Finite Group Databasebeta

← #13140535

Subgroups — #13140535

Order 1728.

470 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 95 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
1D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic
3#11721082C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic
5#11720313C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591446—31nomaximal
9#10591405—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3757⋊C3—21no—
16#3754⋊C3C414yescomplemented
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#88447—21no—
22D4×C3⋊Dic3D311yescharacteristic
23C2×#3754D311yescharacteristic
24C3×D4×Dic3—31no—
25C2×C3×#214—31no—
26#63844—31no—
27#63843—31no—
28C3×#832—31no—
29C3×#832—31no—
30C3×#832—61no—
31#63576—31no—
32#63575—31no—
33#63567—31no—
34#63566—31no—
35#63555—31no—
36#63554—31no—
37C23×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#5164—91no—
52#5018—91no—
53#4757—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C22×C3⋊Dic3—21no—
58C2×#634C2×D311yescharacteristic
59C22×C3×Dic3—61no—
60C2×C3×Dic6—31no—
61#3757—21no—
62#3754Dic314yescomplemented
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×#217—61no—
67C3×#214—34no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×C4×Dic3—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77#3724—61no—
78Dic3⋊Dic3—61no—
79Dic32—61no—
80Dic32—61no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—41no—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×SD16—91no—
93C3×C4×D4—91no—
94Q8×Dic3—91no—
95D4×Dic3—31no—
96D4×Dic3—91no—
97C2×#214—31no—
98C2×#214—91no—
99#887—91no—
100C4×Dic6—91no—
101C3×#102—91no—
102#835—91no—
103#832—31no—
104#832—31no—
105#832—61no—
106#818—91no—
107#816—91no—
108#815—91no—
109#815—91no—
110#810—91no—
111#807—91no—
112#807—91no—
113#803—91no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C32×D4—21no—
118C32×D4—21no—
119C32×D4—22no—
120C32×D4—41no—
121C32×D4C4×D312yes—
122C32×D4C2×Dic312yes—
123C2×C32×C4—21no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—42no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131#634—21no—
132#634C2×Dic312yescomplemented
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—121no—
140C2×C3×Dic3—122no—
141C3×Dic6—32no—
142C3×Dic6—61no—
143C3⋊(C3⋊C8)C2×Dic312yescomplemented
144C3×C3⋊C8—32no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—121no—
148#440—271noSylow
149C2×H(3,𝔽3)—42no—
150C2×H(3,𝔽3)C4×D411yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152C2×H(3,𝔽3)#13711yescharacteristic
153C2×C3×Q8—91no—
154C2×C3×D4—31no—
155C2×C3×D4—31no—
156C2×C3×D4—61no—
157C2×C3×D4D3211yescharacteristic
158C22×C3×C4—181no—
159C22×Dic3—61no—
160C22×Dic3—181no—
161C2×Dic6—31no—
162C2×Dic6—91no—
163C3×SD16—94no—
164C2×C3×C8—91no—
165C2×C3×C8—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C22⋊C4—181no—
169C3×C42—91no—
170#217—61no—
171#217—181no—
172#214—34no—
173#214—94no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C4⋊Dic3—91no—
179C4⋊Dic3—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182C4×Dic3—31no—
183C4×Dic3—91no—
184C4×Dic3—181no—
185C4×Dic3—181no—
186C4×Dic3—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—91no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C32×C4—21no—
205C32×C4—21no—
206C32×C4C2×C4×D311yescharacteristic
207C32×C4#23511yescharacteristic
208C32×C4#23711yescharacteristic
209C32×C4C22×Dic311yescharacteristic
210C3⋊Dic3—21no—
211C3⋊Dic3—22no—
212C3⋊Dic3—21no—
213C3⋊Dic3—41no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—62no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—241no—
221C2×SD16—91no—
222C4×Q8—271no—
223C4×D4—91no—
224#107—271no—
225#103—271no—
226#102—91no—
227#97—271no—
228H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
229C23×C3—21no—
230C23×C3—61no—
231C23×C3—61no—
232C23×C3—121no—
233C3×Q8—92no—
234C3×Q8—181no—
235C3×D4—21no—
236C3×D4—32no—
237C3×D4—32no—
238C3×D4—61no—
239C3×D4—61no—
240C3×D4—62no—
241C3×D4—121no—
242C3×D4D3×Dic312yes—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—61no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—362no—
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—91no—
260C2×Dic3—91no—
261C2×Dic3—121no—
262C2×Dic3—122no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265C2×Dic3—362no—
266Dic6—32no—
267Dic6—61no—
268Dic6—92no—
269Dic6—181no—
270C3×C8—92no—
271C3×C8—181no—
272C3⋊C8—32no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—92no—
276C3⋊C8—121no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32—42no—
282C2×C32—82no—
283C2×C32#91411yescharacteristic
284C2×C32#91511yescharacteristic
285C2×C32C4×#6711yescharacteristic
286C2×C32#93211yescharacteristic
287C2×C32D4×Dic311yescharacteristic
288C2×C32#95111yescharacteristic
289C2×Q8—91no—
290C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C22×C4—181no—
292SD16—94no—
293C2×C8—91no—
294C2×C8—91no—
295C4⋊C4—91no—
296C4⋊C4—91no—
297C4⋊C4—541no—
298C22⋊C4—181no—
299C42—91no—
300C42—541no—
301C22×C3—22no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—122no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3—241no—
315C22×C3—241no—
316C22×C3D3×#6711yescharacteristic
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—181no—
324C3×C4—181no—
325C3×C4—182no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4C2×D3×Dic311yescharacteristic
329C3×C4#380411yescharacteristic
330Dic3—61no—
331Dic3—61no—
332Dic3—62no—
333Dic3—121no—
334Dic3—121no—
335Dic3—121no—
336Dic3—182no—
337Dic3—181no—
338Dic3—181no—
339Dic3—241no—
340Dic3—361no—
341C32—21no—
342C32#501811yescharacteristic, complemented
343C32#516411yescharacteristic, complemented
344C23—21no—
345Q8—92no—
346Q8—181no—
347D4—21no—
348D4((C3⋊D3)⋊C3)⋊C412yes—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—362no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—92no—
357C8—181no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—42no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—122no—
369C2×C3—122no—
370C2×C3—242no—
371C2×C3#6392911yescharacteristic
372C2×C3#6393411yescharacteristic
373C2×C3Dic3×#6711yescharacteristic
374C22—22no—
375C22—41no—
376C22—41no—
377C22#8867611yescharacteristic
378C4—182no—
379C4—181no—
380C4—181no—
381C4—361no—
382C4—361no—
383C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
384C4#8867111yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059144011yescharacteristic
389C2—42no—
390C2#1172119611yescharacteristic
391C2#1172119911yescharacteristic
392C2#1172131411yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721082C2semidirect
5. #11720313C2semidirect
6. #11720312C2semidirect
7. #11720306C2semidirect
16. #3754⋊C3C4semidirect
48. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
62. #3754Dic3semidirect
132. #634C2×Dic3semidirect
143. C3⋊(C3⋊C8)C2×Dic3semidirect
228. H(3,𝔽3)#440semidirect
342. C32#5018semidirect
343. C32#5164semidirect

Maximal subgroups

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

subgroupindex
D4×(C3⋊Dic3)⋊C32
C2×#3754⋊C32
#117210822
#117205782
#117203132
#117203122
#117203062
#105914463
#105914053