Finite Group Databasebeta

← C3×#69021 #11794998

Subgroups — C3×#69021 #11794998

Order 1152.

453 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5131C211yesmaximal, characteristic
2C2×C3×#823C211yesmaximal, characteristic
3C3×#5124C211yesmaximal, characteristic, complemented
4C3×#5001C211yesmaximal, characteristic
5C32×#344C211yesmaximal, characteristic, complemented
6C3×#4559C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#2081—31nomaximal
9#69021C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#823—22no—
18C3×#822—22no—
19C3×#821—22no—
20C3×#809—22no—
21C3×#430—31no—
22C3×#418—31no—
23C3×#418—31no—
24C2×C3×#100—31no—
25#5131C2×C311yescharacteristic
26C2×#823C2×C311yescharacteristic
27#5124C2×C311yescharacteristic, complemented
28#5001C2×C311yescharacteristic
29C3×#344—21no—
30C3×#344—31no—
31C3×#344D311yescharacteristic, complemented
32C3×#344C2×C311yescharacteristic, complemented
33C3×H(3,ℤ/4)—31no—
34#4559C2×C311yescharacteristic, complemented
35#4531C2×C311yescharacteristic, complemented
36C23×C3×D3—21no—
37C22×C32×C4C2311yescharacteristic
38C2×C3×D12—21no—
39C2×C3×D12—21no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12D412yes—
44C32×M16—21no—
45C32×M16—21no—
46C32×M16—21no—
47C32×M16—21no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C32×C4⋊C4—21no—
51C32×C22⋊C4—21no—
52C32×C42—21no—
53C3×#217—21no—
54C3×C4⋊Dic3—21no—
55C3×C4×Dic3—21no—
56C3×#208—21no—
57C3×#208—21no—
58C3×#208—21no—
59C3×#208—21no—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8D412yes—
62#2081—31noSylow
63C22×C3×D4—31no—
64C22×D12C22×C311yescharacteristic
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—31no—
68C2×C3×M16—31no—
69C2×C3×M16C2×D311yescharacteristic
70C2×C3×M16C2×D311yescharacteristic
71C2×C3×M16C22×C311yescharacteristic
72C2×C3×M16C22×C311yescharacteristic
73C3×#117—21no—
74C3×#117—31no—
75C3×#117C2×D311yescharacteristic
76C3×#117C22×C311yescharacteristic
77#926C22×C311yescharacteristic
78C2×#208C22×C311yescharacteristic
79C2×#208C22×C311yescharacteristic
80C3×#108—62no—
81C3×#102—62no—
82C3×#102—62no—
83C3×#100—62no—
84#823—22no—
85#822—22no—
86#821—22no—
87#809—22no—
88C23×C32#4111yescharacteristic
89C22×C3×D3—22no—
90C22×C3×D3—41no—
91C22×C3×D3—41no—
92C22×C3×D3—82no—
93C2×C32×C4—41no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C32×C4#4111yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3×Dic3—41no—
101C3×D12—22no—
102C3×D12—22no—
103C3×D12—41no—
104C3×D12—41no—
105C32×C8—41no—
106C32×C8—41no—
107C3×C3⋊C8—22no—
108C3×C3⋊C8—41no—
109#430—31no—
110#418—31no—
111#418—31no—
112C2×#100—31no—
113#344—31no—
114#344C3×D311yescharacteristic, complemented
115H(3,ℤ/4)—31no—
116C24×C3—61no—
117C23×D3—21no—
118C2×C3×D4—32no—
119C2×C3×D4—61no—
120C2×C3×D4—61no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C22×C3×C4—21no—
125C22×C3×C4C22×D311yescharacteristic
126C22×C3×C4C23×C311yescharacteristic
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12C3×D412yes—
133C3×M16—22no—
134C3×M16—21no—
135C3×M16—21no—
136C3×M16—21no—
137C3×M16—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C3×M16—21no—
142C3×M16—41no—
143C3×M16—41no—
144C3×M16—41no—
145C3×M16—61no—
146C3×M16—61no—
147C3×M16—61no—
148C3×M16—61no—
149C2×C3×C8—22no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8—21no—
154C2×C3×C8—32no—
155C2×C3×C8—41no—
156C2×C3×C8—61no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—61no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—61no—
165C3×C42—21no—
166C3×C42—21no—
167C3×C42—41no—
168C3×C42—61no—
169#217—21no—
170C4⋊Dic3—21no—
171C4×Dic3—21no—
172#208—21no—
173#208—21no—
174#208—21no—
175#208—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8C3×D412yes—
178C22×C32—41no—
179C22×C32#12311yescharacteristic
180C22×C32#12411yescharacteristic
181C22×C32#12611yescharacteristic
182C2×C3×D3—42no—
183C2×C3×D3—82no—
184C2×C3×D3—82no—
185C2×C3×D3—82no—
186C32×C4—81no—
187C32×C4#12111yescharacteristic
188C32×C4#12311yescharacteristic
189C32×C4#12311yescharacteristic
190C32×C4#12611yescharacteristic
191C3×Dic3—81no—
192C22×D4—31no—
193C2×M16—31no—
194C2×M16—31no—
195C2×M16C2×C3×D311yescharacteristic
196C2×M16C2×C3×D311yescharacteristic
197#117—31no—
198#117C2×C3×D311yescharacteristic
199#108—62no—
200#102—62no—
201#102—62no—
202#100—62no—
203C23×C3—21no—
204C23×C3—62no—
205C23×C3—121no—
206C23×C3—121no—
207C23×C3—242no—
208C23×C3#23511yescharacteristic
209C23×C3C3×#4111yescharacteristic
210C22×D3—22no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—82no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—121no—
217C3×D4—121no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—121no—
228C2×C3×C4#23511yescharacteristic
229C2×C3×C4#23511yescharacteristic
230C2×C3×C4D3×D411yescharacteristic
231C2×C3×C4D3×D411yescharacteristic
232C2×C3×C4#23711yescharacteristic
233C2×C3×C4#23911yescharacteristic
234C2×C3×C4C2×C3×D411yescharacteristic
235C2×C3×C4C2×C3×D411yescharacteristic
236C2×C3×C4C3×#4111yescharacteristic
237C2×C3×C4C3×#4111yescharacteristic
238C2×C3×C4C3×#4111yescharacteristic
239C2×C3×C4C3×#4111yescharacteristic
240C2×Dic3—41no—
241D12—22no—
242D12—22no—
243D12—41no—
244D12—41no—
245C3×C8—42no—
246C3×C8—42no—
247C3×C8—41no—
248C3×C8—41no—
249C3×C8—41no—
250C3×C8—41no—
251C3×C8—62no—
252C3×C8—121no—
253C3⋊C8—22no—
254C3⋊C8—41no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32#39711yescharacteristic
259C3×D3—82no—
260C24—61no—
261C2×D4—32no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4C22×C3×D311yescharacteristic
268M16—21no—
269M16—21no—
270M16—21no—
271M16—21no—
272M16—61no—
273M16—61no—
274M16—61no—
275M16—61no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—32no—
279C2×C8—61no—
280C4⋊C4—21no—
281C4⋊C4—61no—
282C22⋊C4—21no—
283C22⋊C4—61no—
284C42—21no—
285C42—61no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—81no—
292C22×C3—122no—
293C22×C3—242no—
294C22×C3—242no—
295C22×C3—242no—
296C22×C3#87611yescharacteristic
297C22×C3#88511yescharacteristic
298C22×C3#89411yescharacteristic
299C22×C3C3×#12311yescharacteristic
300C22×C3C3×#12411yescharacteristic
301C22×C3C3×#12611yescharacteristic
302C2×D3—42no—
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—82no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—82no—
313C3×C4—241no—
314C3×C4#88311yescharacteristic
315C3×C4#88411yescharacteristic
316C3×C4#89411yescharacteristic
317C3×C4#89811yescharacteristic
318C3×C4C3×#12111yescharacteristic
319C3×C4C3×#12311yescharacteristic
320C3×C4C3×#12311yescharacteristic
321C3×C4C3×#12611yescharacteristic
322Dic3—81no—
323C32#208111yescharacteristic, complemented, Sylow
324C23—62no—
325C23—121no—
326C23—121no—
327C23—242no—
328C23C3×#23511yescharacteristic
329D4—62no—
330D4—62no—
331D4—121no—
332D4—121no—
333C2×C4—41no—
334C2×C4—121no—
335C2×C4C3×#23511yescharacteristic
336C2×C4C3×#23511yescharacteristic
337C2×C4C3×D3×D411yescharacteristic
338C2×C4C3×D3×D411yescharacteristic
339C2×C4C3×#23711yescharacteristic
340C2×C4C3×#23911yescharacteristic
341C8—41no—
342C8—41no—
343C8—62no—
344C8—121no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—242no—
356C2×C3#467511yescharacteristic
357C2×C3C3×#39711yescharacteristic
358D3—82no—
359C22—41no—
360C22—122no—
361C22—242no—
362C22—242no—
363C22—242no—
364C22C3×#87611yescharacteristic
365C22C3×#88511yescharacteristic
366C22C3×#89411yescharacteristic
367C4—81no—
368C4—241no—
369C4C3×#88311yescharacteristic
370C4C3×#88411yescharacteristic
371C4C3×#89411yescharacteristic
372C4C3×#89811yescharacteristic
373C3—21no—
374C3#6902111yescharacteristic, direct product factor
375C3C3×#208111yescharacteristic, complemented
376C2—21no—
377C2—21no—
378C2—21no—
379C2—242no—
380C2C3×#467511yescharacteristic
subgroupG/H
centerC2×C3#4675
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#344C2
socleC2×C32#397
prime radicalC22×C3×D12C22
hypercenterC3×#344D3
hypocenterC3C3×#2081
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#344

Complements

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

NH ≅ G/Ntype
3. C3×#5124C2semidirect
5. C32×#344C2semidirect
6. C3×#4559C2semidirect
7. C3×#4531C2semidirect
9. #69021C3direct (N × H)
27. #5124C2×C3semidirect
31. C3×#344D3semidirect
32. C3×#344C2×C3semidirect
34. #4559C2×C3semidirect
35. #4531C2×C3semidirect
114. #344C3×D3semidirect
323. C32#2081semidirect
374. C3#69021direct (N × H)
375. C3C3×#2081semidirect

Maximal subgroups

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

subgroupindex
C3×#51312
C2×C3×#8232
C3×#51242
C3×#50012
C32×#3442
C3×#45592
C3×#45312
C3×#20813
#690213