Finite Group Databasebeta

← C3×#73065 #11920843

Subgroups — C3×#73065 #11920843

Order 1152.

499 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 72 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×GL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4#73065C311yesmaximal, characteristic, direct product factor
5C3×#1880—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×C3×GL(2,𝔽3)—21no—
8C2×C3×GL(2,𝔽3)—21no—
9C2×C3×GL(2,𝔽3)C412yescomplemented
10C2×C3×(C2×D3)⋊C4—41nomaximal
11C3×C4×SL(2,𝔽3)—21no—
12C3×#859—21no—
13C22×GL(2,𝔽3)C2×C311yescharacteristic
14C22×C3×SD16—31no—
15C2×C3×C4×Q8D311yescharacteristic, complemented
16C2×C4×SL(2,𝔽3)—21no—
17C2×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×#859C2×C311yescharacteristic, complemented
19C2×C3×#103—31no—
20C2×C3×#102—31no—
21C3×#390—31no—
22C3×#340—31no—
23C3×#336—31no—
24C23×C3×D3—41no—
25C22×C32×C4—41no—
26C22×C3×Dic3—41no—
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C2×C3×SL(2,𝔽3)D411yescharacteristic
30C3×GL(2,𝔽3)—22no—
31C3×(C2×D3)⋊C4—44no—
32#1880—31noSylow
33C22×C3×Q8C2×D311yescharacteristic
34C22×C3×D4—31no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)C22×C311yescharacteristic
37C2×GL(2,𝔽3)—21no—
38C2×GL(2,𝔽3)—21no—
39C2×GL(2,𝔽3)C3×C412yescomplemented
40C2×C3×SD16—32no—
41C2×C3×SD16—32no—
42C2×C3×SD16—61no—
43C2×C3×SD16—61no—
44C2×C3×SD16—61no—
45C2×C3×SD16—61no—
46C22×C3×C8—31no—
47C22×C3×C8—31no—
48C3×C4×Q8—21no—
49C3×C4×Q8—61no—
50C2×C3×C4⋊C4—31no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C22⋊C4—61no—
53C2×C3×C22⋊C4—61no—
54C2×C3×C42—31no—
55C2×(C2×D3)⋊C4—41no—
56C4×SL(2,𝔽3)—22no—
57C4×SL(2,𝔽3)—21no—
58#859—21no—
59C3×#103—61no—
60C3×#103—61no—
61C3×#102—62no—
62C23×C32—41no—
63C22×C3×D3—42no—
64C22×C3×D3—42no—
65C22×C3×D3—42no—
66C22×C3×D3—81no—
67C22×C3×D3—81no—
68C22×C3×D3—81no—
69C22×C3×D3—81no—
70C2×C32×C4—42no—
71C2×C32×C4—81no—
72C2×C32×C4—81no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—81no—
75C2×C3×Dic3—81no—
76C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
77C22×SD16—31no—
78C2×C4×Q8C3×D311yescharacteristic, complemented
79C2×#103—31no—
80C2×#102—31no—
81#390—31no—
82#340—31no—
83#336—31no—
84C24×C3—61no—
85C23×D3—41no—
86C2×C3×Q8—31no—
87C2×C3×Q8—31no—
88C2×C3×Q8—31no—
89C2×C3×Q8C4×D311yescharacteristic
90C2×C3×Q8D1211yescharacteristic
91C2×C3×Q8#6711yescharacteristic
92C2×C3×D4—32no—
93C2×C3×D4—61no—
94C2×C3×D4—61no—
95C2×C3×D4—61no—
96C2×C3×D4—61no—
97C2×C3×D4—61no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—41no—
101C22×C3×C4—61no—
102C22×C3×C4—81no—
103C22×C3×C4S411yescharacteristic
104C22×Dic3—41no—
105C2×SL(2,𝔽3)—21no—
106C2×SL(2,𝔽3)—21no—
107C2×SL(2,𝔽3)—21no—
108C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
109C2×SL(2,𝔽3)C3×D411yescharacteristic
110C2×SL(2,𝔽3)C3×D411yescharacteristic
111GL(2,𝔽3)—22no—
112C3×SD16—62no—
113C3×SD16—62no—
114C3×SD16—62no—
115C3×SD16—62no—
116C2×C3×C8—32no—
117C2×C3×C8—61no—
118C2×C3×C8—61no—
119C2×C3×C8—61no—
120C2×C3×C8—61no—
121C2×C3×C8—61no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—61no—
126C3×C22⋊C4—122no—
127C3×C22⋊C4—122no—
128C3×C42—61no—
129C3×C42—61no—
130(C2×D3)⋊C4—44no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C2×C3×D3—44no—
139C2×C3×D3—82no—
140C2×C3×D3—82no—
141C2×C3×D3—82no—
142C2×C3×D3—82no—
143C2×C3×D3—82no—
144C2×C3×D3—82no—
145C32×C4—82no—
146C3×Dic3—82no—
147C22×Q8C2×C3×D311yescharacteristic
148C22×D4—31no—
149C2×SD16—32no—
150C2×SD16—32no—
151C2×SD16—61no—
152C2×SD16—61no—
153C2×SD16—61no—
154C2×SD16—61no—
155C22×C8—31no—
156C22×C8—31no—
157C4×Q8—21no—
158C4×Q8—61no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C42—31no—
164#103—61no—
165#103—61no—
166#102—62no—
167C23×C3—41no—
168C23×C3—62no—
169C23×C3—81no—
170C23×C3—121no—
171C23×C3—121no—
172C23×C3—121no—
173C23×C3—121no—
174C23×C3—121no—
175C23×C3—121no—
176C23×C3C2×S411yescharacteristic
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C22×D3—81no—
183C22×D3—81no—
184C3×Q8—31no—
185C3×Q8—31no—
186C3×Q8—31no—
187C3×Q8—61no—
188C3×Q8(C2×D3)⋊C411yescharacteristic, complemented
189C3×D4—62no—
190C3×D4—62no—
191C3×D4—121no—
192C3×D4—121no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—42no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—82no—
208C2×C3×C4—82no—
209C2×C3×C4—82no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4GL(2,𝔽3)12yescomplemented
214C2×Dic3—42no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)C3×C22⋊C411yescharacteristic, complemented
219C3×C8—62no—
220C3×C8—121no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C3×D3—84no—
229C24—61no—
230C2×Q8—31no—
231C2×Q8—31no—
232C2×Q8—31no—
233C2×Q8C3×C4×D311yescharacteristic
234C2×Q8C3×D1211yescharacteristic
235C2×Q8C3×#6711yescharacteristic
236C2×D4—32no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—61no—
245C22×C4C3×S411yescharacteristic
246SD16—62no—
247SD16—62no—
248SD16—62no—
249SD16—62no—
250C2×C8—32no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C4⋊C4—32no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C22⋊C4—122no—
261C22⋊C4—122no—
262C42—61no—
263C42—61no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3—241no—
282C22×C3—241no—
283C22×C3—241no—
284C22×C3—241no—
285C22×C3C4×S411yescharacteristic
286C22×C3C4⋊S411yescharacteristic
287C22×C3C2×GL(2,𝔽3)11yescharacteristic
288C22×C3#98311yescharacteristic
289C22×C3#98511yescharacteristic
290C22×C3#98611yescharacteristic
291C22×C3GL(2,ℤ/4)11yescharacteristic
292C2×D3—44no—
293C2×D3—82no—
294C2×D3—82no—
295C2×D3—82no—
296C2×D3—82no—
297C2×D3—82no—
298C2×D3—82no—
299C3×C4—22no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—82no—
305C3×C4—84no—
306C3×C4—121no—
307C3×C4—241no—
308Dic3—82no—
309C32—41noSylow
310C23—62no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23—121no—
315C23—121no—
316C23—121no—
317C23C2×C3×S411yescharacteristic
318Q8—31no—
319Q8—31no—
320Q8—31no—
321Q8—61no—
322Q8C3×(C2×D3)⋊C411yescharacteristic, complemented
323D4—62no—
324D4—62no—
325D4—121no—
326D4—121no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4C3×GL(2,𝔽3)12yescomplemented
342C8—62no—
343C8—121no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—242no—
359C2×C3C4×GL(2,𝔽3)11yescharacteristic
360C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
361C2×C3#539411yescharacteristic
362C2×C3#539511yescharacteristic
363C2×C3(C2×S4)⋊C411yescharacteristic
364C2×C3#542111yescharacteristic
365C2×C3#542711yescharacteristic
366D3—84no—
367C22—122no—
368C22—122no—
369C22—122no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22—241no—
374C22C3×C4×S411yescharacteristic
375C22C3×C4⋊S411yescharacteristic
376C22C2×C3×GL(2,𝔽3)11yescharacteristic
377C22C3×#98311yescharacteristic
378C22C3×#98511yescharacteristic
379C22C3×#98611yescharacteristic
380C22C3×GL(2,ℤ/4)11yescharacteristic
381C4—22no—
382C4—31no—
383C4—31no—
384C4—31no—
385C4—31no—
386C4—121no—
387C4—241no—
388C3—41no—
389C3—81no—
390C3#7306511yescharacteristic, direct product factor
391C2—242no—
392C2C3×C4×GL(2,𝔽3)11yescharacteristic
393C2C3×C4⋊GL(2,𝔽3)11yescharacteristic
394C2C3×#539411yescharacteristic
395C2C3×#539511yescharacteristic
396C2C3×(C2×S4)⋊C411yescharacteristic
397C2C3×#542111yescharacteristic
398C2C3×#542711yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC23C2×C3×S4
FittingC2×C3×C4×Q8D3
prime radicalC22×C3×GL(2,𝔽3)C2
cosocleC22×SL(2,𝔽3)C22×C3
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C3×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×C3×#859C2semidirect
4. #73065C3direct (N × H)
9. C2×C3×GL(2,𝔽3)C4semidirect
15. C2×C3×C4×Q8D3semidirect
17. C2×C4×SL(2,𝔽3)C2×C3semidirect
18. C2×#859C2×C3semidirect
39. C2×GL(2,𝔽3)C3×C4semidirect
76. C3×SL(2,𝔽3)C22⋊C4semidirect
78. C2×C4×Q8C3×D3semidirect
188. C3×Q8(C2×D3)⋊C4semidirect
213. C2×C3×C4GL(2,𝔽3)semidirect
218. SL(2,𝔽3)C3×C22⋊C4semidirect
322. Q8C3×(C2×D3)⋊C4semidirect
341. C2×C4C3×GL(2,𝔽3)semidirect
390. C3#73065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#8592
#730653
C3×#18803
C2×C3×(C2×D3)⋊C44