Finite Group Databasebeta

← #13137385

Subgroups — #13137385

Order 1728.

430 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 79 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#64084⋊C3C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88433C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic
5C4⋊#6180C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic, complemented
8#10590378—31nomaximal
9#10590368—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#88433—22no—
22#88431—22no—
23#64084D311yescharacteristic
24C4×#634D311yescharacteristic, complemented
25C3×#875—31no—
26C3×C4×Dic6—31no—
27C2×#3719—31no—
28C2×#3719—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63567—31no—
35#63566—31no—
36#63564—31no—
37#63563—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×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51#6180—21no—
52#6180—21no—
53#5054—91no—
54#5028—91no—
55#4845—91no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C32×C42—21no—
63C32×C42C2×D311yescharacteristic
64C32×C42C2×D311yescharacteristic
65#3752—21no—
66#3752—21no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80C2×C3×C3⋊C8—61no—
81#3719—62no—
82#3719—62no—
83#3717—62no—
84#3716—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—21no—
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C3×#124—91no—
97C3×C4×Q8—91no—
98C2×#216—91no—
99C2×#216—91no—
100C2×#215—91no—
101C2×#214—91no—
102C2×Dic12—91no—
103C2×#204—91no—
104#875—31no—
105#875—91no—
106C4×Dic6—31no—
107C4×Dic6—91no—
108C3×C4⋊C8—91no—
109#835—91no—
110#832—91no—
111#821—91no—
112#816—91no—
113#810—91no—
114#809—91no—
115C4⋊(C3⋊C8)—31no—
116C4⋊(C3⋊C8)—31no—
117C4⋊(C3⋊C8)—61no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4#6711yescharacteristic
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134#634—21no—
135#634—21no—
136#634—21no—
137#634D1212yes—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×Dic6—32no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—121no—
152C3×C3⋊C8—121no—
153#464—271noSylow
154C2×H(3,𝔽3)#12111yescharacteristic
155C2×H(3,𝔽3)#13511yescharacteristic
156C2×H(3,𝔽3)#13711yescharacteristic
157(C3⋊D3)⋊C3—81no—
158C2×C3×Q8—91no—
159C2×C3×Q8—91no—
160C2×C3×D4—91no—
161C2×D12—31no—
162C2×D12—91no—
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×Dic6—91no—
166C2×Dic6—91no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—181no—
171C3×C22⋊C4—181no—
172C3×C22⋊C4—181no—
173C3×C42—31no—
174C3×C42—31no—
175C3×C42—61no—
176C3×C42—181no—
177C3×C42D3211yescharacteristic
178#216—182no—
179#216—182no—
180#215—182no—
181#214—182no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—61no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—31no—
187C4⋊Dic3—91no—
188Dic3⋊C4—61no—
189Dic3⋊C4—181no—
190C4×Dic3—61no—
191C4×Dic3—181no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—61no—
198Dic12—182no—
199#204—182no—
200C22×C32—21no—
201C22×C32C2×#6711yescharacteristic
202C22×C32C2×#6711yescharacteristic
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—41no—
217C32×C4C2×D1211yescharacteristic
218C32×C4#23511yescharacteristic
219C32×C4D3×D411yescharacteristic
220C32×C4#23711yescharacteristic
221C3⋊Dic3—22no—
222C3⋊Dic3—41no—
223C3⋊Dic3—41no—
224C3×Dic3—62no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C2×Q16—271no—
228C2×SD16—271no—
229#124—91no—
230C4×Q8—91no—
231C4⋊C8—91no—
232#103—271no—
233#102—271no—
234H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
235C23×C3—181no—
236C22×D3—61no—
237C22×D3—181no—
238C3×Q8—92no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4#62511yescharacteristic
257C2×C3×C4#62511yescharacteristic
258C2×C3×C4C2×D3211yescharacteristic
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265D12—61no—
266D12—61no—
267D12—181no—
268D12—181no—
269Dic6—32no—
270Dic6—61no—
271Dic6—61no—
272Dic6—61no—
273Dic6—92no—
274Dic6—181no—
275Dic6—181no—
276Dic6—181no—
277C3×C8—181no—
278C3×C8—181no—
279C3⋊C8—61no—
280C3⋊C8—61no—
281C3⋊C8—61no—
282C3⋊C8—61no—
283C3⋊C8—121no—
284C3⋊C8—121no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32#93011yescharacteristic
289C2×C32#93811yescharacteristic
290C2×C32#94211yescharacteristic
291C2×C32#95011yescharacteristic
292C2×C32#95011yescharacteristic
293C2×C32#95111yescharacteristic
294C3⋊D3—81no—
295C3×D3—241no—
296C2×Q8—91no—
297C2×Q8—91no—
298C2×D4—91no—
299Q16—542no—
300SD16—542no—
301C2×C8—91no—
302C2×C8—91no—
303C4⋊C4—91no—
304C4⋊C4—181no—
305C22⋊C4—181no—
306C22⋊C4—181no—
307C42—181no—
308C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3C2×#62511yescharacteristic
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—361no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—62no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—121no—
336C3×C4—182no—
337C3×C4—361no—
338C3×C4—361no—
339C3×C4#379511yescharacteristic
340C3×C4D3×D1211yescharacteristic
341Dic3—62no—
342Dic3—121no—
343Dic3—121no—
344Dic3—182no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32#502811yescharacteristic, complemented
349C32#505411yescharacteristic, complemented
350C23—181no—
351Q8—92no—
352Q8—181no—
353Q8—181no—
354Q8—181no—
355D4—181no—
356D4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4#619811yescharacteristic
361C2×C4#619811yescharacteristic
362C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
363C8—181no—
364C8—181no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—721no—
375C2×C3#6382211yescharacteristic
376C2×C3#6382711yescharacteristic
377C2×C3#6390811yescharacteristic
378D3—241no—
379D3—721no—
380C22—361no—
381C22—361no—
382C22—361no—
383C22C2×#619811yescharacteristic
384C4—21no—
385C4—21no—
386C4—182no—
387C4—361no—
388C4—361no—
389C4#8864811yescharacteristic
390C4#8865411yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3#1059026711yescharacteristic
395C2—721no—
396C2#1172105811yescharacteristic
397C2#1172106011yescharacteristic
398C2#1172114111yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
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
2. C4×#634⋊C3C2semidirect
3. C2×#88433C2semidirect
5. C4⋊#6180C2semidirect
7. #11720313C2semidirect
24. C4×#634D3semidirect
234. H(3,𝔽3)#464semidirect
348. C32#5028semidirect
349. C32#5054semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
C4×#634⋊C32
C2×#884332
C2×#884312
C4⋊#61802
#117203182
#117203132
#105903783
#105903683