Finite Group Databasebeta

← C3×#10593659 #13173027

Subgroups — C3×#10593659 #13173027

Order 1728.

507 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 66 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×C32×#41⋊C3C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593659C311yesmaximal, characteristic, direct product factor
5C3×#5425—33nomaximal
6C3×#5239—31nomaximal
7C32×#41⋊C3—21no—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3×C3⋊#226—21no—
10C3×#3750—41nomaximal
11C2×C32×#41D311yescharacteristic
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3—23no—
14C2×C3×#41⋊C3D313yes—
15C2×C3×#41⋊C3C2×C311yescharacteristic
16C2×C3⋊#226C2×C311yescharacteristic, complemented
17C2×C3×#226—33no—
18C2×C3×#216—31no—
19C2×C3×#214—31no—
20C3×#924—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24C3×#832—31no—
25C3×#830—31no—
26C2×C33×C4—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5425—33no—
31C3×#452—91no—
32#5239—31no—
33C32×#41—21no—
34C32×#41—61no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C22×C32×C4—31no—
38C2×C3×Dic6—31no—
39C3×#41⋊C3—22no—
40C3×#41⋊C3—26no—
41C3×#41⋊C3—23no—
42C3×#41⋊C3—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—23no—
45C2×C3×SL(2,𝔽3)C2×D313yes—
46C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
47C3⋊#226—21no—
48C3×#226—63no—
49#3750—41no—
50C3×#217—61no—
51C3×#216—61no—
52C3×#216—61no—
53C3×#214—62no—
54C3×C4⋊Dic3—31no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—123no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—31no—
59C22×C33—41no—
60C33×C4—81no—
61C3×C3⋊Dic3—42no—
62C3×C3⋊Dic3—81no—
63C2×C3×#41—21no—
64C2×C3×#41C3×D311yescharacteristic
65C2×C3×#41C3⋊D311yescharacteristic
66C2×#41⋊C3—23no—
67C2×#41⋊C3C3×D313yes—
68C2×#226—33no—
69C2×C3×Q16—91no—
70C2×C3×SD16—91no—
71C3×C22⋊Q8—91no—
72C2×#216—31no—
73C2×#214—31no—
74#924—31no—
75#859—33no—
76C3×#103—91no—
77C3×#102—91no—
78C3×C22⋊C8—91no—
79#835—31no—
80#832—31no—
81#830—31no—
82C23×C32—31no—
83C32×Q8—31no—
84C32×Q8#6711yescharacteristic
85C32×D4—32no—
86C32×D4—61no—
87C2×C32×C4—31no—
88C2×C32×C4—41no—
89C2×C32×C4—43no—
90C2×C32×C4—61no—
91C2×C32×C4—61no—
92C2×C32×C4—83no—
93C2×C32×C4—81no—
94C2×C32×C4S411yescharacteristic
95C2×C3⋊Dic3—41no—
96C2×C3⋊Dic3—41no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—123no—
100C2×C3×Dic3—123no—
101C3×Dic6—61no—
102C3×Dic6—61no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—23no—
105C3×SL(2,𝔽3)#6713yes—
106C3×SL(2,𝔽3)C3×D411yescharacteristic
107C3×C3⋊C8—61no—
108C3×C3⋊C8—61no—
109#452—91noSylow
110C2×C33—41no—
111C2×C33—41no—
112C2×C33—41no—
113C3×#41—22no—
114C3×#41—21no—
115C3×#41—21no—
116C3×#41—61no—
117C3×#41—61no—
118C3×#41—62no—
119C2×C3×Q8—21no—
120C2×C3×Q8—91no—
121C2×C3×Q8C2×C3×D311yescharacteristic
122C2×C3×Q8C2×C3⋊D311yescharacteristic
123C2×C3×D4—31no—
124C2×C3×D4—31no—
125C2×C3×D4—61no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C2×Dic6—31no—
130#41⋊C3—26no—
131#41⋊C3—23no—
132C2×SL(2,𝔽3)—23no—
133C2×SL(2,𝔽3)C2×C3×D313yes—
134#226—63no—
135C3×Q16—181no—
136C3×Q16—181no—
137C3×SD16—182no—
138C2×C3×C8—91no—
139C2×C3×C8—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C22⋊C4—181no—
143#217—61no—
144#216—61no—
145#216—61no—
146#214—62no—
147C4⋊Dic3—31no—
148Dic3⋊C4—61no—
149Dic3⋊C4—123no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C22×C32—41no—
153C22×C32—43no—
154C22×C32—61no—
155C22×C32—61no—
156C22×C32—61no—
157C22×C32—83no—
158C22×C32—81no—
159C22×C32C2×S411yescharacteristic
160C32×C4—21no—
161C32×C4—31no—
162C32×C4—31no—
163C32×C4—81no—
164C32×C4—83no—
165C32×C4—86no—
166C32×C4—82no—
167C3⋊Dic3—42no—
168C3⋊Dic3—81no—
169C3×Dic3—126no—
170C3×Dic3—121no—
171C3×Dic3—121no—
172C3×Dic3—243no—
173C2×#41C3×C3⋊D311yescharacteristic
174C2×Q16—91no—
175C2×SD16—91no—
176C22⋊Q8—91no—
177#103—91no—
178#102—91no—
179C22⋊C8—91no—
180C33—41noSylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C3×Q8—21no—
185C3×Q8—31no—
186C3×Q8—31no—
187C3×Q8—61no—
188C3×Q8—181no—
189C3×Q8—181no—
190C3×Q8C3×#6711yescharacteristic
191C3×Q8#63811yescharacteristic
192C3×D4—32no—
193C3×D4—32no—
194C3×D4—62no—
195C3×D4—61no—
196C3×D4—61no—
197C3×D4—62no—
198C2×C3×C4—21no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—43no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—83no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C3×S411yescharacteristic
213C2×C3×C4C3⋊S411yescharacteristic
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—123no—
217C2×Dic3—123no—
218Dic6—61no—
219Dic6—61no—
220SL(2,𝔽3)—23no—
221SL(2,𝔽3)C3×#6713yes—
222C3×C8—181no—
223C3×C8—181no—
224C3⋊C8—61no—
225C3⋊C8—61no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—43no—
230C2×C32—43no—
231C2×C32—43no—
232C2×C32—83no—
233C2×C32—83no—
234C2×C32—83no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—121no—
239C2×C32#98411yescharacteristic
240C2×C32#98511yescharacteristic
241C2×C32GL(2,ℤ/4)11yescharacteristic
242#41—21no—
243#41—61no—
244C2×Q8—91no—
245C2×Q8C2×C3×C3⋊D311yescharacteristic
246C2×D4—31no—
247C22×C4—31no—
248Q16—181no—
249Q16—181no—
250SD16—182no—
251C2×C8—91no—
252C2×C8—91no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C22⋊C4—181no—
256C22×C3—21no—
257C22×C3—43no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—83no—
266C22×C3—121no—
267C22×C3—121no—
268C22×C3C2×C3×S411yescharacteristic
269C22×C3C2×C3⋊S411yescharacteristic
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—61no—
278C3×C4—61no—
279C3×C4—83no—
280C3×C4—86no—
281C3×C4—361no—
282C3×C4—361no—
283Dic3—126no—
284Dic3—121no—
285Dic3—121no—
286Dic3—243no—
287C32—41no—
288C32—43no—
289C32—83no—
290C32—81no—
291C32#542511yescharacteristic, complemented
292C23—31no—
293Q8—31no—
294Q8—181no—
295Q8—181no—
296Q8C3×#63811yescharacteristic
297D4—32no—
298D4—61no—
299C2×C4—31no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4C3×C3⋊S411yescharacteristic
305C8—181no—
306C8—181no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—43no—
311C2×C3—43no—
312C2×C3—43no—
313C2×C3—83no—
314C2×C3—83no—
315C2×C3—83no—
316C2×C3—122no—
317C2×C3—121no—
318C2×C3—121no—
319C2×C3C3×#98411yescharacteristic
320C2×C3C3×#98511yescharacteristic
321C2×C3C3×GL(2,ℤ/4)11yescharacteristic
322C2×C3#6426511yescharacteristic
323C2×C3#6426611yescharacteristic
324C2×C3#6426911yescharacteristic
325C22—61no—
326C22—61no—
327C22—61no—
328C22C2×C3×C3⋊S411yescharacteristic
329C4—21no—
330C4—31no—
331C4—31no—
332C4—361no—
333C4—361no—
334C3—21no—
335C3—43no—
336C3—83no—
337C3C3×#542511yescharacteristic, complemented
338C3#1059365911yescharacteristic, direct product factor
339C2—121no—
340C2C3×#6426511yescharacteristic
341C2C3×#6426611yescharacteristic
342C2C3×#6426911yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
prime radicalC2×C32×#41⋊C3C2
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. C3×C3⋊#859C2semidirect
4. #10593659C3direct (N × H)
16. C2×C3⋊#226C2×C3semidirect
21. C3⋊#859C2×C3semidirect
291. C32#5425semidirect
337. C3C3×#5425semidirect
338. C3#10593659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3×C3⋊#2262
C3×C3⋊#8592
#105936593
C3×#54253
C3×#52393
C3×#37504