Finite Group Databasebeta

← #13166093

Subgroups — #13166093

Order 1728.

443 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 46 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⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593211—31nomaximal
5#10593210—31nomaximal
6#10591603—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊D3×SL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10#88814—41nomaximal
11C2×Q8×C3⋊D3D311yescharacteristic
12C2×D3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)—31no—
14C2×D3×SL(2,𝔽3)—61no—
15C2×C3⋊#226D311yescharacteristic, complemented
16C2×C3×#226—31no—
17C2×C3×#226—61no—
18C2×C3×#216—31no—
19#63871—31no—
20C2×#3719—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—31no—
23C3×#859—61no—
24C3×#835—31no—
25#63567—31no—
26#63557—31no—
27C22×C3×C3⋊D3—41no—
28C2×C3×C3⋊Dic3—41no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic
31#5419—91no—
32#5186—91no—
33#4809—91no—
34C2×C32×Q8C2×D311yescharacteristic
35Q8×C3⋊D3—21no—
36Q8×C3⋊D3—61no—
37C2×C4×C3⋊D3—31no—
38C2×#634—31no—
39C2×C3×Dic6—31no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—41no—
44C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46D3×SL(2,𝔽3)—62no—
47D3×SL(2,𝔽3)—61no—
48D3×SL(2,𝔽3)—61no—
49C3⋊#226—21no—
50C3×#226—62no—
51C3×#226—61no—
52C3×#216—61no—
53C3×#216—61no—
54C3×C4⋊Dic3—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57Dic3⋊Dic3—61no—
58#3722—61no—
59(C2×D3)⋊Dic3—121no—
60(C2×D3)⋊Dic3—121no—
61#3719—62no—
62C22×C33—41no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—81no—
65C2×C3×C3⋊D3—81no—
66C3×C3⋊Dic3—81no—
67C3×C3⋊Dic3—81no—
68C2×D3×Q8—31no—
69C2×D3×Q8—31no—
70C2×D3×Q8—61no—
71C22×SL(2,𝔽3)—91no—
72C2×#226—31no—
73C2×#226—61no—
74C2×C3×Q16—91no—
75#946—91no—
76C2×#216—31no—
77C2×#216—91no—
78C2×Dic12—91no—
79#897—91no—
80#859—31no—
81#859—61no—
82C3×#103—91no—
83#835—31no—
84(C2×D3)⋊C8—91no—
85(C2×D3)⋊C8—91no—
86#816—91no—
87#810—91no—
88C22×C3⋊D3S411yescharacteristic
89C22×C3×D3—121no—
90C22×C3×D3—121no—
91C22×C3×D3—241no—
92C32×Q8—31no—
93C32×Q8#6711yescharacteristic
94C2×C32×C4—31no—
95C2×C3⋊Dic3—31no—
96C2×C3⋊Dic3—41no—
97C2×C3⋊Dic3—41no—
98C4×C3⋊D3—61no—
99C4×C3⋊D3—61no—
100#634—32no—
101#634—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—121no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—241no—
107C2×C3×Dic3—241no—
108C3×Dic6—61no—
109C3×Dic6—61no—
110C3×SL(2,𝔽3)—21no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—21no—
113C3×SL(2,𝔽3)—41no—
114C3×SL(2,𝔽3)D1211yescharacteristic
115C3×SL(2,𝔽3)#6711yescharacteristic
116C3×C3⋊C8—61no—
117C3×C3⋊C8—61no—
118#451—271noSylow
119C2×C33—41no—
120C2×C33—41no—
121C2×C33—41no—
122C3×C3⋊D3—82no—
123C2×C3×Q8—21no—
124C2×C3×Q8—91no—
125C2×C3×Q8D3211yescharacteristic
126C2×C3×Q8D3211yescharacteristic
127D3×Q8—62no—
128D3×Q8—61no—
129D3×Q8—61no—
130D3×Q8—181no—
131D3×Q8—181no—
132D3×Q8—182no—
133C2×C4×D3—91no—
134C2×C4×D3—91no—
135C2×C4×D3—181no—
136C2×Dic6—31no—
137C2×Dic6—91no—
138C2×Dic6—91no—
139C2×Dic6—181no—
140C2×SL(2,𝔽3)—21no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)—41no—
143C2×SL(2,𝔽3)—181no—
144C2×SL(2,𝔽3)D3211yescharacteristic
145#226—62no—
146#226—61no—
147C3×Q16—181no—
148C3×Q16—181no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C3×C4⋊C4—91no—
152#217—361no—
153#216—61no—
154#216—61no—
155#216—182no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—31no—
159C4⋊Dic3—181no—
160Dic3⋊C4—181no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—31no—
163Dic12—182no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—81no—
167C22×C32—81no—
168C22×C32—81no—
169C22×C32—161no—
170C22×C32C2×S411yescharacteristic
171C2×C3⋊D3—21no—
172C2×C3⋊D3—21no—
173C2×C3⋊D3#22612yescomplemented
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—242no—
177C2×C3×D3—241no—
178C2×C3×D3—241no—
179C2×C3×D3—241no—
180C2×C3×D3—241no—
181C2×C3×D3—242no—
182C2×C3×D3—242no—
183C32×C4—31no—
184C32×C4—31no—
185C3⋊Dic3—61no—
186C3⋊Dic3—81no—
187C3⋊Dic3—81no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C3×Dic3—241no—
191C3×Dic3—241no—
192C3×Dic3—242no—
193C3×Dic3—242no—
194C22×Q8—91no—
195C2×Q16—91no—
196C2×Q16—271no—
197C22⋊Q8—271no—
198#103—91no—
199#103—271no—
200C22⋊C8—271no—
201C33—41noSylow
202C23×C3—361no—
203C22×D3—31no—
204C22×D3—31no—
205C22×D3—61no—
206C3×Q8—21no—
207C3×Q8—31no—
208C3×Q8—31no—
209C3×Q8—61no—
210C3×Q8—181no—
211C3×Q8—181no—
212C3×Q8#62511yescharacteristic
213C3×Q8#62611yescharacteristic
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—121no—
224C2×Dic3—121no—
225C2×Dic3—181no—
226C2×Dic3—241no—
227C2×Dic3—241no—
228C4×D3—181no—
229C4×D3—181no—
230C4×D3—181no—
231C4×D3—181no—
232C4×D3—182no—
233C4×D3—182no—
234Dic6—61no—
235Dic6—61no—
236Dic6—92no—
237Dic6—92no—
238Dic6—182no—
239Dic6—181no—
240Dic6—181no—
241Dic6—182no—
242SL(2,𝔽3)—21no—
243SL(2,𝔽3)—21no—
244SL(2,𝔽3)—41no—
245SL(2,𝔽3)#62611yescharacteristic
246C3×C8—181no—
247C3×C8—181no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C2×C32—41no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—81no—
257C2×C32—81no—
258C2×C32—81no—
259C2×C32—81no—
260C2×C32—81no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—81no—
264C2×C32—81no—
265C2×C32—161no—
266C2×C32—161no—
267C2×C32—161no—
268C2×C32C2×#22611yescharacteristic
269C2×C32#98311yescharacteristic
270C2×C32GL(2,ℤ/4)11yescharacteristic
271C3⋊D3—22no—
272C3×D3—242no—
273C3×D3—242no—
274C3×D3—244no—
275C2×Q8—91no—
276C2×Q8—181no—
277C2×Q8—271no—
278C2×Q8—541no—
279C2×Q8#111311yescharacteristic
280C22×C4—271no—
281Q16—181no—
282Q16—181no—
283Q16—542no—
284C2×C8—91no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4—541no—
288C22⋊C4—541no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—161no—
294C22×C3—362no—
295C22×C3—721no—
296C22×C3—721no—
297C22×C3D3×S411yescharacteristic
298C22×C3D3×S411yescharacteristic
299C2×D3—32no—
300C2×D3—32no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—61no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—61no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—361no—
315C3×C4—361no—
316Dic3—121no—
317Dic3—121no—
318Dic3—182no—
319Dic3—181no—
320Dic3—181no—
321Dic3—241no—
322Dic3—241no—
323Dic3—242no—
324Dic3—242no—
325C32—41no—
326C32—41no—
327C32—81no—
328C32—81no—
329C32—81no—
330C32—161no—
331C32#541911yescharacteristic, complemented
332C23—91no—
333Q8—31no—
334Q8—181no—
335Q8—181no—
336Q8—272no—
337Q8—541no—
338Q8#629511yescharacteristic
339C2×C4—31no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—271no—
343C2×C4—541no—
344C2×C4—541no—
345C8—181no—
346C8—181no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—161no—
360C2×C3—161no—
361C2×C3—161no—
362C2×C3—722no—
363C2×C3D3×#22611yescharacteristic
364C2×C3D3×#22611yescharacteristic
365C2×C3#6420111yescharacteristic
366C2×C3#6420211yescharacteristic
367C2×C3#6420911yescharacteristic
368C2×C3#6421011yescharacteristic
369D3—64no—
370D3—62no—
371D3—62no—
372C22—92no—
373C22—181no—
374C22—181no—
375C22#8910111yescharacteristic
376C4—31no—
377C4—31no—
378C4—361no—
379C4—361no—
380C4—541no—
381C3—21no—
382C3—41no—
383C3—81no—
384C3—81no—
385C3—161no—
386C3#1059321011yescharacteristic, complemented
387C3#1059321111yescharacteristic, complemented
388C2—182no—
389C2#1172420911yescharacteristic
390C2#1172421011yescharacteristic
391C2#1172421611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. C3×C3⋊#859C2semidirect
15. C2×C3⋊#226D3semidirect
21. C3⋊#859D3semidirect
173. C2×C3⋊D3#226semidirect
331. C32#5419semidirect
386. C3#10593210semidirect
387. C3#10593211semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3×C3⋊#2262
C3×C3⋊#8592
#105932113
#105932103
#105916033
#888144