Finite Group Databasebeta

← C3×#68224 #11777784

Subgroups — C3×#68224 #11777784

Order 1152.

367 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#215C211yesmaximal, characteristic
2C3×#5024C211yesmaximal, characteristic, complemented
3C3×#4838C211yesmaximal, characteristic
4C32×Q8⋊C8C211yesmaximal, characteristic, complemented
5C3×#4480C211yesmaximal, characteristic, complemented
6C3×#4459C211yesmaximal, characteristic
7C3×C8×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#1590—31nomaximal
9#68224C311yesmaximal, characteristic, direct product factor
10C32×C4×Q8C2211yescharacteristic
11C2×C3×#215C411yescharacteristic
12C2×C3×ΓL(1,𝔽25)—21no—
13C3×C4×D12C2211yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C32×C4×C8C2211yescharacteristic
16C3×#835C411yescharacteristic
17C3×(C2×D3)⋊C8—21no—
18C3×#809C411yescharacteristic
19C3×#808C411yescharacteristic
20C3×C4⋊(C3⋊C8)—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#803—21no—
23C3×C4×C3⋊C8C2211yescharacteristic
24C3×C4×C3⋊C8C2211yescharacteristic
25C3×#446—31no—
26C3×C4×SD16—31no—
27C3×#436—31no—
28C4×#215C2×C311yescharacteristic
29#5024C2×C311yescharacteristic, complemented
30#4838C2×C311yescharacteristic
31C3×#334—31no—
32C3×Q8⋊C8—21no—
33C3×Q8⋊C8D311yescharacteristic, complemented
34C3×Q8⋊C8C2×C311yescharacteristic, complemented
35C3×D4⋊C8—31no—
36C3×C82—31no—
37#4480C2×C311yescharacteristic, complemented
38#4459C2×C311yescharacteristic
39C8×C3⋊C8C2×C311yescharacteristic, complemented
40C2×C32×Q8C2×C411yescharacteristic
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C32×C8—21no—
44C2×C32×C8—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4C2×C411yescharacteristic
47C32×C42—21no—
48C32×C42C2311yescharacteristic
49C3×#215—22no—
50C3×(C2×D3)⋊C4—21no—
51C3×C4⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8C2×C411yescharacteristic
54C2×C3×C3⋊C8C2×C411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57C3×ΓL(1,𝔽25)—42no—
58#1590—31noSylow
59C2×C3×SD16—31no—
60C2×C3×M16—61no—
61C3×C4×Q8—21no—
62C3×C4×Q8C2×D311yescharacteristic
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×D4—31no—
65C2×#215C3×C411yescharacteristic
66C2×ΓL(1,𝔽25)—21no—
67C4×D12C22×C311yescharacteristic
68C3×#106—31no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—61no—
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4⋊C8C22×C311yescharacteristic
74C3×#103—31no—
75C3×#102—31no—
76C3×C22⋊C8—61no—
77C3×#97—61no—
78C3×C4×C8—21no—
79C3×C4×C8—31no—
80C3×C4×C8—31no—
81C3×C4×C8C2×D311yescharacteristic
82C3×C4×C8C22×C311yescharacteristic
83#835C3×C411yescharacteristic
84(C2×D3)⋊C8—21no—
85#809C3×C411yescharacteristic
86#808C3×C411yescharacteristic
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)C22×C311yescharacteristic
89#803—21no—
90C4×C3⋊C8C22×C311yescharacteristic
91C4×C3⋊C8C22×C311yescharacteristic
92C22×C3×D3—21no—
93C32×Q8—21no—
94C32×Q8—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4C22×C411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3×Dic3—21no—
101C3×D12—21no—
102C3×D12—21no—
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105C32×C8—22no—
106C32×C8—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8M1612yescomplemented
111C3×C3⋊C8SD1612yescomplemented
112#446—31no—
113C4×SD16—31no—
114#436—31no—
115#334—31no—
116Q8⋊C8C3×D311yescharacteristic, complemented
117D4⋊C8—31no—
118C82—31no—
119C2×C3×Q8—21no—
120C2×C3×Q8C4×D311yescharacteristic
121C2×C3×Q8C2×C3×C411yescharacteristic
122C2×C3×D4—31no—
123C22×C3×C4—61no—
124C2×D12C2×C3×C411yescharacteristic
125C2×C4×D3—21no—
126C3×SD16—62no—
127C3×M16—122no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—61no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4C4×D311yescharacteristic
145C3×C4⋊C4C2×C3×C411yescharacteristic
146C3×C22⋊C4—61no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—41no—
151C3×C42C22×D311yescharacteristic
152C3×C42C23×C311yescharacteristic
153#215—22no—
154(C2×D3)⋊C4—21no—
155C4⋊Dic3C2×C3×C411yescharacteristic
156C2×C3⋊C8—21no—
157C2×C3⋊C8C2×C3×C411yescharacteristic
158C2×C3⋊C8C2×C3×C411yescharacteristic
159C2×C3⋊C8C3×D411yescharacteristic
160C2×C3⋊C8C3×D411yescharacteristic
161ΓL(1,𝔽25)—42no—
162C22×C32C4×D411yescharacteristic
163C2×C3×D3—41no—
164C2×C3×D3—41no—
165C2×C3×D3—41no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—41no—
169C32×C4C2×M1611yescharacteristic
170C32×C4#13111yescharacteristic
171C32×C4C2×SD1611yescharacteristic
172C32×C4#13511yescharacteristic
173C3×Dic3—41no—
174C2×SD16—31no—
175C2×M16—61no—
176C4×Q8C2×C3×D311yescharacteristic
177C4×D4—31no—
178#106—31no—
179C4⋊C8—31no—
180C4⋊C8—61no—
181C4⋊C8C2×C3×D311yescharacteristic
182#103—31no—
183#102—31no—
184C22⋊C8—61no—
185#97—61no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8C2×C3×D311yescharacteristic
189C23×C3—61no—
190C22×D3—21no—
191C3×Q8—22no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—41no—
197C3×D4—61no—
198C3×D4—61no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—61no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4C2×C4×D311yescharacteristic
212C2×C3×C4D3×D411yescharacteristic
213C2×C3×C4#23711yescharacteristic
214C2×C3×C4C22×C3×C411yescharacteristic
215C2×C3×C4C2×C3×D411yescharacteristic
216C2×C3×C4C3×#4111yescharacteristic
217C2×Dic3—21no—
218D12—21no—
219D12—21no—
220C4×D3—41no—
221C4×D3—41no—
222C3×C8—22no—
223C3×C8—22no—
224C3×C8—32no—
225C3×C8—32no—
226C3×C8—42no—
227C3×C8—41no—
228C3×C8—41no—
229C3×C8—61no—
230C3×C8—61no—
231C3×C8—81no—
232C3×C8—121no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—41no—
236C3⋊C8C3×M1612yescomplemented
237C3⋊C8C3×SD1612yescomplemented
238C2×C32#43611yescharacteristic
239C2×C32C4×SD1611yescharacteristic
240C2×C32#44311yescharacteristic
241C3×D3—81no—
242C2×Q8C3×C4×D311yescharacteristic
243C2×D4—31no—
244C22×C4—61no—
245SD16—62no—
246M16—122no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—61no—
254C4⋊C4—21no—
255C4⋊C4—31no—
256C4⋊C4C3×C4×D311yescharacteristic
257C22⋊C4—61no—
258C42—21no—
259C42C22×C3×D311yescharacteristic
260C22×C3—21no—
261C22×C3—121no—
262C22×C3—121no—
263C22×C3—121no—
264C22×C3#67⋊C411yescharacteristic
265C22×C3C3×C4×D411yescharacteristic
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—42no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—81no—
282C3×C4—121no—
283C3×C4#90111yescharacteristic
284C3×C4D3×M1611yescharacteristic
285C3×C4D3×SD1611yescharacteristic
286C3×C4#91911yescharacteristic
287C3×C4C2×C3×M1611yescharacteristic
288C3×C4C3×#13111yescharacteristic
289C3×C4C2×C3×SD1611yescharacteristic
290C3×C4C3×#13511yescharacteristic
291Dic3—41no—
292C32#159011yescharacteristic, complemented, Sylow
293C23—61no—
294Q8—21no—
295Q8—21no—
296D4—61no—
297D4—61no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—61no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4C2×C3×C4×D311yescharacteristic
304C2×C4C3×D3×D411yescharacteristic
305C2×C4C3×#23711yescharacteristic
306C8—22no—
307C8—32no—
308C8—32no—
309C8—41no—
310C8—61no—
311C8—61no—
312C8—121no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—241no—
317C2×C3#472911yescharacteristic
318C2×C3#478811yescharacteristic
319C2×C3#482411yescharacteristic
320C2×C3C3×#43611yescharacteristic
321C2×C3C3×C4×SD1611yescharacteristic
322C2×C3C3×#44311yescharacteristic
323D3—81no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22C3×#67⋊C411yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—121no—
332C4C3×#90111yescharacteristic
333C4C3×D3×M1611yescharacteristic
334C4C3×D3×SD1611yescharacteristic
335C4C3×#91911yescharacteristic
336C3—21no—
337C3#6822411yescharacteristic, direct product factor
338C3C3×#159011yescharacteristic, complemented
339C2—241no—
340C2C3×#472911yescharacteristic
341C2C3×#478811yescharacteristic
342C2C3×#482411yescharacteristic
subgroupG/H
centerC2×C3×C4D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC42C22×C3×D3
FittingC32×Q8⋊C8C2
socleC22×C32C4×D4
prime radicalC2×C3×D12C2×C4
cosocleC3×C42C23×C3
hypercenterC3×Q8⋊C8D3
hypocenterC3C3×#1590
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#5024C2semidirect
4. C32×Q8⋊C8C2semidirect
5. C3×#4480C2semidirect
7. C3×C8×C3⋊C8C2semidirect
9. #68224C3direct (N × H)
29. #5024C2×C3semidirect
33. C3×Q8⋊C8D3semidirect
34. C3×Q8⋊C8C2×C3semidirect
37. #4480C2×C3semidirect
39. C8×C3⋊C8C2×C3semidirect
110. C3×C3⋊C8M16semidirect
111. C3×C3⋊C8SD16semidirect
116. Q8⋊C8C3×D3semidirect
236. C3⋊C8C3×M16semidirect
237. C3⋊C8C3×SD16semidirect
292. C32#1590semidirect
337. C3#68224direct (N × H)
338. C3C3×#1590semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#2152
C3×#50242
C3×#48382
C32×Q8⋊C82
C3×#44802
C3×#44592
C3×C8×C3⋊C82
C3×#15903
#682243