Finite Group Databasebeta

← C3×#69034 #11787101

Subgroups — C3×#69034 #11787101

Order 1152.

565 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic
2C3×#5129C211yesmaximal, characteristic
3C3×#5119C211yesmaximal, characteristic, complemented
4C3×#5006C211yesmaximal, characteristic
5C3×#4999C211yesmaximal, characteristic, complemented
6C32×#344C211yesmaximal, characteristic, complemented
7C3×#4559C211yesmaximal, characteristic, complemented
8C3×#1868—31nomaximal
9#69034C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#907—22no—
18C3×D3×M16—22no—
19C3×#901—22no—
20C3×(C2×D3)⋊C8—22no—
21C3×Dic3⋊C8—22no—
22C3×#806C414yes—
23C3×C4⋊(C3⋊C8)—22no—
24C3×#568—31no—
25#5745C2×C311yescharacteristic
26C3×#424—31no—
27C3×#424—31no—
28C3×#410—31no—
29C3×#408—31no—
30#5129C2×C311yescharacteristic
31#5119C2×C311yescharacteristic, complemented
32#5006C2×C311yescharacteristic
33#4999C2×C311yescharacteristic, complemented
34C3×#344—21no—
35C3×#344—31no—
36C3×#344D311yescharacteristic, complemented
37C3×#344C2×C311yescharacteristic, complemented
38#4559C2×C311yescharacteristic, complemented
39C22×C32×C4C2311yescharacteristic
40C2×C3×#67C2×C412yes—
41C3×#235—21no—
42C3×#235—21no—
43C3×#235—21no—
44C3×#235—21no—
45C2×C3×D12Q811yescharacteristic
46C2×C3×C4×D3D412yes—
47C2×C3×Dic6Q811yescharacteristic
48C32×M16—21no—
49C32×M16—21no—
50C32×M16—21no—
51C32×M16—21no—
52C2×C32×C8—21no—
53C2×C32×C8D412yes—
54C32×C4⋊C4—21no—
55C32×C22⋊C4C2×C412yes—
56C32×C42—21no—
57C3×#217C2×C412yes—
58C3×C4⋊Dic3—21no—
59C3×C4×Dic3—21no—
60C3×#208—21no—
61C3×#208—21no—
62C2×C3×C3⋊C8—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8D412yes—
66C3×ΓL(1,𝔽25)—24no—
67C3×D3×C8—24no—
68#1868—31noSylow
69C2×C3×#41—31no—
70C2×#235C22×C311yescharacteristic
71C3×#131—62no—
72C3×#131—62no—
73C2×C3×M16—21no—
74C2×C3×M16—21no—
75C2×C3×M16—31no—
76C2×C3×M16—62no—
77C2×C3×M16C2×D311yescharacteristic
78C2×C3×M16C2×D311yescharacteristic
79C2×C3×M16C22×C311yescharacteristic
80C2×C3×M16C22×C311yescharacteristic
81C22×C3×C8—31no—
82C3×#117—21no—
83C3×#117—31no—
84C3×#117C2×D311yescharacteristic
85C3×#117C22×C311yescharacteristic
86#926C22×C311yescharacteristic
87C2×#208C22×C311yescharacteristic
88C22×C3⋊C8C22×C311yescharacteristic
89#907—22no—
90D3×M16—22no—
91#901—22no—
92C3×C4⋊C8—62no—
93C3×C4⋊C8—62no—
94C3×#99—34no—
95C3×C22⋊C8—62no—
96(C2×D3)⋊C8—22no—
97Dic3⋊C8—22no—
98#806C3×C414yes—
99C4⋊(C3⋊C8)—22no—
100C23×C32C22×C411yescharacteristic
101C22×C3×D3C4⋊C412yes—
102C2×C32×C4—22no—
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×Q811yescharacteristic
107C2×C32×C4#4111yescharacteristic
108C2×C32×C4#4111yescharacteristic
109C3×#67—22no—
110C3×#67—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3C4⋊C412yes—
113C3×D12—21no—
114C3×D12—21no—
115C3×C4×D3—22no—
116C3×C4×D3—22no—
117C3×Dic6—21no—
118C3×Dic6—21no—
119C32×C8—22no—
120C32×C8—41no—
121C3×C3⋊C8—22no—
122C3×C3⋊C8—41no—
123#568—31no—
124#424—31no—
125#424—31no—
126#410—31no—
127#408—31no—
128#344—31no—
129#344C3×D311yescharacteristic, complemented
130C3×#41—61no—
131C3×#41—61no—
132C3×#41—61no—
133C3×#41—61no—
134C2×C3×Q8—31no—
135C2×C3×D4—32no—
136C2×C3×D4—31no—
137C22×C3×C4—21no—
138C22×C3×C4—32no—
139C22×C3×C4C22×D311yescharacteristic
140C22×C3×C4C23×C311yescharacteristic
141C2×#67C2×C3×C412yes—
142#235—21no—
143#235—21no—
144#235—21no—
145#235—21no—
146C2×D12C3×Q811yescharacteristic
147C2×C4×D3C3×D412yes—
148C2×Dic6C3×Q811yescharacteristic
149C3×M16—21no—
150C3×M16—21no—
151C3×M16—21no—
152C3×M16—21no—
153C3×M16—22no—
154C3×M16—21no—
155C3×M16—21no—
156C3×M16—21no—
157C3×M16—21no—
158C3×M16—41no—
159C3×M16—41no—
160C3×M16—41no—
161C3×M16—64no—
162C3×M16—61no—
163C3×M16—61no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—32no—
168C2×C3×C8—41no—
169C2×C3×C8—64no—
170C2×C3×C8—61no—
171C2×C3×C8—61no—
172C2×C3×C8—61no—
173C2×C3×C8D1212yes—
174C2×C3×C8C3×D412yes—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—61no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—32no—
181C3×C22⋊C4C4×D312yes—
182C3×C22⋊C4C2×C3×C412yes—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—41no—
186C3×C42—61no—
187#217C2×C3×C412yes—
188C4⋊Dic3—21no—
189C4×Dic3—21no—
190#208—21no—
191#208—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8C3×D412yes—
196ΓL(1,𝔽25)—24no—
197D3×C8—24no—
198C22×C32—41no—
199C22×C32C2×C4⋊C411yescharacteristic
200C22×C32C4×D411yescharacteristic
201C22×C32C4×D411yescharacteristic
202C2×C3×D3—22no—
203C2×C3×D3—42no—
204C32×C4—42no—
205C32×C4#12011yescharacteristic
206C32×C4C22⋊Q811yescharacteristic
207C32×C4C22⋊Q811yescharacteristic
208C32×C4#12311yescharacteristic
209C3×Dic3—22no—
210C3×Dic3—42no—
211C2×#41—31no—
212#131—62no—
213#131—62no—
214C2×M16—31no—
215C2×M16—62no—
216C2×M16C2×C3×D311yescharacteristic
217C2×M16C2×C3×D311yescharacteristic
218C22×C8—31no—
219#117—31no—
220#117C2×C3×D311yescharacteristic
221C4⋊C8—62no—
222C4⋊C8—62no—
223#99—34no—
224C22⋊C8—62no—
225C23×C3—21no—
226C23×C3—32no—
227C23×C3C2×C4×D311yescharacteristic
228C23×C3C22×C3×C411yescharacteristic
229C22×D3C3×C4⋊C412yes—
230C3×Q8—61no—
231C3×Q8—61no—
232C3×D4—62no—
233C3×D4—62no—
234C3×D4—61no—
235C3×D4—61no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—32no—
245C2×C3×C4—42no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4C2×D1211yescharacteristic
250C2×C3×C4#23511yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4#23711yescharacteristic
254C2×C3×C4D3×Q811yescharacteristic
255C2×C3×C4C2×C3×D411yescharacteristic
256C2×C3×C4C2×C3×D411yescharacteristic
257C2×C3×C4C2×C3×D411yescharacteristic
258C2×C3×C4C2×C3×Q811yescharacteristic
259C2×C3×C4C3×#4111yescharacteristic
260C2×C3×C4C3×#4111yescharacteristic
261#67—22no—
262#67—22no—
263C2×Dic3—22no—
264C2×Dic3C3×C4⋊C412yes—
265D12—21no—
266D12—21no—
267C4×D3—22no—
268C4×D3—22no—
269Dic6—21no—
270Dic6—21no—
271C3×C8—22no—
272C3×C8—22no—
273C3×C8—42no—
274C3×C8—42no—
275C3×C8—41no—
276C3×C8—41no—
277C3×C8—62no—
278C3×C8—121no—
279C3⋊C8—22no—
280C3⋊C8—41no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32#38511yescharacteristic
285C3×D3—42no—
286#41—61no—
287#41—61no—
288#41—61no—
289#41—61no—
290C2×Q8—31no—
291C2×D4—32no—
292C2×D4—31no—
293C22×C4—32no—
294C22×C4C22×C3×D311yescharacteristic
295M16—21no—
296M16—21no—
297M16—21no—
298M16—21no—
299M16—64no—
300M16—61no—
301M16—61no—
302C2×C8—21no—
303C2×C8—32no—
304C2×C8—64no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8C3×D1212yes—
309C4⋊C4—21no—
310C4⋊C4—61no—
311C22⋊C4—32no—
312C22⋊C4C3×C4×D312yes—
313C42—21no—
314C42—61no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—62no—
321C22×C3—81no—
322C22×C3—122no—
323C22×C3C4×D1211yescharacteristic
324C22×C3#67⋊C411yescharacteristic
325C22×C3D3×C4⋊C411yescharacteristic
326C22×C3C2×C3×C4⋊C411yescharacteristic
327C22×C3C3×C4×D411yescharacteristic
328C22×C3C3×C4×D411yescharacteristic
329C2×D3—22no—
330C2×D3—42no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—62no—
338C3×C4—82no—
339C3×C4—122no—
340C3×C4#88211yescharacteristic
341C3×C4#88311yescharacteristic
342C3×C4#89611yescharacteristic
343C3×C4#89711yescharacteristic
344C3×C4C3×#12011yescharacteristic
345C3×C4C3×C22⋊Q811yescharacteristic
346C3×C4C3×C22⋊Q811yescharacteristic
347C3×C4C3×#12311yescharacteristic
348Dic3—22no—
349Dic3—42no—
350C32#186811yescharacteristic, complemented, Sylow
351C23—32no—
352C23C2×C3×C4×D311yescharacteristic
353Q8—61no—
354Q8—61no—
355D4—62no—
356D4—62no—
357D4—61no—
358D4—61no—
359C2×C4—22no—
360C2×C4—32no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4C2×C3×D1211yescharacteristic
365C2×C4C3×#23511yescharacteristic
366C2×C4C3×D3×D411yescharacteristic
367C2×C4C3×D3×D411yescharacteristic
368C2×C4C3×#23711yescharacteristic
369C2×C4C3×D3×Q811yescharacteristic
370C8—22no—
371C8—41no—
372C8—62no—
373C8—121no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—122no—
385C2×C3#466811yescharacteristic
386C2×C3C3×#38511yescharacteristic
387D3—42no—
388C22—41no—
389C22—62no—
390C22—122no—
391C22C3×C4×D1211yescharacteristic
392C22C3×#67⋊C411yescharacteristic
393C22C3×D3×C4⋊C411yescharacteristic
394C4—42no—
395C4—62no—
396C4—122no—
397C4C3×#88211yescharacteristic
398C4C3×#88311yescharacteristic
399C4C3×#89611yescharacteristic
400C4C3×#89711yescharacteristic
401C3—21no—
402C3#6903411yescharacteristic, direct product factor
403C3C3×#186811yescharacteristic, complemented
404C2—21no—
405C2—21no—
406C2—21no—
407C2—122no—
408C2C3×#466811yescharacteristic
subgroupG/H
centerC3×C4#882
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#344C2
socleC2×C32#385
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#344D3
hypocenterC3C3×#1868
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#344

Complements

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

NH ≅ G/Ntype
3. C3×#5119C2semidirect
5. C3×#4999C2semidirect
6. C32×#344C2semidirect
7. C3×#4559C2semidirect
9. #69034C3direct (N × H)
31. #5119C2×C3semidirect
33. #4999C2×C3semidirect
36. C3×#344D3semidirect
37. C3×#344C2×C3semidirect
38. #4559C2×C3semidirect
129. #344C3×D3semidirect
350. C32#1868semidirect
402. C3#69034direct (N × H)
403. C3C3×#1868semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
C3×#51292
C3×#51192
C3×#50062
C3×#49992
C32×#3442
C3×#45592
C3×#18683
#690343