Finite Group Databasebeta

← #13137939

Subgroups — #13137939

Order 1728.

470 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 75 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#64104⋊C3C211yesmaximal, characteristic
2#11721106C211yesmaximal, characteristic, complemented
3C2×#88431C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#11720530C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590578—31nomaximal
9#10590523—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3752⋊C3—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88449—21no—
21#88431—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64104D311yescharacteristic
24C3×#895—31no—
25#63871—31no—
26#63857—31no—
27C2×#3717—31no—
28C2×#3716—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63637D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#808—31no—
34C3×#808—31no—
35C3×#808—61no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—42no—
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5041—91no—
55#4864—91no—
56#4807—91no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2×D311yescharacteristic
67C32×C4⋊C4C2×D311yescharacteristic
68#3752—21no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78Dic3⋊Dic3—61no—
79Dic3⋊Dic3—61no—
80#3722—61no—
81(C2×D3)⋊Dic3—61no—
82#3717—62no—
83#3716—62no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3SD1612yescomplemented
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C3×#121—91no—
100#946—91no—
101C2×#215—91no—
102C2×#214—91no—
103#924—91no—
104C22×C3⋊C8—91no—
105C2×#204—91no—
106C2×D3×C8—91no—
107C2×D3×C8—91no—
108#897—91no—
109#895—31no—
110#895—91no—
111C3×#106—91no—
112C3×#103—91no—
113#832—91no—
114#821—91no—
115#810—31no—
116#810—91no—
117#809—91no—
118#808—31no—
119#808—31no—
120#808—61no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3#6711yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3#6711yescharacteristic
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—42no—
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—121no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—122no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—32no—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—61no—
157#465—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×SD1611yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161(C3⋊D3)⋊C3—22no—
162(C3⋊D3)⋊C3—81no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—91no—
167C2×D12—31no—
168C2×D12—61no—
169C2×D12—91no—
170C2×D12—181no—
171C2×C4×D3—31no—
172C2×C4×D3—91no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4D3211yescharacteristic
182C3×C22⋊C4—181no—
183#217—181no—
184#215—182no—
185#214—182no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—61no—
193C4⋊Dic3—181no—
194Dic3⋊C4—181no—
195Dic3⋊C4—181no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—182no—
202#204—182no—
203D3×C8—182no—
204D3×C8—182no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3#21412yescomplemented
214C2×C3×D3—32no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—81no—
225C32×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4#23711yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3—41no—
231C3×Dic3—61no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C3×Dic3—241no—
236C2×SD16—271no—
237C22×C8—271no—
238C22⋊Q8—271no—
239#121—91no—
240#106—91no—
241#103—91no—
242#102—271no—
243H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
244C23×C3—91no—
245C23×C3—181no—
246C22×D3—31no—
247C22×D3—61no—
248C22×D3—91no—
249C22×D3—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—181no—
254C3×D4—362no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4C2×D3211yescharacteristic
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—121no—
274C2×Dic3—181no—
275D12—61no—
276D12—61no—
277D12—122no—
278D12—181no—
279D12—181no—
280D12—362no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—181no—
284C4×D3—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—92no—
290C3×C8—181no—
291C3⋊C8—32no—
292C3⋊C8—32no—
293C3⋊C8—62no—
294C3⋊C8—61no—
295C3⋊C8—181no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32#88411yescharacteristic
300C2×C32D3×SD1611yescharacteristic
301C2×C32#91911yescharacteristic
302C2×C32C2×#21411yescharacteristic
303C2×C32#93911yescharacteristic
304C2×C32#95011yescharacteristic
305C3⋊D3—22no—
306C3⋊D3—81no—
307C3×D3—62no—
308C3×D3—241no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—91no—
313SD16—542no—
314C2×C8—91no—
315C2×C8—91no—
316C2×C8—542no—
317C4⋊C4—91no—
318C4⋊C4—541no—
319C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C22⋊C4—181no—
321C22⋊C4—541no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—61no—
325C22×C3—92no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3D3×#6711yescharacteristic
332C2×D3—32no—
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—92no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—41no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—181no—
354C3×C4—241no—
355C3×C4—361no—
356C3×C4—361no—
357C3×C4#379511yescharacteristic
358C3×C4#380211yescharacteristic
359Dic3—61no—
360Dic3—121no—
361Dic3—121no—
362Dic3—121no—
363Dic3—181no—
364Dic3—241no—
365Dic3—361no—
366C32—21no—
367C32#480711yescharacteristic, complemented
368C32#504111yescharacteristic, complemented
369C23—91no—
370C23—181no—
371Q8—181no—
372Q8—181no—
373D4—181no—
374D4—181no—
375D4—362no—
376C2×C4—21no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—92no—
384C8—181no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—182no—
395C2×C3—721no—
396C2×C3#6388511yescharacteristic
397C2×C3D3×#21411yescharacteristic
398C2×C3#6394911yescharacteristic
399D3—62no—
400D3—182no—
401D3—241no—
402D3—721no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22#8867611yescharacteristic
410C4—41no—
411C4—181no—
412C4—361no—
413C4—361no—
414C4#8864811yescharacteristic
415C4#8865511yescharacteristic
416C3—31no—
417C3—31no—
418C3—61no—
419C3#1059051411yescharacteristic
420C2—182no—
421C2—721no—
422C2#1172112611yescharacteristic
423C2#1172119811yescharacteristic
424C2#1172123511yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64104⋊C3C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
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
2. #11721106C2semidirect
3. C2×#88431C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #11720530C2semidirect
6. #63637⋊C3C2semidirect
7. #11720308C2semidirect
31. #63637D3semidirect
52. #6180D4semidirect
92. C2×(C3⋊D3)⋊C3SD16semidirect
213. C2×C3⋊D3#214semidirect
243. H(3,𝔽3)#465semidirect
367. C32#4807semidirect
368. C32#5041semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
#117211062
C2×#884312
C2×((C3⋊D3)⋊C3)⋊C82
#117205302
#63637⋊C32
#117203082
#105905783
#105905233