Finite Group Databasebeta

← C3×#71370 #11780223

Subgroups — C3×#71370 #11780223

Order 1152.

449 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5732C211yesmaximal, characteristic
2C32×#500C211yesmaximal, characteristic, complemented
3C3×#5094C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6C3×#4459C211yesmaximal, characteristic
7C3×#4456C211yesmaximal, characteristic, complemented
8C3×#1650—31nomaximal
9#71370C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4⋊Q8C2211yescharacteristic
12C3×#947—21no—
13C3×Q8×Dic3—21no—
14C2×C3×#215—21no—
15C3×#898—22no—
16C3×#897—21no—
17C3×#892—21no—
18C3×C4×D12C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20C32×#107—22no—
21C32×C4×C8C2211yescharacteristic
22C3×#835—21no—
23C3×#810—21no—
24C3×#808—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×#556—31no—
28#5732C2×C311yescharacteristic
29C3×#500—21no—
30C3×#500D311yescharacteristic, complemented
31C3×#500C2×C311yescharacteristic, complemented
32C3×#475—31no—
33C3×#461—31no—
34#5094C2×C311yescharacteristic, complemented
35#5083C2×C311yescharacteristic
36C3×#332—31no—
37C3×Q8⋊C8—31no—
38C3×D4⋊C8—31no—
39#4547C2×C311yescharacteristic, complemented
40#4459C2×C311yescharacteristic
41#4456C2×C311yescharacteristic, complemented
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C2×C3×D12D411yescharacteristic
45C2×C3×C4×D3—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×C32×C8—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4D412yes—
52C32×C42C2311yescharacteristic
53C3×#215—42no—
54C3×(C2×D3)⋊C4—22no—
55C3×(C2×D3)⋊C4—22no—
56C3×(C2×D3)⋊C4—21no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3D411yescharacteristic
59C3×C4⋊Dic3D411yescharacteristic
60C3×Dic3⋊C4—21no—
61C3×C4×Dic3—22no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65#1650—31noSylow
66C2×C3×SD16—61no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8C2×D311yescharacteristic
70C3×C4⋊Q8C2×D311yescharacteristic
71C3×C4⋊Q8C22×C311yescharacteristic
72C3×C4⋊Q8C22×C311yescharacteristic
73C3×#126—62no—
74C3×#124—61no—
75C3×C22⋊Q8—61no—
76C3×C4×Q8—31no—
77C3×C4×Q8—61no—
78C3×C4×D4—31no—
79C3×#117—61no—
80#947—21no—
81Q8×Dic3—21no—
82C2×#215—21no—
83#898—22no—
84#897—21no—
85#892—21no—
86C4×D12C22×C311yescharacteristic
87C4×Dic6C22×C311yescharacteristic
88C3×#107—24no—
89C3×#107—22no—
90C3×#107—22no—
91C3×#106—61no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8—31no—
94C3×#103—61no—
95C3×#103—61no—
96C3×C4×C8—21no—
97C3×C4×C8C2×D311yescharacteristic
98C3×C4×C8C22×C311yescharacteristic
99#835—21no—
100#810—21no—
101#808—21no—
102C4⋊(C3⋊C8)C22×C311yescharacteristic
103C4⋊(C3⋊C8)C22×C311yescharacteristic
104C22×C3×D3—21no—
105C32×Q8—41no—
106C32×Q8—41no—
107C32×Q8—41no—
108C32×Q8—41no—
109C2×C32×C4—22no—
110C2×C32×C4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C3×D12—21no—
118C3×D12—21no—
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×Dic6—21no—
122C3×Dic6—21no—
123C32×C8—22no—
124C3×C3⋊C8—41no—
125C3×C3⋊C8—41no—
126#556—31no—
127#500C3×D311yescharacteristic, complemented
128#475—31no—
129#461—31no—
130#332—31no—
131Q8⋊C8—31no—
132D4⋊C8—31no—
133C2×C3×Q8—21no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—31no—
138C2×C3×Q8—41no—
139C2×C3×Q8—41no—
140C2×C3×D4—31no—
141C22×C3×C4—61no—
142C2×D12C3×D411yescharacteristic
143C2×C4×D3—21no—
144C2×Dic6C3×D411yescharacteristic
145C3×SD16—122no—
146C2×C3×C8—22no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—61no—
150C2×C3×C8—61no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4#6712yes—
166C3×C4⋊C4C3×D412yes—
167C3×C22⋊C4—62no—
168C3×C22⋊C4—62no—
169C3×C22⋊C4—61no—
170C3×C42—21no—
171C3×C42—62no—
172C3×C42—61no—
173C3×C42C22×D311yescharacteristic
174C3×C42C23×C311yescharacteristic
175#215—42no—
176(C2×D3)⋊C4—22no—
177(C2×D3)⋊C4—22no—
178(C2×D3)⋊C4—21no—
179C4⋊Dic3—22no—
180C4⋊Dic3C3×D411yescharacteristic
181C4⋊Dic3C3×D411yescharacteristic
182Dic3⋊C4—21no—
183C4×Dic3—22no—
184C4×Dic3—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C22×C32#12011yescharacteristic
188C2×C3×D3—41no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C32×C4—21no—
192C32×C4—42no—
193C32×C4—81no—
194C32×C4#13511yescharacteristic
195C32×C4#13511yescharacteristic
196C32×C4AGL(1,ℤ/8)11yescharacteristic
197C32×C4#13711yescharacteristic
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—41no—
201C2×SD16—61no—
202C4⋊Q8C2×C3×D311yescharacteristic
203C4⋊Q8C2×C3×D311yescharacteristic
204#126—62no—
205#124—61no—
206C22⋊Q8—61no—
207C4×Q8—31no—
208C4×Q8—61no—
209C4×D4—31no—
210#117—61no—
211#107—22no—
212#106—61no—
213C4⋊C8—31no—
214C4⋊C8—31no—
215#103—61no—
216#103—61no—
217C4×C8C2×C3×D311yescharacteristic
218C23×C3—61no—
219C22×D3—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—41no—
226C3×Q8—41no—
227C3×Q8—41no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8—81no—
231C3×Q8—81no—
232C3×Q8—81no—
233C3×Q8—81no—
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—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4D3×D411yescharacteristic
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4C2×#6711yescharacteristic
253C2×C3×C4C2×C3×D411yescharacteristic
254C2×C3×C4C2×C3×D411yescharacteristic
255C2×C3×C4C2×C3×D411yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261C4×D3—41no—
262C4×D3—41no—
263Dic6—21no—
264Dic6—21no—
265C3×C8—24no—
266C3×C8—22no—
267C3×C8—22no—
268C3×C8—121no—
269C3×C8—121no—
270C3⋊C8—41no—
271C3⋊C8—41no—
272C2×C32#44911yescharacteristic
273C2×C32#45211yescharacteristic
274C2×C32#45611yescharacteristic
275C3×D3—81no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—31no—
279C2×D4—31no—
280C22×C4—61no—
281SD16—122no—
282C2×C8—21no—
283C2×C8—61no—
284C2×C8—61no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4C3×#6712yes—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—61no—
296C42—62no—
297C42—61no—
298C42C22×C3×D311yescharacteristic
299C22×C3—21no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3#93711yescharacteristic
304C22×C3C3×#12011yescharacteristic
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—42no—
317C3×C4—82no—
318C3×C4—82no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4—121no—
322C3×C4—121no—
323C3×C4—121no—
324C3×C4#91111yescharacteristic
325C3×C4#91211yescharacteristic
326C3×C4#91811yescharacteristic
327C3×C4#91911yescharacteristic
328C3×C4C3×#13511yescharacteristic
329C3×C4C3×#13511yescharacteristic
330C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
331C3×C4C3×#13711yescharacteristic
332Dic3—41no—
333Dic3—41no—
334Dic3—41no—
335C32#165011yescharacteristic, complemented, Sylow
336C23—61no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—41no—
341Q8—61no—
342Q8—61no—
343D4—61no—
344D4—61no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4C3×D3×D411yescharacteristic
353C2×C4C3×D3×D411yescharacteristic
354C2×C4C2×C3×#6711yescharacteristic
355C8—22no—
356C8—121no—
357C8—121no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—241no—
362C2×C3#509511yescharacteristic
363C2×C3#515811yescharacteristic
364C2×C3#518711yescharacteristic
365C2×C3C3×#44911yescharacteristic
366C2×C3C3×#45211yescharacteristic
367C2×C3C3×#45611yescharacteristic
368D3—81no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C3×#93711yescharacteristic
373C4—21no—
374C4—42no—
375C4—81no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4C3×#91111yescharacteristic
380C4C3×#91211yescharacteristic
381C4C3×#91811yescharacteristic
382C4C3×#91911yescharacteristic
383C3—21no—
384C3#7137011yescharacteristic, direct product factor
385C3C3×#165011yescharacteristic, complemented
386C2—241no—
387C2C3×#509511yescharacteristic
388C2C3×#515811yescharacteristic
389C2C3×#518711yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#500C2
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC3×#500D3
hypocenterC3C3×#1650
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#500

Complements

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

NH ≅ G/Ntype
2. C32×#500C2semidirect
3. C3×#5094C2semidirect
5. C3×#4547C2semidirect
7. C3×#4456C2semidirect
9. #71370C3direct (N × H)
30. C3×#500D3semidirect
31. C3×#500C2×C3semidirect
34. #5094C2×C3semidirect
39. #4547C2×C3semidirect
41. #4456C2×C3semidirect
127. #500C3×D3semidirect
335. C32#1650semidirect
384. C3#71370direct (N × H)
385. C3C3×#1650semidirect

Maximal subgroups

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

subgroupindex
C3×#57322
C32×#5002
C3×#50942
C3×#50832
C3×#45472
C3×#44592
C3×#44562
C3×#16503
#713703