Finite Group Databasebeta

← #13138755

Subgroups — #13138755

Order 1728.

355 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721095C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic
4#63645⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic
7#11720314C211yesmaximal, characteristic
8#10590894—31nomaximal
9#10590892—31nomaximal
10C2×#6196—21no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88449—21no—
19#88448—21no—
20#88448—21no—
21#63918—31no—
22#63918—31no—
23#63869—31no—
24#63869—31no—
25#63645D311yescharacteristic
26#63644D311yescharacteristic
27C3×#818—31no—
28C3×#817—31no—
29C3×#815—31no—
30C3×#815—31no—
31C3×#815—61no—
32#63575—31no—
33#63575—31no—
34#63573—31no—
35#63573—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#6196—42no—
44C8×H(3,𝔽3)Q812yescomplemented
45#6180Q812yescomplemented
46#4876—91no—
47#4856—91no—
48#4702—91no—
49C2×#627—61no—
50C2×#627—61no—
51C2×C32×C8—21no—
52C2×C32×C8C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4×Dic3—31no—
63C3×C4×Dic3—31no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—61no—
68#3724—61no—
69#3724—61no—
70Dic3⋊Dic3—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C4×H(3,𝔽3)C2×Q811yescharacteristic
76C4×H(3,𝔽3)C2×Q811yescharacteristic
77#1084—21no—
78#1084—21no—
79(C3⋊Dic3)⋊C3—41no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3—41no—
83#890—91no—
84#890—91no—
85#888—91no—
86#888—91no—
87#870—91no—
88#869—91no—
89C3×#107—91no—
90C3×#106—91no—
91C3×#97—91no—
92#818—31no—
93#818—91no—
94#818—91no—
95#817—31no—
96#817—91no—
97#817—91no—
98#815—31no—
99#815—31no—
100#815—61no—
101#808—91no—
102#808—91no—
103#807—91no—
104#807—91no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3—21no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122#627—122no—
123#627—122no—
124C32×C8—22no—
125C32×C8Dic612yescomplemented
126C32×C8Dic612yescomplemented
127C3×C3⋊C8—32no—
128C3×C3⋊C8—32no—
129C3×C3⋊C8—62no—
130#501—271noSylow
131C2×H(3,𝔽3)C4⋊Q811yescharacteristic
132C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
133C2×H(3,𝔽3)#13711yescharacteristic
134C2×Dic6—181no—
135C2×Dic6—181no—
136C2×Dic6—181no—
137C2×C3×C8—31no—
138C2×C3×C8—31no—
139C2×C3×C8—61no—
140C2×C3×C8—91no—
141C2×C3×C8D3211yescharacteristic
142C3×C4⋊C4—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—91no—
146C3×C42—91no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—91no—
152C4⋊Dic3—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—61no—
170C22×C32—21no—
171C22×C32D3×D411yescharacteristic
172C22×C32D3×D411yescharacteristic
173C32×C4—21no—
174C32×C4—21no—
175C32×C4C2×Dic611yescharacteristic
176C32×C4C2×Dic611yescharacteristic
177C32×C4D3×Q811yescharacteristic
178C32×C4D3×Q811yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3—41no—
181C3⋊Dic3—41no—
182C3⋊Dic3—41no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—121no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C3×Dic3—121no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C4⋊Q8—271no—
194#125—271no—
195#107—91no—
196#107—271no—
197#106—91no—
198#106—271no—
199#97—91no—
200H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—31no—
214C2×Dic3—31no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225Dic6—362no—
226Dic6—362no—
227Dic6—362no—
228C3×C8—32no—
229C3×C8—32no—
230C3×C8—62no—
231C3×C8—92no—
232C3×C8#62712yes—
233C3⋊C8—32no—
234C3⋊C8—32no—
235C3⋊C8—62no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32#88811yescharacteristic
240C2×C32#88811yescharacteristic
241C2×C32#91111yescharacteristic
242C2×C32#91411yescharacteristic
243C2×C32#91511yescharacteristic
244C2×C32#91811yescharacteristic
245C2×Q8—541no—
246C2×C8—91no—
247C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C4⋊C4—91no—
249C4⋊C4—91no—
250C4⋊C4—91no—
251C4⋊C4—91no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4—541no—
255C42—91no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—61no—
259C22×C3#380211yescharacteristic
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—181no—
267C3×C4—181no—
268C3×C4—361no—
269C3×C4—361no—
270C3×C4—361no—
271C3×C4—361no—
272C3×C4#379711yescharacteristic
273C3×C4C2×#62711yescharacteristic
274Dic3—61no—
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—121no—
279Dic3—121no—
280Dic3—121no—
281Dic3—121no—
282Dic3—121no—
283Dic3—121no—
284Dic3—361no—
285Dic3—361no—
286Dic3—361no—
287Dic3—361no—
288C32—21no—
289C32#485611yescharacteristic, complemented
290C32#487611yescharacteristic, complemented
291Q8—1082no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C8—92no—
300C8#619612yescomplemented
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3#6379811yescharacteristic
311C2×C3#6381111yescharacteristic
312C2×C3#6391911yescharacteristic
313C22#8865511yescharacteristic
314C4—181no—
315C4—181no—
316C4—361no—
317C4—361no—
318C4—361no—
319C4—361no—
320C4#8865111yescharacteristic
321C4C2×#619611yescharacteristic
322C3—31no—
323C3—31no—
324C3—61no—
325C3#1059090311yescharacteristic
326C2#1172083911yescharacteristic
327C2#1172084511yescharacteristic
328C2#1172111711yescharacteristic
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)#501
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. C8×H(3,𝔽3)Q8semidirect
45. #6180Q8semidirect
125. C32×C8Dic6semidirect
126. C32×C8Dic6semidirect
200. H(3,𝔽3)#501semidirect
289. C32#4856semidirect
290. C32#4876semidirect
300. C8#6196semidirect

Maximal subgroups

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

subgroupindex
#117211172
#117210952
#117205482
#63645⋊C32
#63644⋊C32
#117203152
#117203142
#105908943
#105908923