Finite Group Databasebeta

← #13137432

Subgroups — #13137432

Order 1728.

462 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C4×#6229C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10590368—31nomaximal
9#10590267—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4×#1084—21no—
17#3752⋊C3—21no—
18#3752⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88439—22no—
22#88438—22no—
23#64084D311yescharacteristic
24C3×#875—31no—
25C3×C4×Dic6—31no—
26C3×C4×Dic6—31no—
27C3×C4×Dic6—61no—
28C2×#3719—31no—
29C2×#3718—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
33C3×C4⋊(C3⋊C8)—31no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084—21no—
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#5054—91no—
54#5028—91no—
55#4845—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×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C2×D311yescharacteristic
66#3752—21no—
67#3752—21no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×Dic3⋊C4—62no—
76C3×Dic3⋊C4—61no—
77C3×Dic3⋊C4—61no—
78C3×C4×Dic3—62no—
79C3×C4×Dic3—61no—
80C3×C4×Dic3—61no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
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)—21no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—22no—
96#1084—41no—
97(C3⋊Dic3)⋊C3—41no—
98C3×#124—91no—
99C3×C4×Q8—91no—
100C2×#216—91no—
101C2×#216—91no—
102C2×#215—91no—
103C2×#214—91no—
104C2×Dic12—91no—
105C2×#204—91no—
106#875—31no—
107#875—91no—
108C4×Dic6—31no—
109C4×Dic6—31no—
110C4×Dic6—61no—
111C3×C4⋊C8—91no—
112#835—91no—
113#832—91no—
114#821—91no—
115#816—91no—
116#810—91no—
117#809—91no—
118C4⋊(C3⋊C8)—31no—
119C4⋊(C3⋊C8)—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4D1211yescharacteristic
126C2×C32×C4D1211yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134#634—21no—
135#634—21no—
136C2×C3×Dic3—62no—
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—91no—
175C3×C4⋊C4—181no—
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178C3×C42—31no—
179C3×C42—31no—
180C3×C42—61no—
181C3×C42—181no—
182C3×C42D3211yescharacteristic
183#216—182no—
184#216—182no—
185#215—182no—
186#214—182no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—61no—
194Dic3⋊C4—61no—
195Dic3⋊C4—61no—
196Dic3⋊C4—62no—
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×C32C2×D1211yescharacteristic
208C22×C32C2×#6711yescharacteristic
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—22no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4D3×D411yescharacteristic
224C32×C4D3×D411yescharacteristic
225C32×C4#23711yescharacteristic
226C32×C4#23911yescharacteristic
227C3⋊Dic3—41no—
228C3×Dic3—64no—
229C3×Dic3—62no—
230C3×Dic3—62no—
231C3×Dic3—122no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C2×Q16—271no—
236C2×SD16—271no—
237#124—91no—
238C4×Q8—91no—
239C4⋊C8—91no—
240#103—271no—
241#102—271no—
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—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4#62611yescharacteristic
265C2×C3×C4#62611yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—62no—
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#89511yescharacteristic
299C2×C32#90411yescharacteristic
300C2×C32#90911yescharacteristic
301C2×C32#93811yescharacteristic
302C2×C32#95011yescharacteristic
303C2×C32#95111yescharacteristic
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—91no—
314C4⋊C4—181no—
315C22⋊C4—181no—
316C22⋊C4—181no—
317C42—181no—
318C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3C2×#62611yescharacteristic
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—361no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—62no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—121no—
346C3×C4—182no—
347C3×C4—361no—
348C3×C4—361no—
349C3×C4#379611yescharacteristic
350C3×C4#380211yescharacteristic
351Dic3—62no—
352Dic3—62no—
353Dic3—64no—
354Dic3—121no—
355Dic3—121no—
356Dic3—122no—
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#484511yescharacteristic, complemented
361C32#505411yescharacteristic, complemented
362C23—181no—
363Q8—92no—
364Q8—181no—
365Q8—181no—
366Q8—181no—
367D4—181no—
368D4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4#620011yescharacteristic
373C2×C4#620011yescharacteristic
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#6382911yescharacteristic
388C2×C3#6383111yescharacteristic
389C2×C3#6391611yescharacteristic
390D3—241no—
391D3—721no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22C2×#620011yescharacteristic
396C4—21no—
397C4—21no—
398C4—182no—
399C4—361no—
400C4—361no—
401C4#8864911yescharacteristic
402C4#8865511yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3#1059037811yescharacteristic
407C2—721no—
408C2#1172107311yescharacteristic
409C2#1172107411yescharacteristic
410C2#1172115911yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6229C2semidirect
3. C2×#88439C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720309C2semidirect
32. C4⋊(C3⋊(C3⋊C8))D3semidirect
242. H(3,𝔽3)#464semidirect
360. C32#4845semidirect
361. C32#5054semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
#64084⋊C32
C2×#884392
C2×#884382
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#117203082
#105903683
#105902673