Finite Group Databasebeta

← C3×#3497⋊C3 #11919744

Subgroups — C3×#3497⋊C3 #11919744

Order 1152.

892 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×C3×2−1+4⋊C3C212yesmaximal, complemented
2C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#548⋊C3C212yesmaximal, complemented
5C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C3×#3497C311yesmaximal, characteristic, complemented
7#3497⋊C3C313yesmaximal, direct product factor
8C3×2−1+4⋊C3—24no—
9C2×C3×#41⋊C3C2212yes—
10C22×C3×SL(2,𝔽3)C2212yes—
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C32×#121—41nomaximal
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×C3×2−1+4C2×C312yescomplemented
16C2×2−1+4⋊C3C2×C316yescomplemented
17C3×#550—31no—
18C3×#550C2×C311yescharacteristic, complemented
19C3×D4×Q8—31no—
20C3×D4×Q8C2×C311yescharacteristic, complemented
21C3×#548—32no—
22C3×#548C2×C312yescomplemented
23C3×#537—32no—
24C3×#532—31no—
25C3×#532—31no—
26C2×C3×C4×Q8C2×C311yescharacteristic, complemented
27C4⋊(#41⋊C3)C2×C313yescomplemented
28D4×SL(2,𝔽3)C2×C313yescomplemented
29#548⋊C3C2×C316yescomplemented
30C2×C4×SL(2,𝔽3)C2×C313yescomplemented
31C2×C32×D4—42no—
32C2×C32×D4—41no—
33C22×C32×C4—41no—
34C3×#41⋊C3—22no—
35C2×C3×SL(2,𝔽3)—22no—
36C2×C3×SL(2,𝔽3)D412yescomplemented
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C32×C4⋊C4—41no—
39C32×C22⋊C4—42no—
40#3497C3211yescharacteristic, complemented, Sylow
41C3×2−1+4—24no—
42C3×2−1+4—64no—
43C2×C3×#41—32no—
44C2×C3×#41—32no—
45C2×C3×#41—32no—
46C2×C3×#41C22×C312yes—
47C22×C3×Q8—32no—
48C22×C3×Q8C22×C312yes—
49C22×C3×Q8C22×C311yescharacteristic
502−1+4⋊C3—212no—
51C2×#41⋊C3C22×C316yes—
52C22×SL(2,𝔽3)C22×C316yes—
53C22×SL(2,𝔽3)C22×C313yes—
54C3×C4⋊Q8—31no—
55C3×C4⋊Q8—31no—
56C3×#127—31no—
57C3×#127—31no—
58C3×#124—32no—
59C3×#124—32no—
60C3×C22⋊Q8—34no—
61C3×C22⋊Q8—32no—
62C3×C22⋊Q8—32no—
63C3×#121—32no—
64C3×#121—32no—
65C3×#121—34no—
66C3×#121—32no—
67C3×#121—31no—
68C3×#121—32no—
69C3×#121—43no—
70C3×#121A411yescharacteristic
71C3×C4×Q8—31no—
72C3×C4×Q8—31no—
73C3×C4×Q8C22×C311yescharacteristic
74C3×C4×Q8C22×C311yescharacteristic
75C3×C4×D4—32no—
76C3×C4×D4—32no—
77C3×C4×D4—31no—
78C3×C4×D4—31no—
79C3×C4×D4—32no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C42—31no—
82C4×SL(2,𝔽3)C22×C313yes—
83C4×SL(2,𝔽3)C22×C313yes—
84C23×C32—41no—
85C23×C32—42no—
86C32×D4—82no—
87C32×D4—84no—
88C2×C32×C4—41no—
89C2×C32×C4—42no—
90C2×C32×C4—41no—
91C2×C32×C4—82no—
92C3×SL(2,𝔽3)C2×D411yescharacteristic
93C2×2−1+4C2×C3212yescomplemented
94#550—31no—
95#550C2×C3211yescharacteristic, complemented
96D4×Q8—31no—
97D4×Q8C2×C3211yescharacteristic, complemented
98#548—32no—
99#548C2×C3212yescomplemented
100#537—32no—
101#532—31no—
102#532—31no—
103C2×C4×Q8C2×C3211yescharacteristic, complemented
104C3×#41—22no—
105C3×#41—64no—
106C3×#41—64no—
107C3×#41—64no—
108C3×#41—64no—
109C3×#41—64no—
110C3×#41—64no—
111C3×#41—62no—
112C2×C3×Q8—22no—
113C2×C3×Q8—32no—
114C2×C3×Q8—31no—
115C2×C3×Q8—32no—
116C2×C3×Q8—32no—
117C2×C3×Q8—64no—
118C2×C3×Q8—64no—
119C2×C3×Q8—62no—
120C2×C3×Q8C3×D412yescomplemented
121C2×C3×Q8C23×C311yescharacteristic
122C2×C3×D4—32no—
123C2×C3×D4—32no—
124C2×C3×D4—32no—
125C2×C3×D4—32no—
126C2×C3×D4—32no—
127C2×C3×D4—31no—
128C2×C3×D4—46no—
129C2×C3×D4—43no—
130C2×C3×D4C2×A411yescharacteristic
131C2×C3×D4C2×A412yes—
132C22×C3×C4—31no—
133C22×C3×C4—34no—
134C22×C3×C4—32no—
135C22×C3×C4—31no—
136C22×C3×C4—32no—
137C22×C3×C4—43no—
138C22×C3×C4C2×A411yescharacteristic
139#41⋊C3—26no—
140C2×SL(2,𝔽3)—26no—
141C2×SL(2,𝔽3)C3×D416yescomplemented
142C2×SL(2,𝔽3)C23×C313yes—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—32no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—43no—
150C3×C4⋊C4C2×A411yescharacteristic
151C3×C22⋊C4—34no—
152C3×C22⋊C4—32no—
153C3×C22⋊C4—32no—
154C3×C22⋊C4—32no—
155C3×C22⋊C4—46no—
156C3×C22⋊C4C2×A412yes—
157C3×C42—31no—
158C3×C42—31no—
159C3×C42—31no—
160C3×C42—31no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—81no—
164C22×C32—81no—
165C22×C32—82no—
166C22×C32—82no—
167C22×C32—82no—
168C32×C4—42no—
169C32×C4—81no—
170C32×C4—82no—
1712−1+4—24no—
1722−1+4—64no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41—32no—
176C2×#41C22×C3212yes—
177C22×Q8—32no—
178C22×Q8C22×C3212yes—
179C22×Q8C22×C3211yescharacteristic
180C4⋊Q8—31no—
181C4⋊Q8—31no—
182#127—31no—
183#127—31no—
184#124—32no—
185#124—32no—
186C22⋊Q8—34no—
187C22⋊Q8—32no—
188C22⋊Q8—32no—
189#121—32no—
190#121—32no—
191#121—34no—
192#121—32no—
193#121—31no—
194#121—32no—
195#121C3×A411yescharacteristic
196C4×Q8—31no—
197C4×Q8—31no—
198C4×Q8C22×C3211yescharacteristic
199C4×Q8C22×C3211yescharacteristic
200C4×D4—32no—
201C4×D4—32no—
202C4×D4—31no—
203C4×D4—31no—
204C4×D4—32no—
205C2×C4⋊C4—31no—
206C2×C42—31no—
207C23×C3—32no—
208C23×C3—43no—
209C23×C3—46no—
210C23×C3C22×A412yes—
211C23×C3C22×A411yescharacteristic
212C3×Q8—32no—
213C3×Q8—31no—
214C3×Q8—32no—
215C3×Q8—64no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8C2×C3×D411yescharacteristic
219C3×D4—24no—
220C3×D4—22no—
221C3×D4—64no—
222C3×D4—62no—
223C3×D4—62no—
224C3×D4—64no—
225C3×D4—62no—
226C3×D4—86no—
227C3×D4—812no—
228C3×D4—122no—
229C3×D4—122no—
230C2×C3×C4—22no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—32no—
238C2×C3×C4—43no—
239C2×C3×C4—43no—
240C2×C3×C4—46no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—68no—
246C2×C3×C4—86no—
247C2×C3×C4—121no—
248C2×C3×C4C22×A411yescharacteristic
249C2×C3×C4C22×A412yes—
250C2×C3×C4C22×A411yescharacteristic
251SL(2,𝔽3)C2×C3×D413yes—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—82no—
256C2×C32—162no—
257#41—22no—
258#41—64no—
259#41—64no—
260#41—64no—
261#41—64no—
262#41—64no—
263#41—64no—
264#41—62no—
265C2×Q8—22no—
266C2×Q8—32no—
267C2×Q8—31no—
268C2×Q8—32no—
269C2×Q8—32no—
270C2×Q8—64no—
271C2×Q8—64no—
272C2×Q8—62no—
273C2×Q8C32×D412yescomplemented
274C2×Q8C23×C3211yescharacteristic
275C2×D4—32no—
276C2×D4—32no—
277C2×D4—32no—
278C2×D4—32no—
279C2×D4—32no—
280C2×D4—31no—
281C2×D4C2×C3×A411yescharacteristic
282C2×D4C2×C3×A412yes—
283C22×C4—31no—
284C22×C4—34no—
285C22×C4—32no—
286C22×C4—31no—
287C22×C4—32no—
288C22×C4C2×C3×A411yescharacteristic
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—31no—
294C4⋊C4—32no—
295C4⋊C4C2×C3×A411yescharacteristic
296C22⋊C4—34no—
297C22⋊C4—32no—
298C22⋊C4—32no—
299C22⋊C4—32no—
300C22⋊C4C2×C3×A412yes—
301C42—31no—
302C42—31no—
303C42—31no—
304C42—31no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—46no—
311C22×C3—43no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—83no—
316C22×C3—83no—
317C22×C3—86no—
318C22×C3—86no—
319C22×C3—86no—
320C22×C3D4×A412yes—
321C22×C3C23×A411yescharacteristic
322C3×C4—22no—
323C3×C4—21no—
324C3×C4—32no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—46no—
328C3×C4—62no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—83no—
332C3×C4—86no—
333C3×C42−1+4⋊C312yescomplemented
334C32—41noSylow
335C23—32no—
336C23C22×C3×A412yes—
337C23C22×C3×A411yescharacteristic
338Q8—32no—
339Q8—31no—
340Q8—32no—
341Q8—64no—
342Q8—62no—
343Q8—62no—
344Q8C2×C32×D411yescharacteristic
345D4—24no—
346D4—22no—
347D4—64no—
348D4—62no—
349D4—62no—
350D4—64no—
351D4—62no—
352D4—122no—
353D4—122no—
354C2×C4—22no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—32no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—68no—
367C2×C4—121no—
368C2×C4C22×C3×A411yescharacteristic
369C2×C4C22×C3×A412yes—
370C2×C4C22×C3×A411yescharacteristic
371C2×C3—22no—
372C2×C3—42no—
373C2×C3—43no—
374C2×C3—43no—
375C2×C3—43no—
376C2×C3—86no—
377C2×C3—122no—
378C2×C3—166no—
379C2×C3C2×D4×A411yescharacteristic
380C2×C3C2×2−1+4⋊C311yescharacteristic
381C2×C3#594511yescharacteristic
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22C3×D4×A412yes—
391C22C23×C3×A411yescharacteristic
392C4—22no—
393C4—21no—
394C4—31no—
395C4—32no—
396C4—31no—
397C4—62no—
398C4—61no—
399C4—61no—
400C4C3×2−1+4⋊C312yescomplemented
401C3—43no—
402C3#3497⋊C311yescharacteristic, direct product factor
403C2—22no—
404C2—42no—
405C2—122no—
406C2C2×C3×D4×A411yescharacteristic
407C2C2×C3×2−1+4⋊C311yescharacteristic
408C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3497C3
hypercenterC3×#121A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4⋊C3C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. C3×#548⋊C3C2semidirect
5. C2×C3×C4×SL(2,𝔽3)C2semidirect
6. C3×#3497C3semidirect
7. #3497⋊C3C3direct (N × H)
15. C2×C3×2−1+4C2×C3semidirect
16. C2×2−1+4⋊C3C2×C3semidirect
18. C3×#550C2×C3semidirect
20. C3×D4×Q8C2×C3semidirect
22. C3×#548C2×C3semidirect
26. C2×C3×C4×Q8C2×C3semidirect
27. C4⋊(#41⋊C3)C2×C3semidirect
28. D4×SL(2,𝔽3)C2×C3semidirect
29. #548⋊C3C2×C3semidirect
30. C2×C4×SL(2,𝔽3)C2×C3semidirect
36. C2×C3×SL(2,𝔽3)D4semidirect
40. #3497C32semidirect
93. C2×2−1+4C2×C32semidirect
95. #550C2×C32semidirect
97. D4×Q8C2×C32semidirect
99. #548C2×C32semidirect
103. C2×C4×Q8C2×C32semidirect
120. C2×C3×Q8C3×D4semidirect
141. C2×SL(2,𝔽3)C3×D4semidirect
273. C2×Q8C32×D4semidirect
333. C3×C42−1+4⋊C3semidirect
400. C4C3×2−1+4⋊C3semidirect
402. C3#3497⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C3×C4⋊(#41⋊C3)2
C3×D4×SL(2,𝔽3)2
C3×#548⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#34973
#3497⋊C33
C32×#1214