C411yescharacteristic
94C23×C32SD1612yes—
95C23×C32Q1612yes—
96C23×C32C2×D411yescharacteristic
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4C22⋊C411yescharacteristic
104C2×C32×C4C22⋊C411yescharacteristic
105C2×C32×C4C22×C411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—42no—
110C2×C3×Dic3—42no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3—42no—
114C3×Dic6—42no—
115C3×Dic6—42no—
116C3⋊(C3⋊C8)—42no—
117C2×C22⋊Q8—31no—
118C2×C22⋊C8—91no—
119#380—31no—
120#328—94no—
121C24×C3—21no—
122C24×C3#6711yescharacteristic
123C24×C3#6711yescharacteristic
124C2×C3×Q8—32no—
125C2×C3×Q8—61no—
126C2×C3×Q8—122no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—44no—
136C22×C3×C4—61no—
137C22×C3×C4—61no—
138C22×C3×C4—62no—
139C22×C3×C4C4×D311yescharacteristic
140C22×C3×C4C4×D311yescharacteristic
141C22×C3×C4D1211yescharacteristic
142C22×C3×C4D1211yescharacteristic
143C22×C3×C4C2×Dic311yescharacteristic
144C22×C3×C4C2×Dic311yescharacteristic
145C22×C3×C4#6711yescharacteristic
146C22×C3×C4#6711yescharacteristic
147C22×C3×C4#6711yescharacteristic
148C22×C3×C4#6711yescharacteristic
149C22×C3×C4C22×D311yescharacteristic
150C22×C3×C4C22×D311yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C2×Dic6C2×Dic312yes—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—62no—
159C3×C4⋊C4—61no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—122no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—122no—
165#217—22no—
166#217—22no—
167#217—42no—
168C4⋊Dic3—21no—
169C4⋊Dic3—42no—
170C4⋊Dic3C2×Dic312yes—
171Dic3⋊C4—22no—
172Dic3⋊C4—22no—
173C2×C3⋊C8—64no—
174C2×C3⋊C8—62no—
175C2×C3⋊C8—62no—
176C2×C3⋊C8—124no—
177C2×C3⋊C8—122no—
178C2×C3⋊C8—122no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—42no—
186C22×C32—42no—
187C22×C32#10312yes—
188C22×C32#10312yes—
189C22×C32C2×C22⋊C411yescharacteristic
190C22×C32C2×SD1611yescharacteristic
191C22×C32C2×Q1611yescharacteristic
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4#9912yes—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C3×Dic3—82no—
199C22×Q8—31no—
200C23×C4D3211yescharacteristic
201C22×C8—181no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—61no—
205C22⋊Q8—61no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C22⋊C8—92no—
212C22⋊C8—181no—
213#95—32no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3(C2×D3)⋊C411yescharacteristic
226C23×C3(C2×D3)⋊C411yescharacteristic
227C23×C3#21412yes—
228C23×C3#21512yes—
229C23×C3#21612yes—
230C23×C3#21612yes—
231C23×C3#21711yescharacteristic
232C23×C3#21711yescharacteristic
233C23×C3C2×#6711yescharacteristic
234C23×C3C2×#6711yescharacteristic
235C3×Q8—122no—
236C3×Q8—122no—
237C2×C3×C4—24no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—122no—
262C2×C3×C4—122no—
263C2×C3×C4—122no—
264C2×C3×C4—122no—
265C2×C3×C4—122no—
266C2×C3×C4(C2×D3)⋊C411yescharacteristic
267C2×C3×C4(C2×D3)⋊C411yescharacteristic
268C2×C3×C4#21711yescharacteristic
269C2×C3×C4#21711yescharacteristic
270C2×C3×C4C2×C4×D311yescharacteristic
271C2×C3×C4C2×D1211yescharacteristic
272C2×C3×C4C22×Dic311yescharacteristic
273C2×C3×C4C2×#6711yescharacteristic
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281Dic6—42no—
282Dic6—42no—
283C3⋊C8—124no—
284C3⋊C8—122no—
285C3⋊C8—122no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32C2×#9911yescharacteristic
291C2×C32C2×#10311yescharacteristic
292C2×C32#42111yescharacteristic
293C24#62511yescharacteristic
294C2×Q8—32no—
295C2×Q8—61no—
296C2×Q8—122no—
297C22×C4—42no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—62no—
301C22×C4D3×Dic311yescharacteristic
302C22×C4D3×Dic311yescharacteristic
303C22×C4#62511yescharacteristic
304C22×C4#62611yescharacteristic
305C22×C4#62611yescharacteristic
306C22×C4C2×D3211yescharacteristic
307C2×C8—182no—
308C2×C8—362no—
309C4⋊C4—32no—
310C4⋊C4—61no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—122no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—122no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—44no—
345C22×C3—44no—
346C22×C3#81012yes—
347C22×C3#81012yes—
348C22×C3#83512yes—
349C22×C3#83512yes—
350C22×C3C2×(C2×D3)⋊C411yescharacteristic
351C22×C3C2×#21411yescharacteristic
352C22×C3C2×#21511yescharacteristic
353C22×C3C2×#21611yescharacteristic
354C22×C3C2×#21611yescharacteristic
355C22×C3C2×#21711yescharacteristic
356C3×C4—22no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—22no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—122no—
367C3×C4—122no—
368C3×C4—242no—
369C3×C4#80612yes—
370C3×C4#83412yes—
371Dic3—42no—
372Dic3—42no—
373Dic3—82no—
374C32#152011yescharacteristic, complemented, Sylow
375C23—41no—
376C23—41no—
377C23#371512yes—
378C23#371812yes—
379C23(C2×D3)⋊Dic311yescharacteristic
380C23(C2×D3)⋊Dic311yescharacteristic
381C23C2×#62511yescharacteristic
382Q8—122no—
383Q8—122no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4(C2×D3)⋊Dic311yescharacteristic
398C2×C4(C2×D3)⋊Dic311yescharacteristic
399C2×C4C2×D3×Dic311yescharacteristic
400C2×C4C2×#62611yescharacteristic
401C8—362no—
402C2×C3—24no—
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—22no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3C2×#80611yescharacteristic
418C2×C3C2×#81011yescharacteristic
419C2×C3#500511yescharacteristic
420C2×C3C2×#83411yescharacteristic
421C2×C3C2×#83511yescharacteristic
422C2×C3#523611yescharacteristic
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—42no—
431C22#6356512yes—
432C22#6356512yes—
433C22C2×#371511yescharacteristic
434C22C2×#371811yescharacteristic
435C22C2×(C2×D3)⋊Dic311yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—122no—
440C4—122no—
441C4—242no—
442C4#6358012yes—
443C3—21no—
444C3#6951011yescharacteristic, complemented
445C3#7169011yescharacteristic, complemented
446C2—21no—
447C2—22no—
448C2—22no—
449C2—21no—
450C2C2×#6356511yescharacteristic
451C2#1059111111yescharacteristic
452C2C2×#6358011yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊Dic3
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC23×C32×C4C22
socleC22×C32C2×C22⋊C4
prime radicalC24×C32D4
cosocleC22×C32×C4C23
hypercenterC23×C4D32
hypocenterC32#1520
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
4. #10589153C2semidirect
8. C22×C3×Dic6C4semidirect
13. C2×C3×C4⋊Dic3C4semidirect
16. C3×#924C4semidirect
60. C22×Dic6Dic3semidirect
73. C2×C4⋊Dic3Dic3semidirect
76. #924Dic3semidirect
374. C32#1520semidirect
444. C3#69510semidirect
445. C3#71690semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
C2×#636572
C3×#52122
#105891532
#716903
#695103