Finite Group Databasebeta

← #13137434

Subgroups — #13137434

Order 1728.

490 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C2×#88438C211yesmaximal, characteristic
4#63630⋊C3C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic, complemented
8#10590362—31nomaximal
9#10590358—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4×#1084—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88438C414yescomplemented
22C4×C4⋊(C3⋊D3)D311yescharacteristic
23C3×C4×D12—31no—
24C3×C4×Dic6—31no—
25C3×C4×Dic6—31no—
26C3×C4×Dic6—61no—
27C2×#3717—31no—
28C2×#3715—31no—
29#63630D311yescharacteristic, complemented
30C3×#803—31no—
31#63576—31no—
32#63575—31no—
33#63566—31no—
34#63565—31no—
35#63564—31no—
36#63562—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×#1084—21no—
43#6229—21no—
44#6229C2×C412yescomplemented
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48C4×(C3⋊D3)⋊C3—42no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#5026—91no—
52#5018—91no—
53#4710—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C3×D12—31no—
57C2×C3×C4×D3—61no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—61no—
61C32×C42—21no—
62C32×C42C2×D311yescharacteristic
63C32×C42C2×D311yescharacteristic
64#3752—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×(C2×D3)⋊C4—61no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×Dic3⋊C4—62no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—62no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81#3717—34no—
82#3715—34no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)—21no—
89C4×H(3,𝔽3)C22×C411yescharacteristic
90C4×H(3,𝔽3)#4111yescharacteristic
91#1084—22no—
92#1084—41no—
93(C3⋊Dic3)⋊C3—41no—
94C3×C4×Q8—91no—
95C3×C4×D4—91no—
96C2×#215—91no—
97C2×#214—91no—
98C2×#204—91no—
99C4×D12—31no—
100C4×D12—91no—
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×Dic6—61no—
104C3×#97—91no—
105#835—91no—
106#832—91no—
107#821—91no—
108#818—91no—
109#816—91no—
110#810—91no—
111#809—91no—
112#807—91no—
113#807—91no—
114#803—31no—
115#803—91no—
116C22×C3⋊D3—21no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4D1211yescharacteristic
122C2×C32×C4D1211yescharacteristic
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C4×D312yes—
130C4×C3⋊D3—42no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C3×D12—32no—
139C3×D12—61no—
140C3×C4×D3—122no—
141C3×Dic6—32no—
142C3×Dic6—32no—
143C3×Dic6—62no—
144C3×Dic6—62no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)C4×D312yescomplemented
149C3×C3⋊C8—32no—
150C3×C3⋊C8—61no—
151#440—271noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)#13711yescharacteristic
155(C3⋊D3)⋊C3—42no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C22×C3×C4—181no—
159C2×D12—31no—
160C2×D12—91no—
161C2×C4×D3—61no—
162C2×C4×D3—181no—
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×Dic6—61no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—181no—
171C3×C22⋊C4—181no—
172C3×C42—31no—
173C3×C42—31no—
174C3×C42—61no—
175C3×C42—181no—
176C3×C42D3211yescharacteristic
177#215—94no—
178#214—94no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—61no—
185C4⋊Dic3—91no—
186Dic3⋊C4—61no—
187Dic3⋊C4—61no—
188Dic3⋊C4—62no—
189C4×Dic3—61no—
190C4×Dic3—61no—
191C4×Dic3—62no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—91no—
196#204—94no—
197C22×C32—21no—
198C22×C32C2×D1211yescharacteristic
199C22×C32C2×#6711yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—62no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—22no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—41no—
214C32×C4C2×C4×D311yescharacteristic
215C32×C4C2×C4×D311yescharacteristic
216C32×C4#23511yescharacteristic
217C32×C4#23511yescharacteristic
218C3⋊Dic3—41no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C3×Dic3—62no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C2×SD16—271no—
227C4×Q8—91no—
228C4×D4—91no—
229#107—271no—
230#103—271no—
231#102—271no—
232#97—91no—
233H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
234C23×C3—181no—
235C22×D3—61no—
236C22×D3—181no—
237C3×Q8—92no—
238C3×Q8—181no—
239C3×D4—92no—
240C3×D4—181no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—362no—
254C2×C3×C4#62611yescharacteristic
255C2×C3×C4#62611yescharacteristic
256C2×C3×C4C2×D3211yescharacteristic
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—62no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—62no—
263C2×Dic3—61no—
264C2×Dic3—181no—
265D12—32no—
266D12—61no—
267D12—92no—
268D12—181no—
269C4×D3—122no—
270C4×D3—362no—
271Dic6—32no—
272Dic6—32no—
273Dic6—62no—
274Dic6—61no—
275Dic6—61no—
276Dic6—62no—
277C3×C8—92no—
278C3×C8—181no—
279C3⋊C8—32no—
280C3⋊C8—61no—
281C3⋊C8—92no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32C4×D1211yescharacteristic
287C2×C32#90811yescharacteristic
288C2×C32#90911yescharacteristic
289C2×C32C4×#6711yescharacteristic
290C2×C32#93211yescharacteristic
291C2×C32#95111yescharacteristic
292C3⋊D3—42no—
293C3×D3—122no—
294C2×Q8—91no—
295C2×D4—91no—
296C22×C4—181no—
297SD16—274no—
298C2×C8—91no—
299C2×C8—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C4⋊C4—181no—
303C22⋊C4—181no—
304C42—181no—
305C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—61no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3C2×#62611yescharacteristic
313C2×D3—62no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—182no—
317C2×D3—361no—
318C2×D3—361no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—62no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—121no—
333C3×C4—182no—
334C3×C4—361no—
335C3×C4—361no—
336C3×C4#379811yescharacteristic
337C3×C4C4×D3211yescharacteristic
338Dic3—62no—
339Dic3—62no—
340Dic3—64no—
341Dic3—121no—
342Dic3—121no—
343Dic3—122no—
344Dic3—121no—
345Dic3—361no—
346C32—21no—
347C32#471011yescharacteristic, complemented
348C32#501811yescharacteristic, complemented
349C23—181no—
350Q8—92no—
351Q8—181no—
352D4—92no—
353D4—181no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—362no—
358C2×C4#620011yescharacteristic
359C2×C4#620011yescharacteristic
360C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C8—92no—
362C8—181no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—362no—
373C2×C3#6383011yescharacteristic
374C2×C3#6383111yescharacteristic
375C2×C3C4×#62611yescharacteristic
376D3—122no—
377D3—362no—
378C22—182no—
379C22—361no—
380C22—361no—
381C22C2×#620011yescharacteristic
382C4—21no—
383C4—21no—
384C4—182no—
385C4—361no—
386C4—361no—
387C4#8865011yescharacteristic
388C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3#1059036011yescharacteristic
393C2—362no—
394C2#1172107111yescharacteristic
395C2#1172107411yescharacteristic
396C2C4×#620011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ 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
4. #63630⋊C3C2semidirect
5. #11720326C2semidirect
7. #11720306C2semidirect
21. #88438C4semidirect
29. #63630D3semidirect
44. #6229C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
148. C3⋊(C3⋊C8)C4×D3semidirect
233. H(3,𝔽3)#440semidirect
347. C32#4710semidirect
348. C32#5018semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884382
#63630⋊C32
#117203262
#117203082
#117203062
#105903623
#105903583