Finite Group Databasebeta

← C2×#5831⋊C3 #11923384

Subgroups — C2×#5831⋊C3 #11923384

Order 1152.

893 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 893 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#5831⋊C3C214yesmaximal, direct product factor
4C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#5831C311yesmaximal, characteristic, complemented
6C2×#548⋊C3—31nomaximal
7C2×C3×#41⋊C3—22no—
8C2×C3×#41⋊C3C2212yescomplemented
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×D3×SL(2,𝔽3)—22no—
11C2×D3×SL(2,𝔽3)C2212yescomplemented
12C2×C3×(C2×D3)⋊C4—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C22×C3×#41C2×C311yescharacteristic, complemented
15C22×D3×Q8C2×C311yescharacteristic
16C22×#41⋊C3—21no—
17C22×#41⋊C3D311yescharacteristic, complemented
18C23×SL(2,𝔽3)—31no—
19#5831—34no—
20#5831C2×C314yescomplemented
21C2×Q8×Dic3C2×C311yescharacteristic, complemented
22C2×#938—31no—
23C2×#936—31no—
24C2×C4×#67—31no—
25C2×#924—31no—
26#548⋊C3—34no—
27C2×C4×SL(2,𝔽3)—31no—
28C23×C3×D3—41no—
29C22×C32×C4—41no—
30C22×C3×Dic3—41no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)D412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)—22no—
35C3×(C2×D3)⋊C4—44no—
36C2×#548—31noSylow
37C2×C3×#41—22no—
38C2×C3×#41—32no—
39C2×C3×#41—62no—
40C2×C3×#41C22×C312yescomplemented
41C22×C3×Q8C22×C311yescharacteristic
42C22×C3×D4—31no—
43C23×C3×C4—31no—
44C22×#67—31no—
45C2×D3×Q8—22no—
46C2×D3×Q8—32no—
47C2×D3×Q8—62no—
48C2×D3×Q8C22×C312yes—
49C22×C4×D3—31no—
50C22×Dic6—31no—
51C2×#41⋊C3—24no—
52C2×#41⋊C3—22no—
53C2×#41⋊C3—22no—
54C2×#41⋊C3C2×D312yescomplemented
55C22×SL(2,𝔽3)—21no—
56C22×SL(2,𝔽3)—32no—
57C22×SL(2,𝔽3)—62no—
58C22×SL(2,𝔽3)C2×D311yescharacteristic
59C2×C3×C22⋊C4—121no—
60C2×#217—32no—
61C2×#217—31no—
62Q8×Dic3—32no—
63Q8×Dic3C22×C312yescomplemented
64#938—34no—
65#938—34no—
66#936—34no—
67#936—34no—
68C4×#67—34no—
69C4×#67—34no—
70C2×(C2×D3)⋊C4A411yescharacteristic
71C2×C4⋊Dic3—31no—
72#924—34no—
73#924—34no—
74C2×Dic3⋊C4—31no—
75C2×C4×Dic3—31no—
76C4×SL(2,𝔽3)—32no—
77C23×C32—41no—
78C22×C3×D3—44no—
79C22×C3×D3—42no—
80C22×C3×D3—82no—
81C22×C3×D3—82no—
82C2×C32×C4—42no—
83C2×C32×C4—82no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—82no—
86C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
87C22×#41C3×D311yescharacteristic, complemented
88C23×Q8—31no—
89#548—34no—
90#548—94no—
91C2×#124—91no—
92C2×C22⋊Q8—91no—
93C2×#121—91no—
94C2×C4×Q8—31no—
95C2×C4×D4—91no—
96C24×C3—31no—
97C24×C3—121no—
98C23×D3C2×A411yescharacteristic
99C3×#41—22no—
100C3×#41—64no—
101C3×#41—64no—
102C3×#41—62no—
103C2×C3×Q8—31no—
104C2×C3×Q8—32no—
105C2×C3×Q8C3×D412yes—
106C2×C3×Q8C23×C311yescharacteristic
107C2×C3×D4—32no—
108C2×C3×D4—32no—
109C2×C3×D4—62no—
110C2×C3×D4—62no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—32no—
114C22×C3×C4—41no—
115C22×C3×C4—62no—
116C22×C3×C4—62no—
117C22×C3×C4—81no—
118C22×C3×C4—121no—
119C22×C3×C4C2×A411yescharacteristic
120C2×#67—34no—
121C2×#67—64no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3C2×A411yescharacteristic
125D3×Q8—22no—
126D3×Q8—64no—
127D3×Q8—64no—
128D3×Q8—62no—
129C2×C4×D3—32no—
130C2×C4×D3—32no—
131C2×C4×D3—62no—
132C2×C4×D3—62no—
133C2×Dic6—32no—
134C2×Dic6—32no—
135C2×Dic6—62no—
136C2×Dic6—62no—
137#41⋊C3—24no—
138#41⋊C3—22no—
139C2×SL(2,𝔽3)—22no—
140C2×SL(2,𝔽3)—21no—
141C2×SL(2,𝔽3)—62no—
142C2×SL(2,𝔽3)#6712yescomplemented
143C2×SL(2,𝔽3)C22×D311yescharacteristic
144C3×C22⋊C4—124no—
145#217—38no—
146#217—34no—
147(C2×D3)⋊C4C2×A414yes—
148C4⋊Dic3—32no—
149C4⋊Dic3—32no—
150Dic3⋊C4—34no—
151C4×Dic3—32no—
152C4×Dic3—32no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—41no—
157C2×C3×D3—44no—
158C2×C3×D3—82no—
159C2×C3×D3—88no—
160C2×C3×D3—82no—
161C32×C4—82no—
162C3×Dic3—82no—
163C2×#41—22no—
164C2×#41—32no—
165C2×#41—62no—
166C2×#41C2×C3×D312yescomplemented
167C22×Q8—32no—
168C22×Q8—62no—
169C22×Q8—91no—
170C22×Q8—92no—
171C22×Q8—182no—
172C22×Q8C2×C3×D311yescharacteristic
173C22×D4—31no—
174C22×D4—91no—
175C23×C4—31no—
176C23×C4—91no—
177#124—94no—
178#124—94no—
179C22⋊Q8—94no—
180C22⋊Q8—94no—
181#121—94no—
182#121—94no—
183C4×Q8—32no—
184C4×Q8—92no—
185C4×D4—94no—
186C4×D4—94no—
187C2×C4⋊C4—91no—
188C2×C4⋊C4—91no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—91no—
191C2×C22⋊C4—92no—
192C2×C42—91no—
193C23×C3—32no—
194C23×C3—41no—
195C23×C3—62no—
196C23×C3—62no—
197C23×C3—62no—
198C23×C3—81no—
199C23×C3—122no—
200C23×C3—124no—
201C23×C3—242no—
202C23×C3—242no—
203C23×C3C22×A411yescharacteristic
204C22×D3—22no—
205C22×D3—22no—
206C22×D3#41⋊C314yescomplemented
207C22×D3C22×A412yes—
208C3×Q8—32no—
209C3×Q8—31no—
210C3×Q8—32no—
211C3×Q8C2×C3×D411yescharacteristic
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—64no—
215C2×C3×C4—22no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—42no—
221C2×C3×C4—62no—
222C2×C3×C4—64no—
223C2×C3×C4—62no—
224C2×C3×C4—82no—
225C2×C3×C4—84no—
226C2×C3×C4—82no—
227C2×C3×C4—122no—
228C2×C3×C4—242no—
229C2×C3×C4C22×A412yes—
230#67—68no—
231C2×Dic3—22no—
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3C22×A412yes—
237C4×D3—62no—
238C4×D3—64no—
239C4×D3—62no—
240Dic6—64no—
241Dic6—62no—
242Dic6—62no—
243SL(2,𝔽3)—21no—
244SL(2,𝔽3)C2×#6711yescharacteristic, complemented
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—44no—
249C3×D3—84no—
250C24—31no—
251C24—31no—
252#41—22no—
253#41—64no—
254#41—64no—
255#41—62no—
256C2×Q8—31no—
257C2×Q8—32no—
258C2×Q8—62no—
259C2×Q8—92no—
260C2×Q8—92no—
261C2×Q8—182no—
262C2×Q8—182no—
263C2×Q8—184no—
264C2×Q8—182no—
265C2×Q8—184no—
266C2×Q8C3×#6712yes—
267C2×Q8C22×C3×D311yescharacteristic
268C2×D4—32no—
269C2×D4—32no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—94no—
273C2×D4—184no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—92no—
281C22×C4—92no—
282C22×C4—91no—
283C22×C4—91no—
284C22×C4—182no—
285C22×C4—182no—
286C22×C4D3×A411yescharacteristic
287C4⋊C4—92no—
288C4⋊C4—92no—
289C4⋊C4—94no—
290C22⋊C4—34no—
291C22⋊C4—94no—
292C22⋊C4—98no—
293C42—92no—
294C42—92no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—62no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—82no—
303C22×C3—82no—
304C22×C3—82no—
305C22×C3—81no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3—242no—
309C22×C3—248no—
310C22×C3—242no—
311C22×C3D4×A412yes—
312C22×C3C2×#41⋊C312yes—
313C22×C32−1+4⋊C312yes—
314C22×C3C23×A411yescharacteristic
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—28no—
318C2×D3C2×#41⋊C314yescomplemented
319C3×C4—22no—
320C3×C4—32no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—82no—
324C3×C4—84no—
325C3×C4—242no—
326Dic3—22no—
327Dic3—62no—
328Dic3—62no—
329C32—41noSylow
330C23—34no—
331C23—32no—
332C23—32no—
333C23—62no—
334C23—62no—
335C23—62no—
336C23—62no—
337C23—62no—
338C23C2×D3×A411yescharacteristic
339Q8—32no—
340Q8—31no—
341Q8—32no—
342Q8—182no—
343Q8—182no—
344Q8—184no—
345Q8C2×C3×#6711yescharacteristic
346D4—62no—
347D4—62no—
348D4—64no—
349D4—188no—
350C2×C4—22no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—32no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—64no—
359C2×C4—62no—
360C2×C4—92no—
361C2×C4—92no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—184no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4C2×D3×A412yes—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—44no—
372C2×C3—81no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3—84no—
376C2×C3—122no—
377C2×C3—244no—
378C2×C3#548⋊C314yescomplemented
379C2×C3C2×D4×A411yescharacteristic
380C2×C3C22×#41⋊C311yescharacteristic
381C2×C3C2×2−1+4⋊C311yescharacteristic
382D3—24no—
383C22—34no—
384C22—62no—
385C22—62no—
386C22—68no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22—124no—
391C22D3×#41⋊C312yes—
392C22#1010⋊C312yes—
393C22A4×#6712yes—
394C22C22×D3×A411yescharacteristic
395C4—22no—
396C4—31no—
397C4—32no—
398C4—31no—
399C4—62no—
400C4—182no—
401C4—182no—
402C3—41no—
403C3—81no—
404C3C2×#548⋊C311yescharacteristic, complemented
405C2—64no—
406C2—122no—
407C2#5831⋊C314yesdirect product factor
408C2C2×D3×#41⋊C311yescharacteristic
409C2C2×#1010⋊C311yescharacteristic
410C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC22×C3×#41C2×C3
socleC23×C3C22×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C22×D3×SL(2,𝔽3)C2semidirect
3. #5831⋊C3C2direct (N × H)
4. C2×Dic3×SL(2,𝔽3)C2semidirect
5. C2×#5831C3semidirect
8. C2×C3×#41⋊C3C22semidirect
11. C2×D3×SL(2,𝔽3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. C22×C3×#41C2×C3semidirect
17. C22×#41⋊C3D3semidirect
20. #5831C2×C3semidirect
21. C2×Q8×Dic3C2×C3semidirect
32. C2×C3×SL(2,𝔽3)D4semidirect
40. C2×C3×#41C22×C3semidirect
54. C2×#41⋊C3C2×D3semidirect
63. Q8×Dic3C22×C3semidirect
86. C3×SL(2,𝔽3)C2×D4semidirect
87. C22×#41C3×D3semidirect
142. C2×SL(2,𝔽3)#67semidirect
166. C2×#41C2×C3×D3semidirect
206. C22×D3#41⋊C3semidirect
244. SL(2,𝔽3)C2×#67semidirect
318. C2×D3C2×#41⋊C3semidirect
378. C2×C3#548⋊C3semidirect
404. C3C2×#548⋊C3semidirect
407. C2#5831⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C22×D3×SL(2,𝔽3)2
#5831⋊C32
C2×Dic3×SL(2,𝔽3)2
C2×#58313
C2×#548⋊C33
C2×C3×(C2×D3)⋊C44