Finite Group Databasebeta

← #13137384

Subgroups — #13137384

Order 1728.

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

normal subgroups only

This group has 460 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×#88431C211yesmaximal, characteristic
4#11720520C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590362—31nomaximal
9#10590360—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#88431C414yescomplemented
22C4×C4⋊(C3⋊D3)D311yescharacteristic
23C4×#634D311yescharacteristic, complemented
24C3×C4×D12—31no—
25C3×C4×Dic6—31no—
26C2×#3717—31no—
27C2×#3716—31no—
28C3×#803—31no—
29C3×#803—31no—
30C3×#803—61no—
31#63575—31no—
32#63575—31no—
33#63567—31no—
34#63566—31no—
35#63564—31no—
36#63563—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×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yes—
46C4×(C3⋊D3)⋊C3—42no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49#6180—21no—
50#6180C2×C412yescomplemented
51#5026—91no—
52#5018—91no—
53#4710—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—31no—
60C32×C42—21no—
61C32×C42C2×D311yescharacteristic
62C32×C42C2×D311yescharacteristic
63#3752—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C3×(C2×D3)⋊C4—61no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—61no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78C2×C3×C3⋊C8—61no—
79#3717—34no—
80#3716—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—21no—
86C4×H(3,𝔽3)—21no—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94C2×#215—91no—
95C2×#214—91no—
96C2×#204—91no—
97C4×D12—31no—
98C4×D12—91no—
99C4×Dic6—31no—
100C4×Dic6—91no—
101C3×#97—91no—
102#835—91no—
103#832—91no—
104#821—91no—
105#818—91no—
106#816—91no—
107#810—91no—
108#809—91no—
109#807—91no—
110#807—91no—
111#803—31no—
112#803—31no—
113#803—61no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C4×D312yes—
130C4×C3⋊D3—42no—
131#634—21no—
132#634C4×D312yescomplemented
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—32no—
137C3×D12—61no—
138C3×C4×D3—122no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141C3×C3⋊C8—32no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#440—271noSylow
148C2×H(3,𝔽3)C4×D411yescharacteristic
149C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
150C2×H(3,𝔽3)#13711yescharacteristic
151(C3⋊D3)⋊C3—42no—
152C2×C3×Q8—91no—
153C2×C3×D4—91no—
154C22×C3×C4—181no—
155C2×D12—31no—
156C2×D12—91no—
157C2×C4×D3—61no—
158C2×C4×D3—181no—
159C2×Dic6—31no—
160C2×Dic6—91no—
161C2×C3×C8—91no—
162C2×C3×C8—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C42—31no—
168C3×C42—31no—
169C3×C42—61no—
170C3×C42—181no—
171C3×C42D3211yescharacteristic
172#215—94no—
173#214—94no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—91no—
179C4⋊Dic3—91no—
180Dic3⋊C4—61no—
181Dic3⋊C4—181no—
182C4×Dic3—61no—
183C4×Dic3—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190#204—94no—
191C22×C32—21no—
192C22×C32C2×#6711yescharacteristic
193C22×C32C2×#6711yescharacteristic
194C2×C3⋊D3—22no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3×D3—62no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4C2×C4×D311yescharacteristic
210C32×C4#23511yescharacteristic
211C32×C4#23511yescharacteristic
212C3⋊Dic3—22no—
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3×Dic3—62no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C2×SD16—271no—
219C4×Q8—91no—
220C4×D4—91no—
221#107—271no—
222#103—271no—
223#102—271no—
224#97—91no—
225H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
226C23×C3—181no—
227C22×D3—61no—
228C22×D3—181no—
229C3×Q8—92no—
230C3×Q8—181no—
231C3×D4—92no—
232C3×D4—181no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—362no—
246C2×C3×C4#62511yescharacteristic
247C2×C3×C4#62511yescharacteristic
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255D12—32no—
256D12—61no—
257D12—92no—
258D12—181no—
259C4×D3—122no—
260C4×D3—362no—
261Dic6—32no—
262Dic6—61no—
263Dic6—92no—
264Dic6—181no—
265C3×C8—92no—
266C3×C8—181no—
267C3⋊C8—32no—
268C3⋊C8—32no—
269C3⋊C8—62no—
270C3⋊C8—61no—
271C3⋊C8—61no—
272C3⋊C8—62no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32C4×#6711yescharacteristic
277C2×C32C4×#6711yescharacteristic
278C2×C32#93211yescharacteristic
279C2×C32#94211yescharacteristic
280C2×C32#94911yescharacteristic
281C2×C32#95111yescharacteristic
282C3⋊D3—42no—
283C3×D3—122no—
284C2×Q8—91no—
285C2×D4—91no—
286C22×C4—181no—
287SD16—274no—
288C2×C8—91no—
289C2×C8—91no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—181no—
293C22⋊C4—181no—
294C42—181no—
295C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—182no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3C2×#62511yescharacteristic
303C2×D3—62no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—182no—
307C2×D3—361no—
308C2×D3—361no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—121no—
322C3×C4—121no—
323C3×C4—182no—
324C3×C4—361no—
325C3×C4—361no—
326C3×C4#379811yescharacteristic
327C3×C4C4×D3211yescharacteristic
328Dic3—62no—
329Dic3—121no—
330Dic3—121no—
331Dic3—182no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#501811yescharacteristic, complemented
336C32#502611yescharacteristic, complemented
337C23—181no—
338Q8—92no—
339Q8—181no—
340D4—92no—
341D4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4#619811yescharacteristic
347C2×C4#619811yescharacteristic
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C8—92no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—362no—
361C2×C3#6382311yescharacteristic
362C2×C3#6382711yescharacteristic
363C2×C3C4×#62511yescharacteristic
364D3—122no—
365D3—362no—
366C22—182no—
367C22—361no—
368C22—361no—
369C22C2×#619811yescharacteristic
370C4—21no—
371C4—21no—
372C4—182no—
373C4—361no—
374C4—361no—
375C4#8865011yescharacteristic
376C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3#1059035811yescharacteristic
381C2—362no—
382C2#1172105811yescharacteristic
383C2#1172106111yescharacteristic
384C2C4×#619811yescharacteristic
subgroupG/H
centerC22C2×#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
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
2. C4×#634⋊C3C2semidirect
4. #11720520C2semidirect
6. #11720315C2semidirect
7. #11720313C2semidirect
21. #88431C4semidirect
23. C4×#634D3semidirect
48. #634⋊C3C2×C4semidirect
50. #6180C2×C4semidirect
132. #634C4×D3semidirect
225. H(3,𝔽3)#440semidirect
335. C32#5018semidirect
336. C32#5026semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
C2×#884312
#117205202
#117203182
#117203152
#117203132
#105903623
#105903603