Finite Group Databasebeta

← #13137944

Subgroups — #13137944

Order 1728.

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

normal subgroups only

This group has 435 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
1#64101⋊C3C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720529C211yesmaximal, characteristic, complemented
5#63635⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590574—31nomaximal
9#10590571—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88447—21no—
21((C3⋊D3)⋊C3)⋊C8C414yescomplemented
22#64101D311yescharacteristic
23C3×#892—31no—
24#63849—31no—
25#63841—31no—
26C2×#3710—31no—
27C2×D3×C3⋊C8—31no—
28#63635D311yescharacteristic, complemented
29C3×#808—31no—
30C3×#807—31no—
31C3×#807—31no—
32C3×#807—61no—
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)⋊C3C2×C412yes—
45C4×(C3⋊D3)⋊C3Q812yes—
46#6180C2×C412yes—
47(C3⋊(C3⋊C8))⋊C3C2×C412yes—
48#4998—91no—
49#4880—91no—
50#4860—91no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—31no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—21no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4×C3⋊Dic3—21no—
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—61no—
72#3722—61no—
73(C2×D3)⋊Dic3—61no—
74Dic32—61no—
75Dic32—61no—
76#3710—34no—
77D3×C3⋊C8—34no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3C4⋊C412yes—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)C2×Q811yescharacteristic
85#1084—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3C4⋊C412yes—
89C3×#117—91no—
90#926—91no—
91C22×C3⋊C8—91no—
92C2×D3×C8—91no—
93C2×D3×C8—91no—
94#892—31no—
95#892—91no—
96#892—91no—
97#892—91no—
98C3×#107—91no—
99C3×#106—91no—
100#818—91no—
101#817—91no—
102#808—31no—
103#808—91no—
104#808—91no—
105#808—91no—
106#807—31no—
107#807—31no—
108#807—61no—
109#807—91no—
110#807—91no—
111C22×C3⋊D3#6711yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3#6711yescharacteristic
122C4×C3⋊D3Dic612yes—
123C4×C3⋊D3C4×D312yes—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130C3×C4×D3—32no—
131C3×C4×D3—32no—
132C3⋊(C3⋊C8)C4×D312yes—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—32no—
135C3×C3⋊C8—32no—
136C3×C3⋊C8—62no—
137#427—271noSylow
138C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)#13511yescharacteristic
141(C3⋊D3)⋊C3—22no—
142C22×C3×C4—91no—
143C2×C4×D3—31no—
144C2×C4×D3—91no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4D3211yescharacteristic
154C3×C22⋊C4—181no—
155C3×C42—181no—
156#217—181no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3—91no—
167C4⋊Dic3—91no—
168C4×Dic3—61no—
169C4×Dic3—181no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—61no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—94no—
179D3×C8—94no—
180D3×C8—94no—
181C22×C32—21no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×#6711yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3Dic3⋊C412yes—
187C2×C3×D3—32no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—81no—
195C32×C4C2×Dic611yescharacteristic
196C32×C4C2×C4×D311yescharacteristic
197C32×C4C2×C4×D311yescharacteristic
198C32×C4D3×Q811yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—41no—
201C3⋊Dic3Dic3⋊C412yes—
202C3×Dic3—32no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—241no—
208C22×C8—271no—
209#117—91no—
210#117—271no—
211#107—91no—
212#107—271no—
213#106—91no—
214#106—271no—
215H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
216C23×C3—91no—
217C22×D3—31no—
218C22×D3—91no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—91no—
228C2×C3×C4—121no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—91no—
239C2×Dic3—121no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C4×D3—32no—
243C4×D3—32no—
244C4×D3—92no—
245C4×D3—92no—
246C3×C8—92no—
247C3×C8—92no—
248C3⋊C8—32no—
249C3⋊C8—32no—
250C3⋊C8—32no—
251C3⋊C8—62no—
252C3⋊C8—92no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32D3×C4⋊C411yescharacteristic
257C2×C32#91611yescharacteristic
258C2×C32#91611yescharacteristic
259C2×C32C2×Dic3⋊C411yescharacteristic
260C2×C32#95011yescharacteristic
261C2×C32#95011yescharacteristic
262C3⋊D3—22no—
263C3×D3—62no—
264C22×C4—91no—
265C2×C8—91no—
266C2×C8—91no—
267C2×C8—274no—
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C22⋊C4—181no—
273C22⋊C4—541no—
274C42—181no—
275C42—541no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3—92no—
280C22×C3—181no—
281C22×C3—181no—
282C22×C3D3×#6711yescharacteristic
283C2×D3—32no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—92no—
287C2×D3—181no—
288C2×D3—181no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—92no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—241no—
300C3×C4—361no—
301C3×C4—361no—
302C3×C4—361no—
303C3×C4D3×Dic611yescharacteristic
304C3×C4C4×D3211yescharacteristic
305Dic3—32no—
306Dic3—92no—
307Dic3—121no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—241no—
312Dic3—361no—
313Dic3—361no—
314C32—21no—
315C32#486011yescharacteristic, complemented
316C32#499811yescharacteristic, complemented
317C23—91no—
318C2×C4—21no—
319C2×C4—91no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C8—92no—
327C8—92no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—182no—
338C2×C3D3×Dic3⋊C411yescharacteristic
339C2×C3#6393511yescharacteristic
340C2×C3#6394711yescharacteristic
341D3—62no—
342D3—182no—
343C22—92no—
344C22—181no—
345C22—181no—
346C22#8867611yescharacteristic
347C4—41no—
348C4—92no—
349C4—361no—
350C4—361no—
351C4—361no—
352C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
353C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C3—31no—
355C3—31no—
356C3—61no—
357C3#1059057211yescharacteristic
358C2—182no—
359C2#1172111911yescharacteristic
360C2#1172119711yescharacteristic
361C2#1172123911yescharacteristic
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)#427
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. #11720529C2semidirect
5. #63635⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720307C2semidirect
21. ((C3⋊D3)⋊C3)⋊C8C4semidirect
28. #63635D3semidirect
215. H(3,𝔽3)#427semidirect
315. C32#4860semidirect
316. C32#4998semidirect

Maximal subgroups

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

subgroupindex
#64101⋊C32
#117210812
C2×((C3⋊D3)⋊C3)⋊C82
#117205292
#63635⋊C32
#117203152
#117203072
#105905743
#105905713