Finite Group Databasebeta

← C3×#71720 #11774022

Subgroups — C3×#71720 #11774022

Order 1152.

677 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#124C211yesmaximal, characteristic
2C3×#4926C211yesmaximal, characteristic, complemented
3C3×#4925C211yesmaximal, characteristic, complemented
4C3×#4525C214yesmaximal, complemented
5C3×#1469—31nomaximal
6#71720C311yesmaximal, characteristic, direct product factor
7C22×C32×Q8C411yescharacteristic
8C22×C32×D4C411yescharacteristic
9C32×#124—21no—
10C32×#124—21no—
11C32×#124—21no—
12C32×#124—21no—
13C2×C32×C22⋊C4C412yes—
14C2×C32×C22⋊C4C2212yes—
15C2×C32×C42C2211yescharacteristic
16C3×#830C2212yes—
17C3×#830C2212yes—
18C3×C4⋊(C3⋊C8)—21no—
19C3×C4⋊(C3⋊C8)—21no—
20C3×#803—21no—
21C3×C4×C3⋊C8—21no—
22C2×C3×#124—21no—
23C2×C3×#124D311yescharacteristic
24C2×C3×#124C2×C311yescharacteristic
25C3×#432—31no—
26C3×#431—31no—
27#4926C2×C311yescharacteristic, complemented
28#4925C2×C311yescharacteristic, complemented
29C3×#323—34no—
30#4525C2×C314yescomplemented
31C24×C32C2×C412yes—
32C2×C32×Q8—21no—
33C2×C32×Q8—21no—
34C2×C32×Q8—42no—
35C2×C32×D4—21no—
36C2×C32×D4—21no—
37C2×C32×D4—42no—
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4C2×C412yes—
41C22×C32×C4C2311yescharacteristic
42C32×C22⋊C4—22no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—22no—
46C32×C42D411yescharacteristic
47C32×C42D411yescharacteristic
48C32×C42D411yescharacteristic
49C32×C42D411yescharacteristic
50C2×C3×C3⋊C8—22no—
51C2×C3×C3⋊C8—22no—
52#1469—31noSylow
53C22×C3×Q8—21no—
54C22×C3×Q8Dic311yescharacteristic
55C22×C3×Q8C3×C411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4Dic311yescharacteristic
58C22×C3×D4C3×C411yescharacteristic
59C3×#124—22no—
60C3×#124—22no—
61C3×#124—22no—
62C3×#124—22no—
63C3×#124—21no—
64C3×#124—21no—
65C3×#124—21no—
66C3×#124—21no—
67C3×#124—21no—
68C3×#124—21no—
69C3×#124—21no—
70C3×#124—21no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4Dic312yes—
74C2×C3×C22⋊C4C3×C412yes—
75C2×C3×C22⋊C4C2×D312yes—
76C2×C3×C22⋊C4C22×C312yes—
77C2×C3×C42—21no—
78C2×C3×C42C2×D311yescharacteristic
79C2×C3×C42C22×C311yescharacteristic
80C3×C4⋊C8—61no—
81C3×C4⋊C8—61no—
82C3×C22⋊C8—32no—
83C3×C22⋊C8—32no—
84C3×#97—61no—
85C3×C4×C8—61no—
86#830C22×C312yes—
87#830C22×C312yes—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91C4×C3⋊C8—21no—
92C23×C32—22no—
93C23×C32—24no—
94C23×C32—42no—
95C23×C32—42no—
96C23×C32C22×C411yescharacteristic
97C32×Q8—42no—
98C32×Q8—42no—
99C32×D4—42no—
100C32×D4—42no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—41no—
105C2×C32×C4—42no—
106C2×C32×C4C22⋊C411yescharacteristic
107C2×C32×C4C22⋊C411yescharacteristic
108C2×C32×C4C22⋊C411yescharacteristic
109C2×C32×C4C22⋊C411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C3×C3⋊C8—42no—
113C3×C3⋊C8—42no—
114C2×#124C3×D311yescharacteristic
115#432—31no—
116#431—31no—
117#323—34no—
118C24×C3—22no—
119C24×C3C2×Dic312yes—
120C24×C3C2×C3×C412yes—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—44no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—44no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4C2×Dic311yescharacteristic
144C22×C3×C4C2×Dic311yescharacteristic
145C22×C3×C4C2×Dic312yes—
146C22×C3×C4C2×C3×C411yescharacteristic
147C22×C3×C4C2×C3×C411yescharacteristic
148C22×C3×C4C2×C3×C412yes—
149C22×C3×C4C22×D311yescharacteristic
150C22×C3×C4C23×C311yescharacteristic
151C2×C3×C8—62no—
152C2×C3×C8—62no—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C42—21no—
166C3×C42—21no—
167C3×C42—21no—
168C3×C42—21no—
169C3×C42#6711yescharacteristic
170C3×C42#6711yescharacteristic
171C3×C42#6711yescharacteristic
172C3×C42#6711yescharacteristic
173C3×C42C3×D411yescharacteristic
174C3×C42C3×D411yescharacteristic
175C3×C42C3×D411yescharacteristic
176C3×C42C3×D411yescharacteristic
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—84no—
185C22×C32C2×C22⋊C411yescharacteristic
186C22×C32#13112yes—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—42no—
193C32×C4#10012yes—
194C22×Q8C3×Dic311yescharacteristic
195C22×D4C3×Dic311yescharacteristic
196#124—21no—
197#124—21no—
198#124—21no—
199#124—21no—
200C2×C22⋊C4C3×Dic312yes—
201C2×C22⋊C4C2×C3×D312yes—
202C2×C42C2×C3×D311yescharacteristic
203C4⋊C8—61no—
204C4⋊C8—61no—
205C22⋊C8—32no—
206C22⋊C8—32no—
207#97—61no—
208C4×C8—61no—
209C23×C3—24no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—24no—
213C23×C3—24no—
214C23×C3—21no—
215C23×C3—44no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3C22×Dic311yescharacteristic
223C23×C3C22×C3×C411yescharacteristic
224C3×Q8—44no—
225C3×Q8—42no—
226C3×Q8—42no—
227C3×Q8—44no—
228C3×Q8—42no—
229C3×Q8—42no—
230C3×D4—44no—
231C3×D4—42no—
232C3×D4—42no—
233C3×D4—44no—
234C3×D4—42no—
235C3×D4—42no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
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—21no—
249C2×C3×C4—21no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4#21711yescharacteristic
258C2×C3×C4#21711yescharacteristic
259C2×C3×C4#21711yescharacteristic
260C2×C3×C4#21711yescharacteristic
261C2×C3×C4C3×C22⋊C411yescharacteristic
262C2×C3×C4C3×C22⋊C411yescharacteristic
263C2×C3×C4C3×C22⋊C411yescharacteristic
264C2×C3×C4C3×C22⋊C411yescharacteristic
265C2×C3×C4C2×#6711yescharacteristic
266C2×C3×C4C2×#6711yescharacteristic
267C2×C3×C4C2×C3×D411yescharacteristic
268C2×C3×C4C2×C3×D411yescharacteristic
269C3×C8—122no—
270C3×C8—122no—
271C3⋊C8—42no—
272C3⋊C8—42no—
273C2×C32—22no—
274C2×C32—82no—
275C2×C32#40811yescharacteristic
276C2×C32#41011yescharacteristic
277C2×C32C2×#10011yescharacteristic
278C24C2×C3×Dic312yes—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—42no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C22×C4C2×C3×Dic311yescharacteristic
286C22×C4C2×C3×Dic311yescharacteristic
287C22×C4C2×C3×Dic312yes—
288C22×C4C22×C3×D311yescharacteristic
289C2×C8—62no—
290C2×C8—62no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C42C3×#6711yescharacteristic
296C42C3×#6711yescharacteristic
297C42C3×#6711yescharacteristic
298C42C3×#6711yescharacteristic
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—22no—
306C22×C3—21no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—84no—
317C22×C3—84no—
318C22×C3—88no—
319C22×C3#94812yes—
320C22×C3C2×#21711yescharacteristic
321C22×C3C2×C3×C22⋊C411yescharacteristic
322C22×C3C3×#13112yes—
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—44no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4#83312yes—
343C3×C4C3×#10012yes—
344C32#146911yescharacteristic, complemented, Sylow
345C23—22no—
346C23—24no—
347C23—42no—
348C23—42no—
349C23C22×C3×Dic311yescharacteristic
350Q8—42no—
351Q8—42no—
352D4—42no—
353D4—42no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4C3×#21711yescharacteristic
360C2×C4C3×#21711yescharacteristic
361C2×C4C3×#21711yescharacteristic
362C2×C4C3×#21711yescharacteristic
363C2×C4C2×C3×#6711yescharacteristic
364C2×C4C2×C3×#6711yescharacteristic
365C8—122no—
366C8—122no—
367C2×C3—24no—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—84no—
374C2×C3—82no—
375C2×C3—82no—
376C2×C3C2×#83311yescharacteristic
377C2×C3#523411yescharacteristic
378C2×C3#524011yescharacteristic
379C2×C3C3×#40811yescharacteristic
380C2×C3C3×#41011yescharacteristic
381C2×C3C2×C3×#10011yescharacteristic
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22C3×#94812yes—
389C22C2×C3×#21711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4C3×#83312yes—
397C3—21no—
398C3#7172011yescharacteristic, direct product factor
399C3C3×#146911yescharacteristic, complemented
400C2—22no—
401C2—82no—
402C2C2×C3×#83311yescharacteristic
403C2C3×#523411yescharacteristic
404C2C3×#524011yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#124C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C32×D4C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#124D3
hypocenterC3C3×#1469
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
2. C3×#4926C2semidirect
3. C3×#4925C2semidirect
4. C3×#4525C2semidirect
6. #71720C3direct (N × H)
27. #4926C2×C3semidirect
28. #4925C2×C3semidirect
30. #4525C2×C3semidirect
344. C32#1469semidirect
398. C3#71720direct (N × H)
399. C3C3×#1469semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1242
C3×#49262
C3×#49252
C3×#45252
C3×#14693
#717203