Finite Group Databasebeta

← #13138756

Subgroups — #13138756

Order 1728.

374 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721117C211yesmaximal, characteristic
2#11721084C211yesmaximal, characteristic
3#11720547C211yesmaximal, characteristic
4#63645⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic
7#11720313C211yesmaximal, characteristic
8#10590805—31nomaximal
9#10590804—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×#6196—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88449—21no—
19#88448—21no—
20#88447—21no—
21#63918—31no—
22#63918—31no—
23#63844—31no—
24Dic3×Dic6—31no—
25#63645D311yescharacteristic
26#63643D311yescharacteristic
27C3×#818—31no—
28C3×#816—31no—
29C3×Dic3⋊C8—31no—
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—61no—
32#63575—31no—
33#63575—31no—
34#63567—31no—
35#63566—31no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084D411yescharacteristic
38C2×#1084D411yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—21no—
44#634⋊C3Q812yescomplemented
45#6196—42no—
46C8×H(3,𝔽3)—21no—
47#6180—21no—
48#4875—91no—
49#4850—91no—
50#4793—91no—
51C2×#634C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C2×#627—61no—
54C2×#627—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—31no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74Dic32—61no—
75Dic32—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79#1084—21no—
80#1084Q1612yes—
81(C3⋊Dic3)⋊C3—22no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3—41no—
85Q8×Dic3—91no—
86#888—91no—
87#888—91no—
88#887—91no—
89#869—91no—
90C4×Dic6—91no—
91C3×#107—91no—
92C3×C4⋊C8—91no—
93C3×#103—91no—
94#835—91no—
95#818—31no—
96#818—91no—
97#818—91no—
98#816—31no—
99#816—91no—
100#816—91no—
101Dic3⋊C8—31no—
102Dic3⋊C8—31no—
103Dic3⋊C8—61no—
104#810—91no—
105#807—91no—
106#807—91no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114#634—21no—
115#634Dic612yescomplemented
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—32no—
127C3×Dic6—61no—
128#627—122no—
129#627—122no—
130C32×C8—21no—
131C32×C8—21no—
132C32×C8—41no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—121no—
136#477—271noSylow
137C2×H(3,𝔽3)C22⋊Q811yescharacteristic
138C2×H(3,𝔽3)C2×Q1611yescharacteristic
139C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
140C2×C3×Q8—91no—
141C2×Dic6—31no—
142C2×Dic6—91no—
143C2×Dic6—181no—
144C2×Dic6—181no—
145C2×Dic6—181no—
146C2×C3×C8—31no—
147C2×C3×C8—31no—
148C2×C3×C8—61no—
149C2×C3×C8—91no—
150C2×C3×C8D3211yescharacteristic
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—91no—
154C3×C42—91no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—91no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167C4×Dic3—31no—
168C4×Dic3—31no—
169C4×Dic3—61no—
170C4×Dic3—181no—
171C4×Dic3—181no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—61no—
176C22×C32—21no—
177C22×C32D3×D411yescharacteristic
178C22×C32D3×D411yescharacteristic
179C32×C4—21no—
180C32×C4—21no—
181C32×C4C2×Dic611yescharacteristic
182C32×C4#23511yescharacteristic
183C32×C4#23711yescharacteristic
184C32×C4D3×Q811yescharacteristic
185C3⋊Dic3—22no—
186C3⋊Dic3—41no—
187C3⋊Dic3—41no—
188C3⋊Dic3—41no—
189C3×Dic3—32no—
190C3×Dic3—32no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C4⋊Q8—271no—
200C4×Q8—271no—
201#107—91no—
202#107—271no—
203C4⋊C8—91no—
204#103—91no—
205#103—271no—
206H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
207C3×Q8—92no—
208C3×Q8—181no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3211yescharacteristic
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233Dic6—32no—
234Dic6—61no—
235Dic6—92no—
236Dic6—181no—
237Dic6—362no—
238Dic6—362no—
239Dic6—362no—
240C3×C8—21no—
241C3×C8—61no—
242C3×C8—61no—
243C3×C8—121no—
244C3×C8—181no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C3⋊C8—121no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32#87811yescharacteristic
252C2×C32#89611yescharacteristic
253C2×C32#91111yescharacteristic
254C2×C32#91411yescharacteristic
255C2×C32D3×Q1611yescharacteristic
256C2×C32D3×Q1611yescharacteristic
257C2×Q8—91no—
258C2×Q8—541no—
259C2×C8—91no—
260C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
261C4⋊C4—91no—
262C4⋊C4—91no—
263C4⋊C4—91no—
264C4⋊C4—541no—
265C4⋊C4—541no—
266C42—91no—
267C42—541no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3#380211yescharacteristic
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—92no—
279C3×C4—182no—
280C3×C4—181no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4—361no—
284C3×C4D3×Dic611yescharacteristic
285C3×C4#380411yescharacteristic
286Dic3—32no—
287Dic3—32no—
288Dic3—62no—
289Dic3—61no—
290Dic3—61no—
291Dic3—62no—
292Dic3—121no—
293Dic3—121no—
294Dic3—121no—
295Dic3—121no—
296Dic3—182no—
297Dic3—361no—
298Dic3—361no—
299Dic3—361no—
300C32—21no—
301C32#479311yescharacteristic, complemented
302C32#487511yescharacteristic, complemented
303Q8—92no—
304Q8—181no—
305Q8—1082no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C8—21no—
314C8—181no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3#6379811yescharacteristic
325C2×C3#6380211yescharacteristic
326C2×C3#6389911yescharacteristic
327C22#8865511yescharacteristic
328C4—92no—
329C4—181no—
330C4—182no—
331C4—361no—
332C4—361no—
333C4—361no—
334C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
335C4#8867111yescharacteristic
336C3—31no—
337C3—31no—
338C3—61no—
339C3#1059081711yescharacteristic
340C2#1172083311yescharacteristic
341C2#1172083911yescharacteristic
342C2#1172113411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#477
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
44. #634⋊C3Q8semidirect
115. #634Dic6semidirect
206. H(3,𝔽3)#477semidirect
301. C32#4793semidirect
302. C32#4875semidirect

Maximal subgroups

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

subgroupindex
#117211172
#117210842
#117205472
#63645⋊C32
#63643⋊C32
#117203152
#117203132
#105908053
#105908043