Finite Group Databasebeta

← #13137889

Subgroups — #13137889

Order 1728.

462 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#11721755C211yesmaximal, characteristic, complemented
2#11721113C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic
5#63636⋊C3C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590536—31nomaximal
9#10590532—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88540—21no—
15C4×#1084—21no—
16C4×#1084C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88450—21no—
21#88439—22no—
22#88438—22no—
23C3×#887—31no—
24C3×#887—31no—
25C3×#887—61no—
26#63864—31no—
27#63860—31no—
28C2×#3719—31no—
29C2×#3718—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63637D311yescharacteristic, complemented
33#63636D311yescharacteristic
34C3×#810—31no—
35C3×#809—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×#1084D411yescharacteristic
43C2×#1084D411yescharacteristic
44#6229—21no—
45#6229D412yes—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5028—91no—
54#4798—91no—
55#4767—91no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—61no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×Dic3⋊C4—61no—
69C3×Dic3⋊C4—61no—
70C3×Dic3⋊C4—62no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—62no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79#3722—61no—
80#3722—61no—
81(C2×D3)⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83#3719—62no—
84#3718—62no—
85#3717—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94#1084—22no—
95#1084—21no—
96#1084—21no—
97(C3⋊Dic3)⋊C3—41no—
98C3×C4×Q8—91no—
99#947—91no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#215—91no—
103#936—91no—
104C2×#214—91no—
105C2×Dic12—91no—
106C2×#204—91no—
107#887—31no—
108#887—31no—
109#887—61no—
110#875—91no—
111C3×#103—91no—
112C3×#102—91no—
113Dic3⋊C8—91no—
114Dic3⋊C8—91no—
115#810—31no—
116#810—91no—
117#809—31no—
118#809—91no—
119C4⋊(C3⋊C8)—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131#634—21no—
132#634—21no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×Dic6—32no—
146C3×Dic6—32no—
147C3×Dic6—62no—
148C3×Dic6—62no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157#464—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#13511yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—81no—
162C2×C3×Q8—91no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×D12—31no—
166C2×D12—91no—
167C2×Dic6—31no—
168C2×Dic6—31no—
169C2×Dic6—31no—
170C2×Dic6—61no—
171C2×Dic6—91no—
172C2×C3×C8—91no—
173C2×C3×C8—91no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—181no—
178C3×C4⋊C4D3211yescharacteristic
179C3×C42—91no—
180C3×C42—181no—
181#217—181no—
182#217—181no—
183#216—182no—
184#216—182no—
185#215—182no—
186#214—182no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191Dic3⋊C4—61no—
192Dic3⋊C4—61no—
193Dic3⋊C4—62no—
194C4×Dic3—31no—
195C4×Dic3—31no—
196C4×Dic3—61no—
197C4×Dic3—61no—
198C4×Dic3—61no—
199C4×Dic3—62no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—91no—
204Dic12—182no—
205#204—182no—
206C22×C32—21no—
207C22×C32D3×D411yescharacteristic
208C22×C32D3×D411yescharacteristic
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4#23511yescharacteristic
221C32×C4#23511yescharacteristic
222C32×C4D3×D411yescharacteristic
223C32×C4D3×D411yescharacteristic
224C3⋊Dic3—41no—
225C3×Dic3—64no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C2×Q16—271no—
236C2×SD16—271no—
237#124—271no—
238C4×Q8—91no—
239C4⋊C8—271no—
240#103—91no—
241#102—91no—
242H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—181no—
246C3×Q8—92no—
247C3×Q8—181no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×D4—181no—
251C3×D4—181no—
252C2×C3×C4—21no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C2×D3211yescharacteristic
264C2×Dic3—31no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—61no—
274C2×Dic3—181no—
275D12—61no—
276D12—61no—
277D12—181no—
278D12—181no—
279Dic6—32no—
280Dic6—32no—
281Dic6—62no—
282Dic6—61no—
283Dic6—61no—
284Dic6—62no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—181no—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—181no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32#88411yescharacteristic
299C2×C32#88411yescharacteristic
300C2×C32#91211yescharacteristic
301C2×C32#91511yescharacteristic
302C2×C32#91611yescharacteristic
303C2×C32#91811yescharacteristic
304C3⋊D3—81no—
305C3×D3—241no—
306C2×Q8—91no—
307C2×Q8—91no—
308C2×D4—91no—
309Q16—542no—
310SD16—542no—
311C2×C8—91no—
312C2×C8—91no—
313C4⋊C4—181no—
314C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C22⋊C4—541no—
316C22⋊C4—541no—
317C42—91no—
318C42—181no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3#381111yescharacteristic
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—361no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—41no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—121no—
340C3×C4—121no—
341C3×C4—122no—
342C3×C4—182no—
343C3×C4—181no—
344C3×C4—181no—
345C3×C4—361no—
346C3×C4#379811yescharacteristic
347C3×C4#380211yescharacteristic
348Dic3—62no—
349Dic3—62no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—61no—
354Dic3—64no—
355Dic3—62no—
356Dic3—121no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#476711yescharacteristic, complemented
361C32#479811yescharacteristic, complemented
362C23—181no—
363Q8—92no—
364Q8—181no—
365Q8—181no—
366Q8—181no—
367D4—181no—
368D4—181no—
369C2×C4—21no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C8—181no—
376C8—181no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—721no—
387C2×C3#6391211yescharacteristic
388C2×C3#6393611yescharacteristic
389C2×C3#6395111yescharacteristic
390D3—241no—
391D3—721no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22#8867711yescharacteristic
396C4—41no—
397C4—181no—
398C4—182no—
399C4—181no—
400C4—361no—
401C4#8865011yescharacteristic
402C4#8865511yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3#1059053811yescharacteristic
407C2—721no—
408C2#1172114211yescharacteristic
409C2#1172119211yescharacteristic
410C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721755C2semidirect
3. C2×#88439C2semidirect
6. #63637⋊C3C2semidirect
7. #11720304C2semidirect
32. #63637D3semidirect
242. H(3,𝔽3)#464semidirect
360. C32#4767semidirect
361. C32#4798semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117211132
C2×#884392
C2×#884382
#63636⋊C32
#63637⋊C32
#117203042
#105905363
#105905323