Finite Group Databasebeta

← #13166998

Subgroups — #13166998

Order 1728.

474 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 55 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#11723695C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic, complemented
3#11723103C211yesmaximal, characteristic, complemented
4#11722778C212yesmaximal, complemented
5#11722772C212yesmaximal, complemented
6#10593361—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6295—21no—
9C2×#6295C2211yescharacteristic
10C2×C3×#624—21no—
11C2×C3⋊#622C2212yes—
12C3×#3752—21no—
13C32×C4⋊Dic3—21no—
14#88815C2211yescharacteristic
15C3×#3724C2212yes—
16C3×D3×C4⋊C4—61no—
17#63912—31no—
18#63906—61no—
19#63900D311yescharacteristic, complemented
20#63778—31no—
21#63742—32no—
22#63739—32no—
23C22×C3×C3⋊D3—21no—
24C22×C3×C3⋊D3D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C3×C4×C3⋊D3—22no—
29C2×C32×Dic3—22no—
30#6295—22no—
31#6295—42no—
32C3×#624—42no—
33C3⋊#622—22no—
34#4968—91nomaximal
35C2×C4×C3⋊D3C2×D311yescharacteristic
36C2×C3×C4×D3—61no—
37C2×C3×C4×D3—62no—
38C2×C3×C4×D3—61no—
39C2×#626—61no—
40C2×#626—61no—
41C2×#626—61no—
42C2×#625—31no—
43C2×#625—61no—
44C2×#624—21no—
45C2×#622—32no—
46C32×C4⋊C4—61no—
47#3752—21no—
48C3×(C2×D3)⋊C4—61no—
49C3×(C2×D3)⋊C4—61no—
50C3×(C2×D3)⋊C4—121no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—41no—
54C3×Dic3⋊C4—62no—
55#3724—31no—
56#3724C2×D312yes—
57Dic3⋊Dic3—61no—
58C22×C33C2×D411yescharacteristic
59C2×C3×C3⋊D3—21no—
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—41no—
62C2×C3×C3⋊D3—41no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3SD1612yes—
65C33×C4—21no—
66C3×C3⋊Dic3—41no—
67C3×C3⋊Dic3#4112yes—
68C32×Dic3—42no—
69C2×C3×C4⋊C4—91no—
70#930—91no—
71D3×C4⋊C4—61no—
72#884—181no—
73#830—91no—
74#809—92no—
75#808—92no—
76C22×C3⋊D3—21no—
77C22×C3⋊D3#6711yescharacteristic
78C22×C3×D3—61no—
79C22×C3×D3—61no—
80C22×C3×D3—61no—
81C22×C3×D3—61no—
82C22×C3×D3—121no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—62no—
88C2×C32×C4#6711yescharacteristic
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3C22×D311yescharacteristic
91C4×C3⋊D3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—121no—
100C3×C4×D3—64no—
101C3×C4×D3—122no—
102C3×C4×D3—124no—
103#626—122no—
104#626—122no—
105#626—122no—
106#625—62no—
107#625—122no—
108#624—42no—
109#622—62no—
110#481—271noSylow
111C2×C33#12311yescharacteristic
112C2×C33C2×SD1611yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C3×C3⋊D3—22no—
115C3×C3⋊D3—81no—
116C22×C3×C4—91no—
117C22×C3×C4—181no—
118C2×#67—181no—
119C2×#67—181no—
120C2×#67—181no—
121C2×D12—91no—
122C2×D12—181no—
123C2×C4×D3—61no—
124C2×C4×D3—61no—
125C2×C4×D3—62no—
126C3×C4⋊C4—61no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—92no—
129C3×C4⋊C4—121no—
130C3×C22⋊C4—181no—
131(C2×D3)⋊C4—61no—
132(C2×D3)⋊C4—61no—
133(C2×D3)⋊C4—121no—
134C4⋊Dic3—21no—
135C4⋊Dic3—91no—
136Dic3⋊C4—62no—
137Dic3⋊C4—181no—
138C2×C3⋊C8—92no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32C2×#6711yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3—41no—
147C2×C3⋊D3—41no—
148C2×C3⋊D3—41no—
149C2×C3⋊D3#21512yes—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C2×C3×D3—122no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C2×C3×D3—241no—
164C2×C3×D3—241no—
165C32×C4—22no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—42no—
169C32×C4—81no—
170C32×C4—122no—
171C3⋊Dic3—41no—
172C3⋊Dic3#23512yes—
173C3×Dic3—42no—
174C3×Dic3—42no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C3×Dic3—82no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180C3×Dic3—241no—
181#121—271no—
182C2×C4⋊C4—91no—
183#106—272no—
184#102—272no—
185C22⋊C8—271no—
186C33#48111yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C23×C3—181no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—61no—
192C22×D3—61no—
193C22×D3—121no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—91no—
201C2×C3×C4—122no—
202C2×C3×C4—182no—
203C2×C3×C4—181no—
204C2×C3×C4—362no—
205C2×C3×C4GO+(4,𝔽2)11yescharacteristic
206#67—362no—
207#67—362no—
208#67—362no—
209C2×Dic3—22no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—121no—
215D12—182no—
216D12—362no—
217C4×D3—64no—
218C4×D3—122no—
219C4×D3—124no—
220C3⋊C8—182no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32#92911yescharacteristic
234C2×C32#93211yescharacteristic
235C2×C32C2×#21511yescharacteristic
236C3⋊D3—22no—
237C3⋊D3—81no—
238C3×D3—64no—
239C3×D3—122no—
240C3×D3—241no—
241C3×D3—242no—
242C3×D3—241no—
243C2×D4—271no—
244C2×D4—541no—
245C22×C4—91no—
246C22×C4—181no—
247C2×C8—272no—
248C4⋊C4—61no—
249C4⋊C4—92no—
250C4⋊C4—271no—
251C22⋊C4—181no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—92no—
257C22×C3—181no—
258C22×C3—181no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3C2×GO+(4,𝔽2)11yescharacteristic
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—122no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—241no—
277C2×D3—241no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—42no—
282C3×C4—81no—
283C3×C4—92no—
284C3×C4—122no—
285C3×C4—122no—
286C3×C4—242no—
287C3×C4—361no—
288Dic3—42no—
289Dic3—62no—
290Dic3—62no—
291Dic3—121no—
292Dic3—121no—
293Dic3—241no—
294C32—21no—
295C32—21no—
296C32—41no—
297C32—41no—
298C32#496811yescharacteristic, complemented
299C23—91no—
300C23—181no—
301D4—542no—
302D4—1082no—
303C2×C4—62no—
304C2×C4—91no—
305C2×C4—182no—
306C2×C4—181no—
307C2×C4—362no—
308C2×C4D32⋊D311yescharacteristic
309C8—542no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—182no—
323C2×C3—721no—
324C2×C3#6422411yescharacteristic
325C2×C3#6422711yescharacteristic
326C2×C3#6423311yescharacteristic
327D3—64no—
328D3—122no—
329D3—241no—
330D3—242no—
331D3—241no—
332C22—92no—
333C22—181no—
334C22—181no—
335C22—361no—
336C22—361no—
337C22—361no—
338C22C2×D32⋊D311yescharacteristic
339C4—21no—
340C4—92no—
341C4—122no—
342C4—361no—
343C3—21no—
344C3—21no—
345C3—41no—
346C3—41no—
347C3#1059336111yescharacteristic, complemented
348C2—182no—
349C2—721no—
350C2#1172430411yescharacteristic
351C2#1172430811yescharacteristic
352C2#1172431711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723695C2
hypercenterC2×C4D32⋊D3
hypocenterC33#481
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63900C2semidirect
3. #11723103C2semidirect
4. #11722778C2semidirect
5. #11722772C2semidirect
19. #63900D3semidirect
186. C33#481semidirect
298. C32#4968semidirect
347. C3#10593361semidirect

Maximal subgroups

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

subgroupindex
#117236952
C3×#639002
#117231032
#117227782
#117227722
#105933613
#49689