Finite Group Databasebeta

← #13166099

Subgroups — #13166099

Order 1728.

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

normal subgroups only

This group has 519 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, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593218—31nomaximal
5#10593217—31nomaximal
6#10591592—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6295—41nomaximal
9C3⋊D3×SL(2,𝔽3)—21no—
10C3×C3⋊GL(2,𝔽3)—21no—
11C2×Q8×C3⋊D3D311yescharacteristic, complemented
12C2×D3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)—31no—
14C2×D3×SL(2,𝔽3)—61no—
15C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)—31no—
17C2×C3×GL(2,𝔽3)—61no—
18C2×C3×#215—31no—
19#63916—31no—
20C2×#3716—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—31no—
23C3×#859—61no—
24C3×#835—31no—
25#63567—31no—
26#63557—31no—
27C22×C3×C3⋊D3—41no—
28C22×C3×C3⋊D3—41no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31#6295—44no—
32#5421—91no—
33#5170—91no—
34#4806—91no—
35C2×C32×Q8C2×D311yescharacteristic
36Q8×C3⋊D3—21no—
37Q8×C3⋊D3—61no—
38C2×C4×C3⋊D3—31no—
39C2×#634—31no—
40C2×C3×D12—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—41no—
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
47C2×#626—61no—
48C2×#626—121no—
49C2×#625—121no—
50D3×SL(2,𝔽3)—62no—
51D3×SL(2,𝔽3)—61no—
52D3×SL(2,𝔽3)—61no—
53C3⋊GL(2,𝔽3)—21no—
54C3×GL(2,𝔽3)—62no—
55C3×GL(2,𝔽3)—61no—
56C3×#215—61no—
57C3×#215—61no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61(C2×D3)⋊Dic3—61no—
62#3716—62no—
63C22×C33—41no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—42no—
66C2×C3×C3⋊D3—81no—
67C2×C3×C3⋊D3—81no—
68C2×C3×C3⋊D3—81no—
69C2×C3×C3⋊D3—81no—
70C3×C3⋊Dic3—42no—
71C2×D3×Q8—31no—
72C2×D3×Q8—31no—
73C2×D3×Q8—61no—
74C22×SL(2,𝔽3)—91no—
75C2×GL(2,𝔽3)—31no—
76C2×GL(2,𝔽3)—61no—
77C2×C3×SD16—91no—
78C2×#215—31no—
79#938—91no—
80C2×#214—91no—
81C2×#204—91no—
82#895—91no—
83#859—31no—
84#859—61no—
85C3×#103—91no—
86#835—31no—
87(C2×D3)⋊C8—91no—
88(C2×D3)⋊C8—91no—
89#816—91no—
90#810—91no—
91C22×C3⋊D3—41no—
92C22×C3⋊D3S411yescharacteristic
93C22×C3×D3—61no—
94C22×C3×D3—121no—
95C22×C3×D3—121no—
96C22×C3×D3—121no—
97C22×C3×D3—241no—
98C22×C3×D3—241no—
99C32×Q8—31no—
100C32×Q8#6711yescharacteristic, complemented
101C2×C32×C4—31no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—41no—
104C4×C3⋊D3—61no—
105C4×C3⋊D3—61no—
106#634—32no—
107#634—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—121no—
110C2×C3×Dic3—241no—
111C3×D12—61no—
112C3×D12—61no—
113C3×SL(2,𝔽3)—21no—
114C3×SL(2,𝔽3)—21no—
115C3×SL(2,𝔽3)—21no—
116C3×SL(2,𝔽3)—41no—
117C3×SL(2,𝔽3)D1211yescharacteristic, complemented
118C3×SL(2,𝔽3)#6711yescharacteristic, complemented
119#626—124no—
120#626—122no—
121#625—124no—
122C3×C3⋊C8—61no—
123C3×C3⋊C8—61no—
124#448—271noSylow
125C2×C33—41no—
126C2×C33—41no—
127C2×C33—41no—
128C3×C3⋊D3—82no—
129C3×C3⋊D3—82no—
130C2×C3×Q8—21no—
131C2×C3×Q8D3211yescharacteristic
132C2×C3×Q8D3211yescharacteristic
133C2×C3×D4—91no—
134C2×#67—181no—
135C2×#67—361no—
136D3×Q8—62no—
137D3×Q8—61no—
138D3×Q8—61no—
139D3×Q8—181no—
140D3×Q8—181no—
141D3×Q8—182no—
142C2×D12—31no—
143C2×D12—181no—
144C2×C4×D3—91no—
145C2×C4×D3—91no—
146C2×C4×D3—181no—
147C2×Dic6—91no—
148C2×Dic6—91no—
149C2×Dic6—181no—
150C2×SL(2,𝔽3)—21no—
151C2×SL(2,𝔽3)—21no—
152C2×SL(2,𝔽3)—41no—
153C2×SL(2,𝔽3)—181no—
154C2×SL(2,𝔽3)D3211yescharacteristic
155GL(2,𝔽3)—62no—
156GL(2,𝔽3)—61no—
157C3×SD16—181no—
158C3×SD16—181no—
159C2×C3×C8—91no—
160C2×C3×C8—91no—
161C3×C4⋊C4—91no—
162#217—181no—
163#215—61no—
164#215—61no—
165#214—182no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—31no—
170#204—182no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—81no—
174C22×C32—81no—
175C22×C32—81no—
176C22×C32—161no—
177C22×C32C2×S411yescharacteristic
178C2×C3⋊D3—21no—
179C2×C3⋊D3—21no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—81no—
182C2×C3⋊D3—81no—
183C2×C3⋊D3GL(2,𝔽3)12yescomplemented
184C2×C3×D3—122no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—242no—
191C2×C3×D3—241no—
192C2×C3×D3—241no—
193C2×C3×D3—241no—
194C2×C3×D3—241no—
195C2×C3×D3—242no—
196C2×C3×D3—242no—
197C2×C3×D3—241no—
198C2×C3×D3—241no—
199C2×C3×D3—242no—
200C2×C3×D3—242no—
201C2×C3×D3—242no—
202C32×C4—31no—
203C32×C4—31no—
204C3⋊Dic3—42no—
205C3⋊Dic3—61no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—242no—
209C22×Q8—91no—
210C2×SD16—91no—
211C2×SD16—271no—
212#121—271no—
213#103—91no—
214#103—271no—
215C22⋊C8—271no—
216C33—41noSylow
217C23×C3—181no—
218C23×C3—361no—
219C22×D3—31no—
220C22×D3—31no—
221C22×D3—61no—
222C22×D3—61no—
223C22×D3—121no—
224C22×D3—241no—
225C3×Q8—21no—
226C3×Q8—31no—
227C3×Q8—31no—
228C3×Q8—61no—
229C3×Q8#62511yescharacteristic, complemented
230C3×Q8#62611yescharacteristic, complemented
231C3×D4—181no—
232C3×D4—181no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—61no—
236C2×C3×C4—181no—
237#67—362no—
238#67—364no—
239C2×Dic3—61no—
240C2×Dic3—91no—
241C2×Dic3—91no—
242C2×Dic3—121no—
243C2×Dic3—181no—
244C2×Dic3—241no—
245D12—61no—
246D12—61no—
247D12—362no—
248C4×D3—181no—
249C4×D3—181no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—182no—
253C4×D3—182no—
254Dic6—92no—
255Dic6—92no—
256Dic6—182no—
257Dic6—181no—
258Dic6—181no—
259Dic6—182no—
260SL(2,𝔽3)—21no—
261SL(2,𝔽3)—21no—
262SL(2,𝔽3)—41no—
263SL(2,𝔽3)#62611yescharacteristic, complemented
264C3×C8—181no—
265C3×C8—181no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—81no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32—81no—
278C2×C32—81no—
279C2×C32—81no—
280C2×C32—81no—
281C2×C32—81no—
282C2×C32—81no—
283C2×C32—161no—
284C2×C32—161no—
285C2×C32—161no—
286C2×C32C2×GL(2,𝔽3)11yescharacteristic
287C2×C32#98311yescharacteristic
288C2×C32GL(2,ℤ/4)11yescharacteristic
289C3⋊D3—22no—
290C3⋊D3—82no—
291C3×D3—242no—
292C3×D3—242no—
293C3×D3—244no—
294C3×D3—242no—
295C3×D3—244no—
296C3×D3—241no—
297C2×Q8—181no—
298C2×Q8—271no—
299C2×Q8—541no—
300C2×Q8#111311yescharacteristic
301C2×D4—91no—
302C2×D4—541no—
303C22×C4—271no—
304SD16—181no—
305SD16—181no—
306SD16—542no—
307C2×C8—91no—
308C2×C8—91no—
309C4⋊C4—91no—
310C22⋊C4—541no—
311C22×C3—21no—
312C22×C3—41no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—161no—
316C22×C3—362no—
317C22×C3—361no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3—721no—
321C22×C3—721no—
322C22×C3D3×S411yescharacteristic
323C22×C3D3×S411yescharacteristic
324C2×D3—32no—
325C2×D3—32no—
326C2×D3—62no—
327C2×D3—62no—
328C2×D3—62no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—122no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—241no—
338C2×D3—241no—
339C2×D3—242no—
340C2×D3—242no—
341C2×D3—242no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—361no—
349Dic3—122no—
350Dic3—121no—
351Dic3—182no—
352Dic3—181no—
353Dic3—181no—
354Dic3—242no—
355C32—41no—
356C32—41no—
357C32—81no—
358C32—81no—
359C32—81no—
360C32—161no—
361C32#542111yescharacteristic, complemented
362C23—91no—
363C23—181no—
364Q8—31no—
365Q8—272no—
366Q8—541no—
367Q8#629511yescharacteristic, complemented
368D4—181no—
369D4—181no—
370D4—1082no—
371C2×C4—31no—
372C2×C4—181no—
373C2×C4—271no—
374C2×C4—541no—
375C2×C4—541no—
376C8—181no—
377C8—181no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—81no—
385C2×C3—81no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3—81no—
390C2×C3—161no—
391C2×C3—161no—
392C2×C3—161no—
393C2×C3—722no—
394C2×C3—721no—
395C2×C3#6420111yescharacteristic
396C2×C3#6420211yescharacteristic
397C2×C3D3×GL(2,𝔽3)11yescharacteristic
398C2×C3D3×GL(2,𝔽3)11yescharacteristic
399C2×C3#6420911yescharacteristic
400C2×C3#6421011yescharacteristic
401D3—64no—
402D3—62no—
403D3—62no—
404D3—242no—
405D3—244no—
406D3—241no—
407C22—92no—
408C22—181no—
409C22—181no—
410C22—361no—
411C22—361no—
412C22—361no—
413C22#8910111yescharacteristic
414C4—31no—
415C4—31no—
416C4—361no—
417C4—541no—
418C3—21no—
419C3—41no—
420C3—81no—
421C3—81no—
422C3—161no—
423C3#1059321711yescharacteristic, complemented
424C3#1059321811yescharacteristic, complemented
425C2—182no—
426C2—721no—
427C2#1172421011yescharacteristic
428C2#1172421111yescharacteristic
429C2#1172421611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
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
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
11. C2×Q8×C3⋊D3D3semidirect
15. C2×C3⋊GL(2,𝔽3)D3semidirect
21. C3⋊#859D3semidirect
30. C32×SL(2,𝔽3)D4semidirect
100. C32×Q8#67semidirect
117. C3×SL(2,𝔽3)D12semidirect
118. C3×SL(2,𝔽3)#67semidirect
183. C2×C3⋊D3GL(2,𝔽3)semidirect
229. C3×Q8#625semidirect
230. C3×Q8#626semidirect
263. SL(2,𝔽3)#626semidirect
361. C32#5421semidirect
367. Q8#6295semidirect
423. C3#10593217semidirect
424. C3#10593218semidirect

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⋊GL(2,𝔽3)2
C3×C3⋊#8592
#105932183
#105932173
#105915923
C2×#62954