Finite Group Databasebeta

← C2×#11724192 #13180803

Subgroups — C2×#11724192 #13180803

Order 1728.

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

normal subgroups only

This group has 901 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
1C2×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11724192C214yesmaximal, direct product factor
5C2×#64266—31nomaximal
6C2×#64197—33nomaximal
7C2×#63947—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×Dic3×C3⋊D3—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yescomplemented
11C3⋊(C3⋊#226)C2212yescomplemented
12C3×C3⋊GL(2,𝔽3)C2212yescomplemented
13C2×C3×#239D311yescharacteristic, complemented
14C2×C3×#41⋊C3—31no—
15C2×#3795—31no—
16C2×C3⋊(#41⋊C3)D313yescomplemented
17#64266—34no—
18C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
19C2×C3⋊#226—31no—
20C2×C3⋊#226—33no—
21C2×C3⋊#226—61no—
22C2×C3⋊#226—63no—
23C2×C3×GL(2,𝔽3)—33no—
24#64197—312no—
25C2×#3756—31no—
26C2×C3×#215—31no—
27#63947—34no—
28#63947—34no—
29C2×#3716—31no—
30C2×#3713—31no—
31C2×D3×C3⋊C8—31no—
32C22×C3×C3⋊D3—41no—
33C2×C3⋊(C3⋊Dic3)—41no—
34C2×C32×Dic3—41no—
35C32×SL(2,𝔽3)C2311yescharacteristic
36Dic3×C3⋊D3—44no—
37C2×#985—93no—
38C2×#950—91no—
39C2×#916—91no—
40C2×C32×Q8C2×D311yescharacteristic
41C2×C4×C3⋊D3—121no—
42C2×#634—31no—
43C3×#239—32no—
44C3×#239C2×D312yescomplemented
45C2×C3×D12—31no—
46C2×C3×D12—31no—
47C2×C3×C4×D3—31no—
48C3×#41⋊C3—32no—
49C2×C3×SL(2,𝔽3)—21no—
50C2×C3×SL(2,𝔽3)—23no—
51C2×C3×SL(2,𝔽3)C2×D313yes—
52C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
53C2×#625—61no—
54C2×D3×Dic3—61no—
55C2×D3×Dic3—123no—
56#3795—34no—
57#3795—62no—
58C3⋊(#41⋊C3)C2×D316yescomplemented
59C3⋊GL(2,𝔽3)C2×D312yescomplemented
60C3⋊#226—32no—
61C3⋊#226—36no—
62C3⋊#226—62no—
63C3⋊#226—66no—
64C3×GL(2,𝔽3)—36no—
65#3756—32no—
66#3756—32no—
67C2×C3⋊(C3⋊C8)—31no—
68C3×#215—32no—
69C3×#215—32no—
70C2×C3×C3⋊C8—31no—
71#3716—34no—
72#3713—34no—
73D3×C3⋊C8—34no—
74C22×C33—41no—
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—44no—
77C3⋊(C3⋊Dic3)—42no—
78C32×Dic3—42no—
79C2×C3×#41—31no—
80C2×#239C3⋊D311yescharacteristic, complemented
81C2×#237—91no—
82C2×#235—91no—
83C2×#41⋊C3—33no—
84#985—912no—
85C2×GL(2,𝔽3)—33no—
86C2×#226—93no—
87C2×#226—183no—
88C2×C3×SD16—91no—
89#950—94no—
90#950—94no—
91C2×#216—91no—
92C2×#216—91no—
93C2×#216—181no—
94C2×#215—31no—
95C2×#214—91no—
96C2×#213—91no—
97C2×#213—91no—
98C22×C3⋊C8—91no—
99#916—94no—
100#916—94no—
101C2×#204—91no—
102C2×D3×C8—91no—
103C22×C3⋊D3—41no—
104C22×C3×D3—31no—
105C22×C3×D3—61no—
106C22×C3×D3—123no—
107C32×Q8—31no—
108C32×Q8C22×D311yescharacteristic
109C2×C32×C4—31no—
110C2×C32×C4—121no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—121no—
113C2×C3⋊Dic3—123no—
114C2×C3⋊Dic3—241no—
115C2×C3⋊Dic3—243no—
116C4×C3⋊D3—124no—
117#634—32no—
118#634—61no—
119C2×C3×Dic3—43no—
120C2×C3×Dic3S411yescharacteristic
121C3×D12—32no—
122C3×D12—32no—
123C3×D12—61no—
124C3×D12—61no—
125C3×C4×D3—32no—
126C3×C4×D3—32no—
127C3×SL(2,𝔽3)—21no—
128C3×SL(2,𝔽3)—23no—
129C3×SL(2,𝔽3)C22×D313yes—
130C3×SL(2,𝔽3)C22×D311yescharacteristic
131#625—64no—
132D3×Dic3—64no—
133D3×Dic3—1212no—
134C3⋊(C3⋊C8)—32no—
135C3×C3⋊C8—32no—
136C2×#135—271noSylow
137C2×C33—42no—
138C2×C33—41no—
139C3×C3⋊D3—44no—
140C3×#41—32no—
141C3×#41—92no—
142C2×C3×Q8—21no—
143C2×C3×Q8D3211yescharacteristic
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×C3×D4—91no—
146C2×C3×D4—91no—
147C22×C3×C4—91no—
148C2×#67—181no—
149C2×#67—181no—
150C22×Dic3—181no—
151#239—32no—
152#239C2×C3⋊D312yescomplemented
153#237—94no—
154#237—182no—
155#235—94no—
156#235—182no—
157C2×D12—31no—
158C2×D12—31no—
159C2×C4×D3—31no—
160C2×C4×D3—181no—
161C2×C4×D3—363no—
162C2×Dic6—91no—
163C2×Dic6—91no—
164C2×Dic6—181no—
165#41⋊C3—36no—
166C2×SL(2,𝔽3)—23no—
167C2×SL(2,𝔽3)D3213yes—
168GL(2,𝔽3)—36no—
169#226—96no—
170#226—186no—
171C3×SD16—92no—
172C3×SD16—92no—
173C2×C3×C8—91no—
174#216—92no—
175#216—92no—
176#216—92no—
177#216—92no—
178#216—182no—
179#216—182no—
180#215—32no—
181#215—32no—
182#214—94no—
183#213—94no—
184#213—94no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—181no—
190#204—94no—
191D3×C8—94no—
192C22×C32—41no—
193C22×C32—43no—
194C22×C32—83no—
195C22×C32—81no—
196C22×C32C2×S411yescharacteristic
197C2×C3⋊D3—42no—
198C2×C3⋊D3—44no—
199C2×C3×D3—32no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C2×C3×D3—126no—
203C2×C3×D3—1212no—
204C2×C3×D3—122no—
205C32×C4—31no—
206C32×C4—31no—
207C32×C4—122no—
208C3⋊Dic3—62no—
209C3⋊Dic3—122no—
210C3⋊Dic3—126no—
211C3⋊Dic3—242no—
212C3⋊Dic3—246no—
213C3×Dic3—46no—
214C3×Dic3C2×S412yes—
215C2×#41—31no—
216C2×#41—271no—
217#135—274no—
218#135—274no—
219C2×Q16—271no—
220C2×SD16—91no—
221C2×SD16—271no—
222C2×D8—271no—
223C22×C8—271no—
224C33—41noSylow
225C23×C3—91no—
226C23×C3—181no—
227C22×D3—31no—
228C22×D3—61no—
229C22×D3—123no—
230C3×Q8—21no—
231C3×Q8—31no—
232C3×Q8—31no—
233C3×Q8—61no—
234C3×Q8C2×D3211yescharacteristic
235C3×Q8C22×C3⋊D311yescharacteristic
236C3×D4—92no—
237C3×D4—92no—
238C3×D4—181no—
239C3×D4—181no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—61no—
244C2×C3×C4—92no—
245C2×C3×C4—92no—
246C2×C3×C4—123no—
247#67—184no—
248#67—184no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—184no—
252C2×Dic3—361no—
253C2×Dic3—363no—
254C2×Dic3—723no—
255C2×Dic3C3⋊S411yescharacteristic
256D12—32no—
257D12—32no—
258D12—61no—
259D12—61no—
260C4×D3—32no—
261C4×D3—32no—
262C4×D3—184no—
263C4×D3—3612no—
264Dic6—92no—
265Dic6—92no—
266Dic6—182no—
267Dic6—181no—
268Dic6—181no—
269Dic6—361no—
270SL(2,𝔽3)—23no—
271SL(2,𝔽3)C2×D3213yes—
272C3×C8—92no—
273C3⋊C8—32no—
274C3⋊C8—92no—
275C3⋊C8—92no—
276C3⋊C8—182no—
277C2×C32—41no—
278C2×C32—42no—
279C2×C32—46no—
280C2×C32—43no—
281C2×C32—83no—
282C2×C32—86no—
283C2×C32—81no—
284C2×C32—82no—
285C2×C32#98512yescomplemented
286C2×C32C22×S411yescharacteristic
287C3⋊D3—44no—
288C3×D3—62no—
289C3×D3—1212no—
290C3×D3—122no—
291#41—32no—
292#41—92no—
293#41—274no—
294#41—542no—
295C2×Q8—271no—
296C2×Q8D3×C3⋊D311yescharacteristic
297C2×D4—91no—
298C2×D4—91no—
299C2×D4—541no—
300C22×C4—91no—
301C22×C4—541no—
302Q16—272no—
303Q16—272no—
304SD16—92no—
305SD16—92no—
306SD16—274no—
307D8—274no—
308C2×C8—91no—
309C2×C8—274no—
310C2×C8—271no—
311C22×C3—21no—
312C22×C3—43no—
313C22×C3—83no—
314C22×C3—92no—
315C22×C3—182no—
316C22×C3—182no—
317C22×C3—362no—
318C22×C3D3×S411yescharacteristic
319C22×C3C2×C3⋊S411yescharacteristic
320C2×D3—32no—
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—126no—
324C2×D3—1212no—
325C2×D3—122no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—32no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—126no—
334Dic3—182no—
335Dic3—182no—
336Dic3—362no—
337Dic3—366no—
338Dic3—726no—
339Dic3C2×C3⋊S412yes—
340C32—41no—
341C32—43no—
342C32—83no—
343C32—81no—
344C32C2×#98511yescharacteristic, complemented
345C23—91no—
346C23—181no—
347Q8—31no—
348Q8—272no—
349Q8—541no—
350Q8C2×D3×C3⋊D311yescharacteristic
351D4—92no—
352D4—92no—
353D4—181no—
354D4—181no—
355D4—544no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—92no—
359C2×C4—92no—
360C2×C4—544no—
361C2×C4—541no—
362C8—92no—
363C8—272no—
364C2×C3—22no—
365C2×C3—21no—
366C2×C3—46no—
367C2×C3—43no—
368C2×C3—86no—
369C2×C3—83no—
370C2×C3—182no—
371C2×C3—362no—
372C2×C3#6419712yescomplemented
373C2×C3#6426612yescomplemented
374C2×C3C2×D3×S411yescharacteristic
375C2×C3C22×C3⋊S411yescharacteristic
376D3—62no—
377D3—1212no—
378D3—122no—
379C22—92no—
380C22—182no—
381C22—182no—
382C22—362no—
383C22D3×C3⋊S411yescharacteristic
384C4—31no—
385C4—32no—
386C4—31no—
387C4—542no—
388C3—21no—
389C3—43no—
390C3—83no—
391C3C2×#6419711yescharacteristic, complemented
392C3C2×#6426611yescharacteristic, complemented
393C2—182no—
394C2—362no—
395C2#1172419212yesdirect product factor
396C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×D3×C3⋊S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. C2×C3⋊(C3⋊#226)C2semidirect
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. #11724192C2direct (N × H)
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C3⋊(C3⋊#226)C22semidirect
12. C3×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3×#239D3semidirect
16. C2×C3⋊(#41⋊C3)D3semidirect
18. C2×C3⋊GL(2,𝔽3)D3semidirect
44. C3×#239C2×D3semidirect
58. C3⋊(#41⋊C3)C2×D3semidirect
59. C3⋊GL(2,𝔽3)C2×D3semidirect
80. C2×#239C3⋊D3semidirect
152. #239C2×C3⋊D3semidirect
285. C2×C32#985semidirect
344. C32C2×#985semidirect
372. C2×C3#64197semidirect
373. C2×C3#64266semidirect
391. C3C2×#64197semidirect
392. C3C2×#64266semidirect
395. C2#11724192direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3⋊(C3⋊#226)2
C2×C3×C3⋊GL(2,𝔽3)2
#117241922
C2×#642663
C2×#641973
C2×#639473
C2×Dic3×C3⋊D34