Finite Group Databasebeta

← #13137948

Subgroups — #13137948

Order 1728.

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

normal subgroups only

This group has 449 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
1C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic
4#11720530C211yesmaximal, characteristic, complemented
5#63634⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590583—31nomaximal
9#10590581—31nomaximal
10C2×C4×(C3⋊D3)⋊C3—21no—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88447—21no—
21#88430C414yescomplemented
22C4⋊C4×C3⋊D3D311yescharacteristic
23C3×D3×C4⋊C4—31no—
24#63849—31no—
25#63841—31no—
26C2×#3712—31no—
27C2×#3711—31no—
28#63634D311yescharacteristic, complemented
29C3×#808—31no—
30C3×#808—31no—
31C3×#808—61no—
32C3×#807—31no—
33#63575—31no—
34#63575—31no—
35#63574—31no—
36#63573—31no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4×(C3⋊D3)⋊C3—42no—
45C4×(C3⋊D3)⋊C3C2×C412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47#6180C2×C412yescomplemented
48(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
49#4964—91no—
50#4881—91no—
51#4861—91no—
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—61no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—21no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×Dic3⋊C4—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
73#3722—61no—
74(C2×D3)⋊Dic3—61no—
75Dic32—61no—
76Dic32—61no—
77#3712—34no—
78#3711—34no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3C4⋊C412yes—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)C22×C411yescharacteristic
85C4×H(3,𝔽3)C2×Q811yescharacteristic
86#1084—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3C4⋊C412yes—
90C2×C3×C4⋊C4—91no—
91#926—91no—
92C2×#208—91no—
93C2×ΓL(1,𝔽25)—91no—
94C2×ΓL(1,𝔽25)—91no—
95#892—91no—
96#892—91no—
97D3×C4⋊C4—31no—
98D3×C4⋊C4—91no—
99C3×#107—91no—
100C3×#106—91no—
101#818—91no—
102#817—91no—
103#808—31no—
104#808—31no—
105#808—61no—
106#808—91no—
107#808—91no—
108#807—31no—
109#807—91no—
110#807—91no—
111#807—91no—
112C22×C3⋊D3#6711yescharacteristic
113C22×C3×D3—31no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3#6711yescharacteristic
123C4×C3⋊D3—42no—
124C4×C3⋊D3Dic612yes—
125C4×C3⋊D3C4×D312yes—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×C4×D3—32no—
133C3×C4×D3—32no—
134C3×C4×D3—122no—
135C3⋊(C3⋊C8)C4×D312yescomplemented
136C3×C3⋊C8—32no—
137C3×C3⋊C8—32no—
138C3×C3⋊C8—32no—
139C3×C3⋊C8—62no—
140#428—271noSylow
141C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
142C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—22no—
145C22×C3×C4—91no—
146C22×C3×C4—181no—
147C2×C4×D3—31no—
148C2×C4×D3—61no—
149C2×C4×D3—91no—
150C2×C4×D3—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4D3211yescharacteristic
161#217—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—61no—
169C4⋊Dic3—91no—
170C4⋊Dic3—91no—
171Dic3⋊C4—61no—
172Dic3⋊C4—181no—
173C4×Dic3—181no—
174C4×Dic3—181no—
175C4×Dic3—181no—
176#208—94no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—91no—
182ΓL(1,𝔽25)—94no—
183ΓL(1,𝔽25)—94no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×#6711yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3Dic3⋊C412yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—81no—
198C32×C4C2×Dic611yescharacteristic
199C32×C4C2×C4×D311yescharacteristic
200C32×C4C2×C4×D311yescharacteristic
201C32×C4D3×Q811yescharacteristic
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3⋊Dic3Dic3⋊C412yes—
205C3×Dic3—32no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—241no—
211C2×M16—271no—
212#117—271no—
213C2×C4⋊C4—91no—
214#107—91no—
215#107—271no—
216#106—91no—
217#106—271no—
218H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C2×C3×C4—21no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—362no—
236C2×C3×C4C2×D3211yescharacteristic
237C2×Dic3—31no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—91no—
243C2×Dic3—121no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C4×D3—32no—
247C4×D3—32no—
248C4×D3—92no—
249C4×D3—92no—
250C4×D3—122no—
251C4×D3—362no—
252C3×C8—92no—
253C3×C8—92no—
254C3⋊C8—32no—
255C3⋊C8—32no—
256C3⋊C8—32no—
257C3⋊C8—62no—
258C3⋊C8—92no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32D3×C4⋊C411yescharacteristic
263C2×C32#91111yescharacteristic
264C2×C32#91811yescharacteristic
265C2×C32C2×Dic3⋊C411yescharacteristic
266C2×C32#93211yescharacteristic
267C2×C32#94211yescharacteristic
268C3⋊D3—22no—
269C3×D3—62no—
270C22×C4—91no—
271C22×C4—181no—
272M16—274no—
273C2×C8—91no—
274C2×C8—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—181no—
279C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C22⋊C4—541no—
281C42—541no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—92no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3D3×#6711yescharacteristic
289C2×D3—32no—
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—92no—
293C2×D3—181no—
294C2×D3—181no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—41no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—92no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4—241no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4D3×Dic611yescharacteristic
310C3×C4C4×D3211yescharacteristic
311Dic3—32no—
312Dic3—92no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—241no—
318Dic3—361no—
319Dic3—361no—
320C32—21no—
321C32#488111yescharacteristic, complemented
322C32#496411yescharacteristic, complemented
323C23—91no—
324C2×C4—21no—
325C2×C4—91no—
326C2×C4—92no—
327C2×C4—92no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—362no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—92no—
334C8—92no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—182no—
345C2×C3D3×Dic3⋊C411yescharacteristic
346C2×C3#6393211yescharacteristic
347C2×C3#6394811yescharacteristic
348D3—62no—
349D3—182no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22#8867611yescharacteristic
354C4—41no—
355C4—92no—
356C4—361no—
357C4—361no—
358C4—361no—
359C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
360C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3#1059058011yescharacteristic
365C2—182no—
366C2#1172111911yescharacteristic
367C2#1172120011yescharacteristic
368C2#1172123411yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
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. #11720530C2semidirect
5. #63634⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720307C2semidirect
21. #88430C4semidirect
28. #63634D3semidirect
47. #6180C2×C4semidirect
48. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
135. C3⋊(C3⋊C8)C4×D3semidirect
218. H(3,𝔽3)#428semidirect
321. C32#4881semidirect
322. C32#4964semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×(C3⋊D3)⋊C32
#117210812
C2×#884302
#117205302
#63634⋊C32
#117203152
#117203072
#105905833
#105905813