Finite Group Databasebeta

← #13160985

Subgroups — #13160985

Order 1728.

1168 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#1004C211yesmaximal, characteristic
2C3×#64144C211yesmaximal, characteristic, complemented
3#11723602C212yesmaximal, complemented
4#11722204C211yesmaximal, characteristic, complemented
5#11722062C212yesmaximal, complemented
6#10592942—31nomaximal
7#10591533—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
10C2×C32×#67—21no—
11C2×C32×#67C2212yes—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235—21no—
15#88864C2212yescomplemented
16C3×#3757—21no—
17C3×#3757C2211yescharacteristic, complemented
18C3×#3750—21no—
19#88806—22no—
20(C2×D3)⋊(C3⋊Dic3)—22no—
21C32×H(5,𝔽2)—31no—
22C3×#1004D314yes—
23#64144—32no—
24#64144D311yescharacteristic, complemented
25C3×#935—34no—
26#63961—38no—
27#63661—31no—
28#63661—32no—
29#63661—34no—
30#63661—64no—
31#63580—38no—
32C23×C33D411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3D412yes—
36C33×D4—22no—
37C2×C33×C4D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)—22no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—22no—
42C32×#67—22no—
43C32×#67—22no—
44C32×#67—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D412yes—
47C32×D12—21no—
48C32×C4×D3—22no—
49C32×Dic6—21no—
50#5243—94no—
51#5053—91no—
52C32×#41—62no—
53C32×#41—61no—
54C2×C32×D4—24no—
55C2×C32×D4—32no—
56C2×C32×D4—62no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C22×C3⋊Dic3C2×D311yescharacteristic, complemented
61C2×C3×#67—24no—
62C2×C3×#67C2×D318yes—
63C22×C3×Dic3—34no—
64C3×#237—28no—
65C3×D3×D4—28no—
66C3×#235—24no—
67#3757—21no—
68#3757—32no—
69#3757—38no—
70#3757—62no—
71#3757—68no—
72#3757C2×D311yescharacteristic, complemented
73#3750—21no—
74#3750—62no—
75C3×#217—34no—
76C3×#217—64no—
77C3×Dic3⋊C4—64no—
78#3724—68no—
79(C2×D3)⋊Dic3—68no—
80C22×C33—21no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33C2×D412yes—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—22no—
86C2×C32×D3—42no—
87C2×C32×D3—81no—
88C33×C4—21no—
89C3⋊(C3⋊Dic3)—42no—
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3—41no—
92C32×Dic3—22no—
93C3×H(5,𝔽2)—34no—
94#1004C3⋊D311yescharacteristic
95C3×#123—91no—
96#935—34no—
97#935—98no—
98#929—92no—
99#834—98no—
100#834—94no—
101#834—91no—
102#834—184no—
103#806—92no—
104C23×C32—24no—
105C23×C32—24no—
106C23×C32—32no—
107C23×C32—61no—
108C23×C32#6714yes—
109C23×C32#6711yescharacteristic
110C23×C32C22×D314yes—
111C23×C32C22×D311yescharacteristic
112C22×C3×D3—24no—
113C22×C3×D3#6718yes—
114C32×Q8—61no—
115C32×D4—22no—
116C32×D4—28no—
117C32×D4—48no—
118C32×D4—61no—
119C32×D4—62no—
120C32×D4—62no—
121C32×D4—62no—
122C2×C32×C4—24no—
123C2×C32×C4—32no—
124C2×C32×C4—62no—
125C2×C32×C4—61no—
126C2×C32×C4#6714yes—
127C2×C32×C4#6711yescharacteristic
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—62no—
132C2×C3⋊Dic3—68no—
133C2×C3⋊Dic3—128no—
134C3×#67—28no—
135C3×#67—28no—
136C3×#67—28no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—68no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3#6718yes—
142C3×D12—24no—
143C3×C4×D3—28no—
144C3×Dic6—24no—
145#458—271noSylow
146C2×C33—21no—
147C2×C33—22no—
148C2×C33—41no—
149C2×C33#12011yescharacteristic
150C32×D3—42no—
151C3×#41—68no—
152C3×#41—64no—
153C2×C3×D4—24no—
154C2×C3×D4—38no—
155C2×C3×D4—68no—
156C2×C3×D4—64no—
157C2×C3×D4D3214yes—
158C2×C3×D4C2×C3⋊D311yescharacteristic
159C22×C3×C4—91no—
160C2×#67—21no—
161C2×#67C2×C3⋊D312yes—
162C22×Dic3—34no—
163#237—22no—
164D3×D4—22no—
165#235—21no—
166C3×C4⋊C4—181no—
167C3×C22⋊C4—91no—
168C3×C22⋊C4—181no—
169#217—34no—
170#217—64no—
171#217—98no—
172#217—92no—
173#217—188no—
174#217—188no—
175(C2×D3)⋊C4—182no—
176Dic3⋊C4—64no—
177Dic3⋊C4—188no—
178Dic3⋊C4—182no—
179C22×C32—21no—
180C22×C32—24no—
181C22×C32—28no—
182C22×C32—24no—
183C22×C32—44no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32—62no—
189C22×C32—84no—
190C22×C32—84no—
191C22×C32—122no—
192C22×C32—241no—
193C22×C32D3×D412yes—
194C22×C32C2×#6714yes—
195C22×C32C2×#6718yes—
196C22×C32C2×#6711yescharacteristic
197C2×C3×D3—28no—
198C2×C3×D3—48no—
199C2×C3×D3—84no—
200C32×C4—21no—
201C32×C4—24no—
202C32×C4—44no—
203C32×C4—62no—
204C3⋊Dic3—41no—
205C3⋊Dic3—41no—
206C3⋊Dic3—122no—
207C3⋊Dic3—128no—
208C3⋊Dic3—248no—
209C3×Dic3—28no—
210C3×Dic3—124no—
211C3×Dic3—124no—
212H(5,𝔽2)—31no—
213#123—91no—
214#123—272no—
215#99—272no—
216#99—271no—
217C33#45811yescharacteristic, complemented, Sylow
218C23×C3—24no—
219C23×C3—24no—
220C23×C3—38no—
221C23×C3—64no—
222C23×C3#62514yes—
223C23×C3#63811yescharacteristic
224C23×C3C2×D3214yes—
225C23×C3C22×C3⋊D311yescharacteristic
226C22×D3—21no—
227C22×D3#63812yes—
228C3×Q8—64no—
229C3×D4—28no—
230C3×D4—22no—
231C3×D4—48no—
232C3×D4—64no—
233C3×D4—68no—
234C3×D4—68no—
235C3×D4—68no—
236C2×C3×C4—24no—
237C2×C3×C4—38no—
238C2×C3×C4—68no—
239C2×C3×C4—64no—
240C2×C3×C4—182no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4#62514yes—
244C2×C3×C4#63811yescharacteristic
245#67—22no—
246#67—22no—
247#67—22no—
248C2×Dic3—21no—
249C2×Dic3—64no—
250C2×Dic3—64no—
251C2×Dic3—68no—
252C2×Dic3—188no—
253C2×Dic3—182no—
254C2×Dic3—368no—
255C2×Dic3#63812yes—
256D12—21no—
257C4×D3—22no—
258Dic6—21no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—28no—
264C2×C32—48no—
265C2×C32—44no—
266C2×C32—41no—
267C2×C32—44no—
268C2×C32—84no—
269C2×C32—122no—
270C2×C32#93711yescharacteristic
271C2×C32#95314yes—
272C3×D3—48no—
273#41—62no—
274#41—61no—
275C2×D4—32no—
276C2×D4—62no—
277C2×D4—61no—
278C2×D4D3×C3⋊D311yescharacteristic
279C22×C4—91no—
280C4⋊C4—181no—
281C4⋊C4—542no—
282C22⋊C4—91no—
283C22⋊C4—181no—
284C22⋊C4—272no—
285C22⋊C4—542no—
286C22×C3—21no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—28no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—41no—
293C22×C3—44no—
294C22×C3—41no—
295C22×C3—68no—
296C22×C3—84no—
297C22×C3—84no—
298C22×C3—128no—
299C22×C3—244no—
300C22×C3C2×#62514yes—
301C22×C3D3×#6718yes—
302C22×C3C2×#63811yescharacteristic
303C22×C3C2×#63812yes—
304C2×D3—22no—
305C2×D3—42no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—24no—
309C3×C4—44no—
310C3×C4—68no—
311C3×C4—361no—
312C3×C4—361no—
313Dic3—22no—
314Dic3—124no—
315Dic3—124no—
316Dic3—362no—
317Dic3—368no—
318Dic3—728no—
319C32—24no—
320C32#505311yescharacteristic, complemented
321C32#524314yescomplemented
322C23—32no—
323C23—61no—
324C23#629011yescharacteristic
325C23C2×D3×C3⋊D311yescharacteristic
326Q8—61no—
327D4—22no—
328D4—61no—
329D4—62no—
330D4—62no—
331D4—62no—
332C2×C4—32no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—182no—
338C2×C4—542no—
339C2×C4#629011yescharacteristic
340C2×C3—24no—
341C2×C3—28no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—24no—
345C2×C3—48no—
346C2×C3—44no—
347C2×C3—44no—
348C2×C3—41no—
349C2×C3—84no—
350C2×C3—128no—
351C2×C3#6397614yes—
352C2×C3#6416211yescharacteristic
353D3—42no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—62no—
358C22—122no—
359C22—241no—
360C22C2×#629011yescharacteristic
361C22D3×#63812yes—
362C4—21no—
363C4—62no—
364C4—361no—
365C4—361no—
366C4—1082no—
367C3—24no—
368C3#1059153314yescomplemented
369C3#1059294211yescharacteristic, complemented
370C2—21no—
371C2—22no—
372C2—41no—
373C2—122no—
374C2#1172362911yescharacteristic
subgroupG/H
centerC2#11723629
commutator, cosocleC23×C33C23
FrattiniC23C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#458
LCSG ▷ C23×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. #11723602C2semidirect
4. #11722204C2semidirect
5. #11722062C2semidirect
9. C22×C3×C3⋊Dic3C22semidirect
15. #88864C22semidirect
17. C3×#3757C22semidirect
24. #64144D3semidirect
60. C22×C3⋊Dic3C2×D3semidirect
72. #3757C2×D3semidirect
217. C33#458semidirect
320. C32#5053semidirect
321. C32#5243semidirect
368. C3#10591533semidirect
369. C3#10592942semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C3×#641442
#117236022
#117222042
#117220622
#105929423
#105915333