Finite Group Databasebeta

← #13161314

Subgroups — #13161314

Order 1728.

1016 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1016 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
1#11723985C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic
3C2×C32×#216C211yesmaximal, characteristic, complemented
4#11723436C211yesmaximal, characteristic
5C2×#88799C211yesmaximal, characteristic, complemented
6C2×#88796C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic, complemented
8#10592889—31nomaximal
9#10591611—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88857—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3755—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—22no—
21#88799—22no—
22#88796—22no—
23C2×C32×Q16—31no—
24#64156—31no—
25#64153—31no—
26#64153—34no—
27#64153—64no—
28C2×#3756—31no—
29C2×#3755D311yescharacteristic
30C2×#3744—31no—
31C2×C3×#216D314yescomplemented
32C2×C3×#215—34no—
33#63863—34no—
34C2×#3719—34no—
35C2×#3716—34no—
36#63642—31no—
37#63555—34no—
38C22×C3×C3⋊D3—21no—
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)D412yes—
45#6308—21no—
46#6308—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3×C3⋊(C3⋊C8)D412yes—
53C32×C3⋊C8D412yes—
54#5190—94no—
55#5167—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12—34no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—62no—
67C2×C32×C8—31no—
68#3756—62no—
69#3755—22no—
70#3752—62no—
71#3750—62no—
72#3750—68no—
73#3750—616no—
74C4×C3⋊Dic3—31no—
75C4×C3⋊Dic3—34no—
76C4×C3⋊Dic3—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3744—62no—
79C3×#216—28no—
80C3×#215—68no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—68no—
84#3719—68no—
85#3716—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—42no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4C2×D411yescharacteristic
92C3⋊(C3⋊Dic3)—22no—
93C3⋊(C3⋊Dic3)—41no—
94C32×Dic3—41no—
95C2×C3×Q16—34no—
96C2×C3×SD16—91no—
97#947—94no—
98#944—94no—
99#944—91no—
100#944—184no—
101C2×#216—94no—
102C2×#216C3⋊D311yescharacteristic, complemented
103C2×#215—34no—
104#936—91no—
105C2×#214—91no—
106C2×Dic12—91no—
107C2×#204—94no—
108#815—94no—
109#815—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C32×Q8—21no—
113C32×Q8—21no—
114C32×Q8—24no—
115C32×Q8—24no—
116C32×Q8—44no—
117C32×Q8—44no—
118C32×Q8—61no—
119C32×Q8—61no—
120C2×C32×C4—28no—
121C2×C32×C4—24no—
122C2×C32×C4—21no—
123C2×C32×C4—24no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—32no—
128C2×C3⋊Dic3—38no—
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—68no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—61no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135#634—61no—
136#634—61no—
137#634—64no—
138#634—64no—
139#634—124no—
140#634—124no—
141C2×C3×Dic3—24no—
142C3×D12—64no—
143C3×D12—64no—
144C3×Dic6—24no—
145C3×Dic6—24no—
146C32×C8—32no—
147C3⋊(C3⋊C8)#6712yes—
148C3×C3⋊C8—38no—
149C3×C3⋊C8#6718yes—
150#497—271noSylow
151C2×C33#12711yescharacteristic
152C2×C33#13712yes—
153C3×C3⋊D3—81no—
154C2×C3×Q8—24no—
155C2×C3×Q8—34no—
156C2×C3×Q8D3214yes—
157C2×C3×Q8C2×C3⋊D311yescharacteristic
158C2×C3×D4—91no—
159C2×D12—34no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C3×Q16—68no—
165C3×SD16—182no—
166C2×C3×C8—34no—
167C2×C3×C8—91no—
168#217—182no—
169#216—22no—
170#216—188no—
171#215—68no—
172#214—182no—
173(C2×D3)⋊C4—188no—
174Dic3⋊C4—188no—
175Dic3⋊C4—182no—
176Dic3⋊C4—1816no—
177C4×Dic3—94no—
178C4×Dic3—91no—
179C4×Dic3—184no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182Dic12—182no—
183#204—188no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—42no—
188C2×C3⋊D3—41no—
189C2×C3×D3—128no—
190C2×C3×D3—124no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—48no—
194C32×C4—44no—
195C32×C4—41no—
196C32×C4—121no—
197C32×C4D3×D414yes—
198C32×C4D3×D411yescharacteristic
199C32×C4C2×#6714yes—
200C32×C4C2×#6711yescharacteristic
201C3⋊Dic3—616no—
202C3⋊Dic3—62no—
203C3⋊Dic3—68no—
204C3⋊Dic3—128no—
205C3⋊Dic3—124no—
206C3⋊Dic3—121no—
207C3×Dic3—44no—
208C2×Q16—31no—
209C2×Q16—271no—
210C2×SD16—91no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213#124—271no—
214#97—271no—
215C33#49711yescharacteristic, complemented, Sylow
216C23×C3—181no—
217C22×D3—64no—
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—44no—
223C3×Q8—44no—
224C3×Q8—64no—
225C3×Q8—64no—
226C3×D4—181no—
227C3×D4—181no—
228C2×C3×C4—28no—
229C2×C3×C4—24no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—64no—
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235C2×Dic3—21no—
236C2×Dic3—98no—
237C2×Dic3—92no—
238C2×Dic3—188no—
239C2×Dic3—188no—
240C2×Dic3—184no—
241C2×Dic3—181no—
242D12—64no—
243D12—64no—
244Dic6—21no—
245Dic6—21no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251Dic6—364no—
252C3×C8—38no—
253C3×C8—92no—
254C3⋊C8—38no—
255C3⋊C8#63812yes—
256C2×C32—28no—
257C2×C32—24no—
258C2×C32#91512yes—
259C2×C32#91818yes—
260C2×C32#94014yes—
261C2×C32#94011yescharacteristic
262C3⋊D3—81no—
263C3×D3—244no—
264C2×Q8—31no—
265C2×Q8—271no—
266C2×Q8D3×C3⋊D311yescharacteristic
267C2×D4—91no—
268Q16—62no—
269Q16—542no—
270SD16—182no—
271SD16—542no—
272C2×C8—31no—
273C2×C8—91no—
274C4⋊C4—542no—
275C22⋊C4—542no—
276C42—271no—
277C22×C3—24no—
278C22×C3—362no—
279C22×C3—361no—
280C22×C3#381114yes—
281C22×C3D4×C3⋊D311yescharacteristic
282C2×D3—128no—
283C2×D3—124no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—41no—
288C3×C4—44no—
289C3×C4—124no—
290C3×C4D3×#6714yes—
291C3×C4D3×#6714yes—
292C3×C4D4×C3⋊D311yescharacteristic
293C3×C4C2×#63811yescharacteristic
294Dic3—41no—
295Dic3—1816no—
296Dic3—182no—
297Dic3—188no—
298Dic3—368no—
299Dic3—361no—
300Dic3—364no—
301C32—24no—
302C32#516711yescharacteristic, complemented
303C32#519014yescomplemented
304C23—181no—
305Q8—21no—
306Q8—21no—
307Q8—61no—
308Q8—61no—
309Q8—541no—
310Q8—541no—
311D4—181no—
312D4—181no—
313C2×C4—21no—
314C2×C4—61no—
315C2×C4—272no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—32no—
319C8—92no—
320C2×C3—28no—
321C2×C3—24no—
322C2×C3—721no—
323C2×C3#6395118yes—
324C2×C3#6397914yes—
325C2×C3#6412712yes—
326C2×C3#6414911yescharacteristic
327D3—244no—
328C22—362no—
329C22—361no—
330C22#8903911yescharacteristic
331C4—41no—
332C4—121no—
333C4—542no—
334C4—1081no—
335C4D3×#63811yescharacteristic
336C4C3⋊D3×#6711yescharacteristic
337C3—24no—
338C3#1059161114yescomplemented
339C3#1059288911yescharacteristic, complemented
340C2—721no—
341C2#1172359012yes—
342C2#1172363311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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. #11723985C2semidirect
3. C2×C32×#216C2semidirect
5. C2×#88799C2semidirect
7. #11722021C2semidirect
31. C2×C3×#216D3semidirect
102. C2×#216C3⋊D3semidirect
215. C33#497semidirect
302. C32#5167semidirect
303. C32#5190semidirect
338. C3#10591611semidirect
339. C3#10592889semidirect

Maximal subgroups

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

subgroupindex
#117239852
C2×C3×#37552
C2×C32×#2162
#117234362
C2×#887992
C2×#887962
#117220212
#105928893
#105916113