Finite Group Databasebeta

← C3×#69154 #11798638

Subgroups — C3×#69154 #11798638

Order 1152.

575 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5683C211yesmaximal, characteristic, complemented
2C3×#5061C211yesmaximal, characteristic, complemented
3C3×#4741C212yesmaximal, complemented
4C32×#354C211yesmaximal, characteristic, complemented
5C3×#4471C212yesmaximal, complemented
6C3×#2199—31nomaximal
7#69154C311yesmaximal, characteristic, direct product factor
8C3×#1004C2211yescharacteristic
9C32×#124C2211yescharacteristic
10C3×#947—21no—
11C3×#942—21no—
12C3×#937C2212yes—
13C3×#885—22no—
14C3×#882—22no—
15C3×#882—22no—
16C3×#875—21no—
17C32×#99C2212yes—
18C3×#833C2211yescharacteristic
19C3×#806—22no—
20C3×#805—21no—
21C3×H(3,𝔽4)—31no—
22#5683C2×C311yescharacteristic, complemented
23C3×B(4,𝔽2)—32no—
24C3×#455—31no—
25#5061C2×C311yescharacteristic, complemented
26#4741C2×C312yescomplemented
27C3×#354—21no—
28C3×#354D311yescharacteristic, complemented
29C3×#354C2×C311yescharacteristic, complemented
30C3×AΣL(1,𝔽16)—32no—
31#4471C2×C312yescomplemented
32C23×C3×D3D412yescomplemented
33C2×C32×Q8D411yescharacteristic, complemented
34C2×C32×D4C2311yescharacteristic
35C2×C3×#67—22no—
36C2×C3×#67—22no—
37C3×#237—41no—
38C3×D3×D4—41no—
39C3×#235—21no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12—42no—
43C2×C3×Dic6—21no—
44C32×C22⋊C4—22no—
45C32×C22⋊C4—22no—
46C32×C42D411yescharacteristic, complemented
47C3×#217D412yescomplemented
48C3×#216—41no—
49C3×#215—41no—
50C3×(C2×D3)⋊C4—22no—
51C3×(C2×D3)⋊C4—22no—
52C3×(C2×D3)⋊C4—22no—
53C3×(C2×D3)⋊C4—42no—
54C3×C4×Dic3—21no—
55C3×#208—21no—
56#2199—31noSylow
57C3×H(5,𝔽2)—31no—
58#1004C22×C311yescharacteristic
59C3×#137—61no—
60C3×#124—21no—
61C3×#124—62no—
62C3×#124—61no—
63C3×#124—61no—
64C3×#124C2×D311yescharacteristic
65C3×#124C22×C311yescharacteristic
66C3×#120—32no—
67C3×#120—62no—
68C3×#120—62no—
69#947—21no—
70#942—21no—
71#937C22×C312yes—
72#885—22no—
73#882—22no—
74#882—22no—
75#875—21no—
76C3×#104—61no—
77C3×#100—31no—
78C3×#99—22no—
79C3×#99—62no—
80C3×#99C2×D312yes—
81C3×#99C22×C312yes—
82#833C22×C311yescharacteristic
83#806—22no—
84#805—21no—
85C23×C32C2×D412yes—
86C22×C3×D3—22no—
87C22×C3×D3—22no—
88C22×C3×D3—42no—
89C22×C3×D3—82no—
90C32×Q8—41no—
91C32×D4—41no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—42no—
95C2×C32×C4C2×D411yescharacteristic
96C3×#67—42no—
97C3×#67—41no—
98C3×#67—82no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—22no—
101C3×D12—21no—
102C3×D12—81no—
103C3×D12—81no—
104C3×D12—84no—
105C3×C4×D3—41no—
106C3×Dic6—21no—
107C3×Dic6—81no—
108C3×C3⋊C8—41no—
109H(3,𝔽4)—31no—
110B(4,𝔽2)—32no—
111#455—31no—
112#354C3×D311yescharacteristic, complemented
113AΣL(1,𝔽16)—32no—
114C24×C3—32no—
115C23×D3C3×D412yescomplemented
116C3×#41—61no—
117C3×#41—121no—
118C2×C3×Q8—21no—
119C2×C3×Q8—61no—
120C2×C3×Q8D1211yescharacteristic, complemented
121C2×C3×Q8C3×D411yescharacteristic, complemented
122C2×C3×D4—21no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—62no—
126C2×C3×D4—62no—
127C2×C3×D4—122no—
128C2×C3×D4—121no—
129C2×C3×D4C22×D311yescharacteristic
130C2×C3×D4C23×C311yescharacteristic
131C2×#67—22no—
132C2×#67—22no—
133#237—41no—
134D3×D4—41no—
135#235—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—42no—
139C2×Dic6—21no—
140C3×Q16—121no—
141C3×SD16—121no—
142C3×M16—61no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—32no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—62no—
151C3×C22⋊C4—62no—
152C3×C22⋊C4—62no—
153C3×C22⋊C4—122no—
154C3×C42—21no—
155C3×C42—61no—
156C3×C42D1211yescharacteristic, complemented
157C3×C42C3×D411yescharacteristic, complemented
158#217C3×D412yescomplemented
159#216—41no—
160#215—41no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—22no—
164(C2×D3)⋊C4—42no—
165C4×Dic3—21no—
166#208—21no—
167C22×C32—22no—
168C22×C32—42no—
169C22×C32#12011yescharacteristic
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—44no—
173C2×C3×D3—41no—
174C2×C3×D3—42no—
175C2×C3×D3—82no—
176C2×C3×D3—82no—
177C32×C4—21no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—82no—
181C3×Dic3—42no—
182C3×Dic3—41no—
183H(5,𝔽2)—31no—
184#137—61no—
185#124—62no—
186#124—61no—
187#124—61no—
188#124C2×C3×D311yescharacteristic
189#120—32no—
190#120—62no—
191#120—62no—
192#104—61no—
193#100—31no—
194#99—62no—
195#99C2×C3×D312yes—
196C23×C3—22no—
197C23×C3—62no—
198C23×C3—62no—
199C23×C3—122no—
200C23×C3—242no—
201C23×C3D3×D412yes—
202C23×C3C2×C3×D412yes—
203C22×D3—22no—
204C22×D3—22no—
205C22×D3—42no—
206C22×D3—82no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—61no—
210C3×Q8—81no—
211C3×Q8—241no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—61no—
215C3×D4—81no—
216C3×D4—122no—
217C3×D4—121no—
218C3×D4—241no—
219C3×D4—241no—
220C3×D4—244no—
221C3×D4—242no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—82no—
234C2×C3×C4—121no—
235C2×C3×C4C2×D1211yescharacteristic
236C2×C3×C4C2×C3×D411yescharacteristic
237#67—42no—
238#67—41no—
239#67—82no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242D12—21no—
243D12—81no—
244D12—81no—
245D12—84no—
246C4×D3—41no—
247Dic6—21no—
248Dic6—81no—
249C3×C8—121no—
250C3⋊C8—41no—
251C2×C32—21no—
252C2×C32—42no—
253C2×C32B(4,𝔽2)11yescharacteristic
254C3×D3—82no—
255C3×D3—81no—
256C24—32no—
257#41—61no—
258#41—121no—
259C2×Q8—61no—
260C2×Q8C3×D1211yescharacteristic, complemented
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—122no—
266C2×D4—121no—
267C2×D4C22×C3×D311yescharacteristic
268Q16—121no—
269SD16—121no—
270M16—61no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—32no—
274C22⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C22⋊C4—122no—
278C42—61no—
279C42C3×D1211yescharacteristic, complemented
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—21no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—82no—
287C22×C3—122no—
288C22×C3—122no—
289C22×C3—124no—
290C22×C3—121no—
291C22×C3—122no—
292C22×C3—242no—
293C22×C3—242no—
294C22×C3#88211yescharacteristic
295C22×C3C3×#12011yescharacteristic
296C2×D3—42no—
297C2×D3—42no—
298C2×D3—44no—
299C2×D3—41no—
300C2×D3—42no—
301C2×D3—82no—
302C2×D3—82no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—82no—
313C3×C4—82no—
314C3×C4—122no—
315C3×C4—121no—
316C3×C4—162no—
317Dic3—42no—
318Dic3—41no—
319C32#219911yescharacteristic, complemented, Sylow
320C23—62no—
321C23—62no—
322C23—122no—
323C23—242no—
324C23C3×D3×D412yes—
325Q8—41no—
326Q8—61no—
327Q8—241no—
328D4—41no—
329D4—61no—
330D4—122no—
331D4—121no—
332D4—241no—
333D4—241no—
334D4—244no—
335D4—242no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—121no—
342C2×C4C2×C3×D1211yescharacteristic
343C8—121no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—82no—
351C2×C3—242no—
352C2×C3—241no—
353C2×C3#474111yescharacteristic
354C2×C3C3×B(4,𝔽2)11yescharacteristic
355D3—82no—
356D3—81no—
357C22—22no—
358C22—42no—
359C22—122no—
360C22—122no—
361C22—124no—
362C22—121no—
363C22—122no—
364C22—242no—
365C22—242no—
366C22C3×#88211yescharacteristic
367C4—21no—
368C4—41no—
369C4—41no—
370C4—82no—
371C4—122no—
372C4—121no—
373C3—21no—
374C3#6915411yescharacteristic, direct product factor
375C3C3×#219911yescharacteristic, complemented
376C2—21no—
377C2—42no—
378C2—242no—
379C2—241no—
380C2C3×#474111yescharacteristic
subgroupG/H
centerC2×C3#4741
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#354C2
socleC2×C32B(4,𝔽2)
hypercenterC3×#354D3
hypocenterC3C3×#2199
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#354

Complements

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

NH ≅ G/Ntype
1. C3×#5683C2semidirect
2. C3×#5061C2semidirect
3. C3×#4741C2semidirect
4. C32×#354C2semidirect
5. C3×#4471C2semidirect
7. #69154C3direct (N × H)
22. #5683C2×C3semidirect
25. #5061C2×C3semidirect
26. #4741C2×C3semidirect
28. C3×#354D3semidirect
29. C3×#354C2×C3semidirect
31. #4471C2×C3semidirect
32. C23×C3×D3D4semidirect
33. C2×C32×Q8D4semidirect
46. C32×C42D4semidirect
47. C3×#217D4semidirect
112. #354C3×D3semidirect
115. C23×D3C3×D4semidirect
120. C2×C3×Q8D12semidirect
121. C2×C3×Q8C3×D4semidirect
156. C3×C42D12semidirect
157. C3×C42C3×D4semidirect
158. #217C3×D4semidirect
260. C2×Q8C3×D12semidirect
279. C42C3×D12semidirect
319. C32#2199semidirect
374. C3#69154direct (N × H)
375. C3C3×#2199semidirect

Maximal subgroups

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

subgroupindex
C3×#56832
C3×#50612
C3×#47412
C32×#3542
C3×#44712
C3×#21993
#691543