Finite Group Databasebeta

← #13137965

Subgroups — #13137965

Order 1728.

505 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 87 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×#272C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#63634⋊C3C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590581—31nomaximal
9#10590580—31nomaximal
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88447—21no—
21#88435C414yescomplemented
22C3×D3×C4⋊C4—31no—
23C3×D3×C4⋊C4—31no—
24C3×D3×C4⋊C4—61no—
25#63841—31no—
26#63841—31no—
27C2×#3711—31no—
28C2×#3711—31no—
29#63635D311yescharacteristic, complemented
30#63634D311yescharacteristic, complemented
31C3×#808—31no—
32C3×#807—31no—
33#63576—31no—
34#63575—31no—
35#63574—31no—
36#63573—31no—
37C22×#272D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272—42no—
43C4×#272C2×C412yes—
44C4×#272Q812yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
48(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
49#4964—91no—
50#4881—91no—
51#4861—91no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—62no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—62no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77Dic32—61no—
78Dic32—61no—
79#3711—34no—
80#3711—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×#272—21no—
83C2×#272—21no—
84C2×#272C4⋊C412yes—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)C2×Q811yescharacteristic
88#1084—41no—
89#1084C4⋊C412yes—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×C4⋊C4—91no—
93#926—91no—
94C2×#208—91no—
95C2×ΓL(1,𝔽25)—91no—
96C2×ΓL(1,𝔽25)—91no—
97#892—91no—
98#892—91no—
99D3×C4⋊C4—31no—
100D3×C4⋊C4—31no—
101D3×C4⋊C4—61no—
102C3×#107—91no—
103C3×#106—91no—
104#818—91no—
105#817—91no—
106#808—31no—
107#808—91no—
108#808—91no—
109#808—91no—
110#807—31no—
111#807—91no—
112#807—91no—
113#807—91no—
114C22×C3×D3—31no—
115C22×C3×D3—31no—
116C22×C3×D3—61no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×C4×D3—32no—
134C3×C4×D3—32no—
135C3×C4×D3—32no—
136C3×C4×D3—32no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—124no—
140C3×C4×D3—122no—
141C3×C4×D3—122no—
142C3⋊(C3⋊C8)C4×D312yescomplemented
143C3⋊(C3⋊C8)C4×D312yescomplemented
144C3×C3⋊C8—32no—
145C3×C3⋊C8—32no—
146#428—271noSylow
147C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150#272—22no—
151C22×C3×C4—91no—
152C22×C3×C4—181no—
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×C3×C8—91no—
160C2×C3×C8—91no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—181no—
168C3×C4⋊C4D3211yescharacteristic
169#217—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—61no—
177C4⋊Dic3—91no—
178C4⋊Dic3—91no—
179Dic3⋊C4—61no—
180Dic3⋊C4—61no—
181Dic3⋊C4—62no—
182C4×Dic3—181no—
183C4×Dic3—181no—
184C4×Dic3—181no—
185#208—94no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190ΓL(1,𝔽25)—94no—
191ΓL(1,𝔽25)—94no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3×D3—32no—
196C2×C3×D3—32no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—42no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4C2×C4×D311yescharacteristic
211C32×C4D3×Q811yescharacteristic
212C32×C4D3×Q811yescharacteristic
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3×Dic3—32no—
216C3×Dic3—32no—
217C3×Dic3—62no—
218C3×Dic3—122no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C2×M16—271no—
224#117—271no—
225C2×C4⋊C4—91no—
226#107—91no—
227#107—271no—
228#106—91no—
229#106—271no—
230H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
231C23×C3—91no—
232C22×D3—31no—
233C22×D3—31no—
234C22×D3—61no—
235C2×C3×C4—21no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—362no—
249C2×C3×C4C2×D3211yescharacteristic
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—181no—
259C2×Dic3—181no—
260C4×D3—32no—
261C4×D3—32no—
262C4×D3—32no—
263C4×D3—32no—
264C4×D3—62no—
265C4×D3—62no—
266C4×D3—122no—
267C4×D3—122no—
268C4×D3—124no—
269C3×C8—92no—
270C3×C8—92no—
271C3⋊C8—32no—
272C3⋊C8—32no—
273C3⋊C8—92no—
274C3⋊C8—92no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32D3×C4⋊C411yescharacteristic
279C2×C32D3×C4⋊C411yescharacteristic
280C2×C32#91111yescharacteristic
281C2×C32#91411yescharacteristic
282C2×C32#91511yescharacteristic
283C2×C32#91811yescharacteristic
284C3×D3—64no—
285C3×D3—62no—
286C3×D3—62no—
287C22×C4—91no—
288C22×C4—181no—
289M16—274no—
290C2×C8—91no—
291C2×C8—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4—181no—
296C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C22⋊C4—541no—
298C42—541no—
299C22×C3—31no—
300C22×C3—31no—
301C22×C3—61no—
302C22×C3—92no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3#381111yescharacteristic
306C2×D3—32no—
307C2×D3—32no—
308C2×D3—62no—
309C2×D3—61no—
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—62no—
314C2×D3—62no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—41no—
320C3×C4—61no—
321C3×C4—61no—
322C3×C4—92no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—122no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4—361no—
329C3×C4#379711yescharacteristic
330C3×C4C4×D3211yescharacteristic
331Dic3—32no—
332Dic3—32no—
333Dic3—62no—
334Dic3—121no—
335Dic3—121no—
336Dic3—122no—
337Dic3—121no—
338Dic3—121no—
339Dic3—361no—
340Dic3—361no—
341C32—21no—
342C32#486111yescharacteristic, complemented
343C32#488111yescharacteristic, complemented
344C23—91no—
345C2×C4—21no—
346C2×C4—91no—
347C2×C4—92no—
348C2×C4—92no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4—181no—
352C2×C4—362no—
353C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C8—92no—
355C8—92no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—182no—
366C2×C3#6388311yescharacteristic
367C2×C3#6393711yescharacteristic
368C2×C3#6395111yescharacteristic
369D3—62no—
370D3—62no—
371D3—64no—
372C22—92no—
373C22—181no—
374C22—181no—
375C22#8867711yescharacteristic
376C4—41no—
377C4—92no—
378C4—361no—
379C4—361no—
380C4—361no—
381C4#8865111yescharacteristic
382C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C3—31no—
384C3—31no—
385C3—61no—
386C3#1059058311yescharacteristic
387C2—182no—
388C2#1172114711yescharacteristic
389C2#1172119311yescharacteristic
390C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#428
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63635⋊C3C2semidirect
5. #63634⋊C3C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
21. #88435C4semidirect
29. #63635D3semidirect
30. #63634D3semidirect
47. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
48. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
142. C3⋊(C3⋊C8)C4×D3semidirect
143. C3⋊(C3⋊C8)C4×D3semidirect
230. H(3,𝔽3)#428semidirect
342. C32#4861semidirect
343. C32#4881semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#117210892
C2×#884352
#63635⋊C32
#63634⋊C32
#117203072
#117203062
#105905813
#105905803