Finite Group Databasebeta

← #13155606

Subgroups — #13155606

Order 1728.

690 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C4×#6295C211yesmaximal, characteristic
2C2×#88787C212yesmaximal, complemented
3C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63557C212yesmaximal, complemented
6#10590734—31nomaximal
7#10590732—32nomaximal
8C2×C3×C4×C3⋊D3C2212yes—
9C2×#6295C411yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C32×C3⋊C8C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14#88815C411yescharacteristic
15#88814C412yes—
16#88787—24no—
17C2×#3743—32no—
18C4×#626—32no—
19C4×#625—31no—
20C2×#3711—32no—
21C2×#3711—32no—
22C2×#3711—62no—
23C8×C3⋊Dic3—31no—
24C3×(C2×D3)⋊C8—32no—
25C3×(C2×D3)⋊C8—32no—
26C3×Dic3⋊C8—32no—
27#63572D311yescharacteristic, complemented
28#63557D312yescomplemented
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C3×C3⋊D3C2×C412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3—22no—
38#6295—22no—
39C3⋊(C3⋊(C3⋊C8))D412yescomplemented
40C32×C3⋊C8—22no—
41C32×C3⋊C8—21no—
42#4838—91no—
43#4729—92no—
44C2×C4×C3⋊D3C2×D312yes—
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—62no—
48C2×#626—32no—
49C2×#625—31no—
50C2×C32×C8—32no—
51C2×C32×C8—31no—
52C4×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊(C3⋊C8)—32no—
54C2×C3⋊(C3⋊C8)—31no—
55C2×C3⋊(C3⋊C8)—61no—
56C2×C3⋊(C3⋊C8)—62no—
57C2×C3⋊(C3⋊C8)—121no—
58#3743—64no—
59C3×C4×Dic3—32no—
60C3×C4×Dic3—61no—
61C2×C3×C3⋊C8—21no—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C2×D312yes—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—31no—
67Dic3⋊Dic3—32no—
68#3722—32no—
69(C2×D3)⋊Dic3—32no—
70(C2×D3)⋊Dic3—32no—
71#3711—64no—
72#3711—64no—
73#3711—68no—
74C22×C33C22×C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—22no—
78C33×C4C2×D411yescharacteristic
79C33×C4#4111yescharacteristic
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3M1612yescomplemented
83C4×#67—92no—
84C2×#208—92no—
85C2×ΓL(1,𝔽25)—92no—
86C2×ΓL(1,𝔽25)—92no—
87C2×ΓL(1,𝔽25)—182no—
88C4×D12—91no—
89C3×C4⋊C8—91no—
90C3×C22⋊C8—92no—
91(C2×D3)⋊C8—32no—
92(C2×D3)⋊C8—32no—
93Dic3⋊C8—32no—
94C8×Dic3—92no—
95C8×Dic3—181no—
96C4×C3⋊C8—91no—
97C22×C3⋊D3C4×D312yes—
98C22×C3×D3—32no—
99C22×C3×D3—32no—
100C22×C3×D3—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4C22×D312yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3C4×D311yescharacteristic
107C2×C3⋊Dic3C4×D312yes—
108C2×C3⋊Dic3C4×D311yescharacteristic
109C4×C3⋊D3—22no—
110C4×C3⋊D3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124#626—64no—
125#625—62no—
126C32×C8—62no—
127C32×C8—61no—
128C3⋊(C3⋊C8)—32no—
129C3⋊(C3⋊C8)—34no—
130C3⋊(C3⋊C8)—64no—
131C3⋊(C3⋊C8)—62no—
132C3⋊(C3⋊C8)—122no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8—22no—
137C3×C3⋊C8—22no—
138#436—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×M1611yescharacteristic
141C2×C33#13111yescharacteristic
142C3×C3⋊D3—42no—
143C22×C3×C4—92no—
144C2×#67—92no—
145C2×D12—91no—
146C2×C4×D3—32no—
147C2×C4×D3—32no—
148C2×C4×D3—62no—
149C2×C3×C8—32no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—61no—
153C2×C3×C8—62no—
154C3×C42—91no—
155#217—92no—
156(C2×D3)⋊C4—92no—
157(C2×D3)⋊C4—92no—
158C4⋊Dic3—91no—
159Dic3⋊C4—92no—
160C4×Dic3—32no—
161C4×Dic3—61no—
162#208—184no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—92no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8—182no—
167C2×C3⋊C8—361no—
168C2×C3⋊C8D3211yescharacteristic
169C2×C3⋊C8D3212yes—
170ΓL(1,𝔽25)—184no—
171ΓL(1,𝔽25)—184no—
172ΓL(1,𝔽25)—188no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32C2×C4×D312yes—
177C22×C32C2×C4×D311yescharacteristic
178C2×C3⋊D3—22no—
179C2×C3⋊D3—22no—
180C2×C3⋊D3—22no—
181C2×C3×D3—64no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—122no—
189C2×C3×D3—122no—
190C32×C4—22no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4D3×D412yes—
197C32×C4D3×D411yescharacteristic
198C32×C4#23712yes—
199C32×C4#23911yescharacteristic
200C3⋊Dic3—22no—
201C3⋊Dic3—21no—
202C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
203C3×Dic3—34no—
204C3×Dic3—62no—
205C3×Dic3—64no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C2×M16—272no—
211C4×D4—271no—
212C4⋊C8—91no—
213C22⋊C8—92no—
214C4×C8—271no—
215C33#43611yescharacteristic, complemented, Sylow
216C23×C3—92no—
217C22×D3—32no—
218C22×D3—32no—
219C22×D3—62no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—91no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4C2×D3212yes—
229C2×C3×C4C2×D3211yescharacteristic
230#67—184no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238D12—182no—
239C4×D3—64no—
240C4×D3—64no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—62no—
244C4×D3—62no—
245C3×C8—64no—
246C3×C8—62no—
247C3×C8—62no—
248C3×C8—62no—
249C3×C8—62no—
250C3⋊C8—21no—
251C3⋊C8—22no—
252C3⋊C8—92no—
253C3⋊C8—94no—
254C3⋊C8—184no—
255C3⋊C8—182no—
256C3⋊C8—362no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32#67⋊C412yes—
267C2×C32#89311yescharacteristic
268C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
269C2×C32#90112yes—
270C2×C32D3×M1612yes—
271C2×C32#90711yescharacteristic
272C3⋊D3—42no—
273C3×D3—124no—
274C3×D3—122no—
275C3×D3—122no—
276C2×D4—271no—
277C22×C4—92no—
278M16—544no—
279C2×C8—31no—
280C2×C8—32no—
281C2×C8—271no—
282C4⋊C4—271no—
283C22⋊C4—272no—
284C42—91no—
285C22×C3—21no—
286C22×C3—22no—
287C22×C3—41no—
288C22×C3—182no—
289C22×C3—182no—
290C22×C3—182no—
291C22×C3C4×D3212yes—
292C22×C3C4×D3211yescharacteristic
293C2×D3—64no—
294C2×D3—62no—
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—122no—
301C2×D3—122no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—92no—
309C3×C4—181no—
310C3×C4—182no—
311C3×C4#379612yes—
312C3×C4#380211yescharacteristic
313C3×C4#380511yescharacteristic
314C3×C4#381112yes—
315Dic3—34no—
316Dic3—62no—
317Dic3—64no—
318Dic3—62no—
319Dic3—62no—
320Dic3—62no—
321Dic3—62no—
322C32—22no—
323C32—21no—
324C32—41no—
325C32#472912yescomplemented
326C32#483811yescharacteristic, complemented
327C23—92no—
328D4—542no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4D3311yescharacteristic
335C8—62no—
336C8—61no—
337C8—272no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—362no—
348C2×C3D3×ΓL(1,𝔽25)12yes—
349C2×C3#6379111yescharacteristic
350C2×C3#6380311yescharacteristic
351C2×C3#6380412yes—
352C2×C3#6388112yes—
353C2×C3#6390211yescharacteristic
354D3—124no—
355D3—122no—
356D3—122no—
357C22—182no—
358C22—182no—
359C22—182no—
360C22#8894911yescharacteristic
361C4—92no—
362C4—181no—
363C4—182no—
364C4#8895411yescharacteristic
365C4#8896411yescharacteristic
366C3—22no—
367C3—21no—
368C3—41no—
369C3#1059073212yescomplemented
370C3#1059073411yescharacteristic, complemented
371C2—362no—
372C2#1172282011yescharacteristic
373C2#1172282611yescharacteristic
374C2#1172293811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×#6295C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C3⋊C8×C3⋊Dic3C2semidirect
4. C3×#63572C2semidirect
5. C3×#63557C2semidirect
27. #63572D3semidirect
28. #63557D3semidirect
39. C3⋊(C3⋊(C3⋊C8))D4semidirect
82. C3×C3⋊Dic3M16semidirect
202. C3⋊Dic3ΓL(1,𝔽25)semidirect
215. C33#436semidirect
325. C32#4729semidirect
326. C32#4838semidirect
369. C3#10590732semidirect
370. C3#10590734semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
C2×#887872
C3⋊C8×C3⋊Dic32
C3×#635722
C3×#635572
#105907343
#105907323