Finite Group Databasebeta

← #13161217

Subgroups — #13161217

Order 1728.

984 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#88863C211yesmaximal, characteristic, complemented
2C3×#64153C211yesmaximal, characteristic, complemented
3C2×C32×#215C211yesmaximal, characteristic
4#11723429C211yesmaximal, characteristic
5C2×#88800C211yesmaximal, characteristic, complemented
6C2×#88792C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592866—31nomaximal
9#10591605—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14#88863—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3750—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#215—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—22no—
21#88800—22no—
22#88792—22no—
23C2×C32×SD16—31no—
24#64153D311yescharacteristic, complemented
25C2×#3756—31no—
26C2×#3756—34no—
27C2×#3756—64no—
28#64145—31no—
29C2×#3754—31no—
30C2×#3746—31no—
31C3×#944—34no—
32C2×C3×#215D314yes—
33#63860—34no—
34C2×#3719—34no—
35C2×#3715—34no—
36#63642—31no—
37#63554—34no—
38C22×C32×D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D412yes—
48C3×#634—21no—
49C3×#634—21no—
50C32×D12—21no—
51C32×D12—21no—
52C3⋊(C3⋊(C3⋊C8))D412yes—
53C32×C3⋊C8D412yes—
54#5167—94no—
55#5086—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×D4—31no—
60C2×#634—31no—
61C2×#634—34no—
62C2×#634—64no—
63C2×#634C2×D311yescharacteristic
64C2×C3×D12C2×D314yes—
65C2×C3×Dic6—34no—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—62no—
69#3756—62no—
70#3756—68no—
71#3756—616no—
72#3754—62no—
73#3750—22no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3746—62no—
79C3×#215—28no—
80C3×Dic3⋊C4—68no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—68no—
84#3719—68no—
85#3715—68no—
86C22×C33C2×D411yescharacteristic
87C2×C32×D3—41no—
88C2×C32×D3—42no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4C2×D411yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3—41no—
95C2×C3×SD16—34no—
96C3×C4⋊Q8—91no—
97#944—34no—
98C2×#216—94no—
99C2×#216—91no—
100C2×#216—91no—
101C2×#216—184no—
102C2×#215—91no—
103C2×#215C3⋊D311yescharacteristic
104#936—94no—
105C2×#214—94no—
106C2×Dic12—94no—
107#875—91no—
108#815—94no—
109#803—91no—
110C23×C32—61no—
111C22×C3×D3—24no—
112C32×Q8—28no—
113C32×Q8—24no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×D4—61no—
119C32×D4—61no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—44no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3#6712yes—
131#634—21no—
132#634—21no—
133#634—68no—
134#634—61no—
135#634—64no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—38no—
140C2×C3×Dic3—64no—
141C3×D12—24no—
142C3×D12—24no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145C32×C8—32no—
146C3⋊(C3⋊C8)—32no—
147C3⋊(C3⋊C8)—38no—
148C3⋊(C3⋊C8)—68no—
149C3×C3⋊C8#6718yes—
150#497—271noSylow
151C2×C33#12711yescharacteristic
152C2×C33#13712yes—
153C32×D3—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—91no—
156C2×C3×Q8D3214yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—34no—
159C2×D12C2×C3⋊D311yescharacteristic
160C2×Dic6—34no—
161C2×Dic6—94no—
162C2×Dic6—91no—
163C2×Dic6—184no—
164C3×SD16—68no—
165C2×C3×C8—34no—
166C3×C4⋊C4—182no—
167C3×C42—91no—
168#217—188no—
169#216—188no—
170#216—182no—
171#216—1816no—
172#216—182no—
173#215—22no—
174#215—182no—
175#214—188no—
176(C2×D3)⋊C4—182no—
177Dic3⋊C4—68no—
178C4×Dic3—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183Dic12—188no—
184C22×C32—24no—
185C22×C32—121no—
186C22×C32—122no—
187C22×C32D3×D414yes—
188C22×C32C2×#6711yescharacteristic
189C2×C3×D3—48no—
190C2×C3×D3—44no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—48no—
194C32×C4—44no—
195C32×C4—41no—
196C32×C4D3×D414yes—
197C32×C4D3×D411yescharacteristic
198C32×C4D3×D411yescharacteristic
199C32×C4C2×#6714yes—
200C3⋊Dic3—22no—
201C3⋊Dic3—41no—
202C3⋊Dic3—128no—
203C3⋊Dic3—124no—
204C3⋊Dic3—121no—
205C3×Dic3—68no—
206C3×Dic3—124no—
207C2×Q16—271no—
208C2×Q16—271no—
209C2×SD16—31no—
210C2×SD16—271no—
211C4⋊Q8—91no—
212#124—271no—
213#97—271no—
214C33#49711yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C22×D3—21no—
217C3×Q8—28no—
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—44no—
223C3×Q8—181no—
224C3×Q8—181no—
225C3×D4—64no—
226C3×D4—64no—
227C2×C3×C4—24no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—44no—
231C2×C3×C4—92no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235C2×Dic3—38no—
236C2×Dic3—64no—
237C2×Dic3—184no—
238C2×Dic3—181no—
239C2×Dic3—364no—
240D12—21no—
241D12—21no—
242Dic6—64no—
243Dic6—64no—
244Dic6—188no—
245Dic6—184no—
246Dic6—184no—
247Dic6—181no—
248Dic6—181no—
249Dic6—364no—
250C3×C8—38no—
251C3⋊C8—98no—
252C3⋊C8—92no—
253C3⋊C8—188no—
254C3⋊C8#63812yes—
255C2×C32—28no—
256C2×C32—24no—
257C2×C32—241no—
258C2×C32#91518yes—
259C2×C32#94014yes—
260C2×C32#94011yescharacteristic
261C2×C32#94212yes—
262C3×D3—84no—
263C2×Q8—91no—
264C2×Q8—271no—
265C2×Q8D3×C3⋊D311yescharacteristic
266C2×D4—31no—
267Q16—542no—
268Q16—542no—
269SD16—62no—
270SD16—542no—
271C2×C8—31no—
272C2×C8—271no—
273C4⋊C4—182no—
274C22⋊C4—542no—
275C42—91no—
276C22×C3—24no—
277C22×C3—128no—
278C22×C3—124no—
279C22×C3D3×#6714yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—42no—
282C2×D3—41no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—48no—
286C3×C4—41no—
287C3×C4—44no—
288C3×C4—182no—
289C3×C4—361no—
290C3×C4D3×#6714yes—
291C3×C4#381114yes—
292C3×C4D4×C3⋊D311yescharacteristic
293C3×C4C2×#63811yescharacteristic
294Dic3—68no—
295Dic3—124no—
296Dic3—368no—
297Dic3—361no—
298Dic3—364no—
299C32—24no—
300C32#508611yescharacteristic, complemented
301C32#516714yescomplemented
302C23—61no—
303Q8—21no—
304Q8—21no—
305Q8—181no—
306Q8—181no—
307Q8—541no—
308Q8—541no—
309D4—61no—
310D4—61no—
311C2×C4—21no—
312C2×C4—92no—
313C2×C4—181no—
314C2×C4—541no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—32no—
317C8—272no—
318C2×C3—28no—
319C2×C3—24no—
320C2×C3—244no—
321C2×C3#6394618yes—
322C2×C3#6397914yes—
323C2×C3#6412412yes—
324C2×C3#6414911yescharacteristic
325D3—81no—
326C22—122no—
327C22—121no—
328C22C3⋊D3×#6711yescharacteristic
329C4—41no—
330C4—182no—
331C4—361no—
332C4—1081no—
333C4D3×#63811yescharacteristic
334C4#8903911yescharacteristic
335C3—24no—
336C3#1059160514yescomplemented
337C3#1059286611yescharacteristic, complemented
338C2—241no—
339C2#1172358412yes—
340C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#497
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88863C2semidirect
2. C3×#64153C2semidirect
5. C2×#88800C2semidirect
7. #11722020C2semidirect
24. #64153D3semidirect
214. C33#497semidirect
300. C32#5086semidirect
301. C32#5167semidirect
336. C3#10591605semidirect
337. C3#10592866semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×#641532
C2×C32×#2152
#117234292
C2×#888002
C2×#887922
#117220202
#105928663
#105916053