Finite Group Databasebeta

← #13155599

Subgroups — #13155599

Order 1728.

654 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 122 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×C3×#3712C212yesmaximal, complemented
3#11722023C212yesmaximal, complemented
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6#10590733—31nomaximal
7#10590731—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—
16C3×#3712—24no—
17C2×C3×ΓL(1,𝔽25)—32no—
18C2×C3×ΓL(1,𝔽25)—32no—
19C4×#626—32no—
20C4×#625—31no—
21C2×#3712D312yescomplemented
22#63647—32no—
23C8×C3⋊Dic3—31no—
24C3×Dic3⋊C8—32no—
25#63572D311yescharacteristic, complemented
26(C2×D3)⋊(C3⋊C8)—32no—
27(C2×D3)⋊(C3⋊C8)—32no—
28(C2×D3)⋊(C3⋊C8)—62no—
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))—21no—
40C32×C3⋊C8—22no—
41C32×C3⋊C8D412yescomplemented
42#5128—92no—
43#4838—91no—
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—31no—
51C2×C32×C8—32no—
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—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8C2×D312yes—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C2×D312yes—
65C3×ΓL(1,𝔽25)—64no—
66C3×ΓL(1,𝔽25)—64no—
67#3724—31no—
68Dic3⋊Dic3—32no—
69#3722—32no—
70(C2×D3)⋊Dic3—32no—
71(C2×D3)⋊Dic3—32no—
72#3712—24no—
73C22×C33C22×C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—22no—
77C33×C4C2×D411yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3M1612yescomplemented
82C2×C3×M16—92no—
83C4×#67—92no—
84C2×ΓL(1,𝔽25)—32no—
85C2×ΓL(1,𝔽25)—32no—
86C4×D12—91no—
87C3×C4⋊C8—91no—
88#830—92no—
89(C2×D3)⋊C8—92no—
90(C2×D3)⋊C8—92no—
91(C2×D3)⋊C8—182no—
92Dic3⋊C8—32no—
93C8×Dic3—92no—
94C8×Dic3—181no—
95C4×C3⋊C8—91no—
96C22×C3⋊D3C4×D312yes—
97C22×C3×D3—32no—
98C22×C3×D3—32no—
99C22×C3×D3—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—41no—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3C4×D311yescharacteristic
106C2×C3⋊Dic3C4×D312yes—
107C2×C3⋊Dic3C4×D311yescharacteristic
108C4×C3⋊D3—22no—
109C4×C3⋊D3—22no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—122no—
122C3×C4×D3—122no—
123#626—64no—
124#625—62no—
125C32×C8—32no—
126C32×C8—62no—
127C3⋊(C3⋊C8)—61no—
128C3⋊(C3⋊C8)—62no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—122no—
131C3⋊(C3⋊C8)—122no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—42no—
136C3×C3⋊C8#6714yescomplemented
137#436—271noSylow
138C2×C33C4×D411yescharacteristic
139C2×C33C2×M1611yescharacteristic
140C2×C33#13111yescharacteristic
141C3×C3⋊D3—42no—
142C22×C3×C4—92no—
143C2×#67—92no—
144C2×D12—91no—
145C2×C4×D3—32no—
146C2×C4×D3—32no—
147C2×C4×D3—62no—
148C3×M16—184no—
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—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—92no—
164C2×C3⋊C8—181no—
165C2×C3⋊C8—182no—
166C2×C3⋊C8—361no—
167C2×C3⋊C8D3212yes—
168C2×C3⋊C8D3211yescharacteristic
169ΓL(1,𝔽25)—64no—
170ΓL(1,𝔽25)—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32C2×C4×D311yescharacteristic
175C22×C32C2×C4×D312yes—
176C2×C3⋊D3—22no—
177C2×C3⋊D3—22no—
178C2×C3⋊D3—22no—
179C2×C3×D3—64no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—122no—
187C2×C3×D3—122no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—21no—
191C32×C4—22no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4#23512yes—
195C32×C4D3×D411yescharacteristic
196C32×C4#23911yescharacteristic
197C32×C4C2×#6712yes—
198C3⋊Dic3—21no—
199C3⋊Dic3—22no—
200C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
201C3×Dic3—34no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—122no—
208C2×M16—92no—
209C4×D4—271no—
210C4⋊C8—91no—
211C22⋊C8—272no—
212C4×C8—271no—
213C33#43611yescharacteristic, complemented, Sylow
214C23×C3—92no—
215C22×D3—32no—
216C22×D3—32no—
217C22×D3—62no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—91no—
222C2×C3×C4—92no—
223C2×C3×C4—91no—
224C2×C3×C4—182no—
225C2×C3×C4—182no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C2×D3212yes—
228#67—184no—
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—62no—
236D12—182no—
237C4×D3—62no—
238C4×D3—62no—
239C4×D3—62no—
240C4×D3—62no—
241C4×D3—122no—
242C4×D3—122no—
243C3×C8—34no—
244C3×C8—62no—
245C3×C8—62no—
246C3×C8—62no—
247C3×C8—122no—
248C3⋊C8—22no—
249C3⋊C8—182no—
250C3⋊C8—181no—
251C3⋊C8—182no—
252C3⋊C8—362no—
253C3⋊C8—362no—
254C3⋊C8#62512yescomplemented
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32#89311yescharacteristic
265C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
266C2×C32D3×M1612yes—
267C2×C32#90712yes—
268C2×C32#90711yescharacteristic
269C2×C32C4×#6712yes—
270C3⋊D3—42no—
271C3×D3—124no—
272C3×D3—122no—
273C3×D3—122no—
274C2×D4—271no—
275C22×C4—92no—
276M16—184no—
277C2×C8—31no—
278C2×C8—32no—
279C2×C8—271no—
280C4⋊C4—271no—
281C22⋊C4—272no—
282C42—91no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—41no—
286C22×C3—182no—
287C22×C3—182no—
288C22×C3—182no—
289C22×C3C4×D3212yes—
290C22×C3C4×D3211yescharacteristic
291C2×D3—64no—
292C2×D3—62no—
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—62no—
298C2×D3—122no—
299C2×D3—122no—
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—92no—
307C3×C4—181no—
308C3×C4—182no—
309C3×C4#379811yescharacteristic
310C3×C4#379912yes—
311C3×C4C2×#62511yescharacteristic
312C3×C4D3×#6712yes—
313Dic3—34no—
314Dic3—62no—
315Dic3—62no—
316Dic3—62no—
317Dic3—62no—
318Dic3—62no—
319Dic3—122no—
320C32—22no—
321C32—21no—
322C32—41no—
323C32#483811yescharacteristic, complemented
324C32#512812yescomplemented
325C23—92no—
326D4—542no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—92no—
330C2×C4—182no—
331C2×C4—182no—
332C2×C4D3311yescharacteristic
333C8—32no—
334C8—62no—
335C8—541no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—362no—
346C2×C3D3×ΓL(1,𝔽25)12yes—
347C2×C3#6379111yescharacteristic
348C2×C3#6380512yes—
349C2×C3#6380511yescharacteristic
350C2×C3C4×#62511yescharacteristic
351C2×C3#6390412yes—
352D3—124no—
353D3—122no—
354D3—122no—
355C22—182no—
356C22—182no—
357C22—182no—
358C22#8894911yescharacteristic
359C4—92no—
360C4—181no—
361C4—182no—
362C4D3×#62511yescharacteristic
363C4#8896511yescharacteristic
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059073112yescomplemented
368C3#1059073311yescharacteristic, complemented
369C2—362no—
370C2#1172282011yescharacteristic
371C2#1172282711yescharacteristic
372C2#1172293111yescharacteristic
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×C3×#3712C2semidirect
3. #11722023C2semidirect
4. C3⋊C8×C3⋊Dic3C2semidirect
5. C3×#63572C2semidirect
21. C2×#3712D3semidirect
25. #63572D3semidirect
41. C32×C3⋊C8D4semidirect
81. C3×C3⋊Dic3M16semidirect
136. C3×C3⋊C8#67semidirect
200. C3⋊Dic3ΓL(1,𝔽25)semidirect
213. C33#436semidirect
254. C3⋊C8#625semidirect
323. C32#4838semidirect
324. C32#5128semidirect
367. C3#10590731semidirect
368. C3#10590733semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
C2×C3×#37122
#117220232
C3⋊C8×C3⋊Dic32
C3×#635722
#105907333
#105907313