Finite Group Databasebeta

← #13167512

Subgroups — #13167512

Order 1728.

514 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#88942C212yesmaximal, complemented
2#11723697C211yesmaximal, characteristic
3#11723665C211yesmaximal, characteristic, complemented
4C3×#63779C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593355—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×#6296C2211yescharacteristic
9C2×#6295—21no—
10C2×#6295C2212yes—
11C2×C3×#622C2212yes—
12#88942—24no—
13C3×#3750—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15#88815C2211yescharacteristic
16#88813—21no—
17C2×#3775—32no—
18#63914—31no—
19#63911—61no—
20#63838—31no—
21#63837—61no—
22#63779D311yescharacteristic, complemented
23#63739—32no—
24C22×C3×C3⋊D3—22no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4⋊(C3⋊D3)—42no—
31#6296—21no—
32#6296D412yescomplemented
33#6295—22no—
34#6295—22no—
35#6295—42no—
36C3×#622—22no—
37#4848—91nomaximal
38C2×C4⋊(C3⋊D3)—21no—
39C2×C3×D12—61no—
40C2×C3×D12—61no—
41C2×C3×D12—121no—
42C2×#627—31no—
43C2×#627—61no—
44C2×#626—62no—
45C2×#626—62no—
46C2×#625—32no—
47C2×#625—61no—
48C2×#622C2×D312yes—
49#3775—64no—
50#3750—21no—
51C4×C3⋊Dic3C2×D311yescharacteristic
52C3×Dic3⋊C4—61no—
53C3×Dic3⋊C4—61no—
54C3×Dic3⋊C4—121no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57#3724—31no—
58Dic3⋊Dic3—61no—
59Dic32—61no—
60Dic32—61no—
61Dic32—61no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—42no—
66C33×C4—21no—
67C3×C3⋊Dic3—22no—
68C3×C3⋊Dic3—41no—
69C3×C3⋊Dic3SD1612yescomplemented
70C3×C3⋊Dic3C2×D411yescharacteristic
71C3×C3⋊Dic3#4111yescharacteristic
72#940—181no—
73C2×#204—92no—
74#887—181no—
75#874—91no—
76C4×Dic6—91no—
77C3×C4⋊C8—91no—
78#821—92no—
79C22×C3⋊D3—22no—
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—122no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4D1211yescharacteristic
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3D1211yescharacteristic
91C2×C3⋊Dic3C22×D311yescharacteristic
92C4⋊(C3⋊D3)—42no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—121no—
102C2×C3×Dic3—121no—
103C3×D12—122no—
104C3×D12—122no—
105C3×D12—124no—
106#627—32no—
107#627—62no—
108#627—62no—
109#627—61no—
110#626—122no—
111#626—122no—
112#626—124no—
113#625—62no—
114#625—62no—
115#625—122no—
116#622—22no—
117#460—271noSylow
118C2×C33#12111yescharacteristic
119C2×C33C2×SD1611yescharacteristic
120C2×C33AGL(1,ℤ/8)11yescharacteristic
121C3×C3⋊D3—82no—
122C2×C3×D4—181no—
123C2×#67—182no—
124C2×#67—182no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—92no—
128C2×D12—121no—
129C2×D12—181no—
130C2×Dic6—91no—
131C2×Dic6—181no—
132C2×C3×C8—92no—
133C3×C4⋊C4—181no—
134C3×C42—91no—
135C4⋊Dic3—91no—
136Dic3⋊C4—61no—
137Dic3⋊C4—61no—
138Dic3⋊C4—121no—
139Dic3⋊C4—181no—
140C4×Dic3—61no—
141C4×Dic3—61no—
142C4×Dic3—181no—
143C4×Dic3—181no—
144C4×Dic3—181no—
145#204—184no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32C2×D1211yescharacteristic
151C2×C3⋊D3—42no—
152C2×C3⋊D3—42no—
153C2×C3⋊D3—42no—
154C2×C3×D3—122no—
155C2×C3×D3—122no—
156C2×C3×D3—122no—
157C2×C3×D3—124no—
158C2×C3×D3—122no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—242no—
162C2×C3×D3—242no—
163C32×C4—22no—
164C32×C4—21no—
165C32×C4—41no—
166C32×C4—42no—
167C32×C4—81no—
168C3⋊Dic3—22no—
169C3⋊Dic3—41no—
170C3⋊Dic3#20412yescomplemented
171C3⋊Dic3D3×D411yescharacteristic
172C3⋊Dic3#23911yescharacteristic
173C3×Dic3—62no—
174C3×Dic3—62no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—122no—
182C3×Dic3—121no—
183C3×Dic3—121no—
184C3×Dic3—241no—
185C2×SD16—272no—
186#127—271no—
187C4×Q8—271no—
188C4⋊C8—91no—
189#102—272no—
190C33#46011yescharacteristic, complemented, Sylow
191C23×C3—182no—
192C22×D3—62no—
193C22×D3—62no—
194C22×D3—122no—
195C3×D4—362no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4GO+(4,𝔽2)11yescharacteristic
205#67—362no—
206#67—362no—
207#67—364no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—121no—
217C2×Dic3—121no—
218D12—122no—
219D12—122no—
220D12—124no—
221D12—182no—
222D12—182no—
223D12—362no—
224Dic6—92no—
225Dic6—182no—
226Dic6—181no—
227Dic6—182no—
228C3×C8—182no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32#89511yescharacteristic
242C2×C32C2×#20411yescharacteristic
243C2×C32#90811yescharacteristic
244C3⋊D3—82no—
245C3×D3—242no—
246C3×D3—244no—
247C3×D3—242no—
248C2×Q8—271no—
249C2×D4—181no—
250C2×D4—272no—
251C2×D4—541no—
252SD16—544no—
253C2×C8—92no—
254C4⋊C4—181no—
255C4⋊C4—271no—
256C42—91no—
257C42—541no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—362no—
263C22×C3—362no—
264C22×C3—362no—
265C22×C3C2×GO+(4,𝔽2)11yescharacteristic
266C2×D3—122no—
267C2×D3—122no—
268C2×D3—122no—
269C2×D3—124no—
270C2×D3—122no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—242no—
274C2×D3—242no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—42no—
279C3×C4—81no—
280C3×C4—92no—
281C3×C4—91no—
282C3×C4—91no—
283C3×C4—182no—
284C3×C4—361no—
285Dic3—61no—
286Dic3—61no—
287Dic3—62no—
288Dic3—61no—
289Dic3—61no—
290Dic3—62no—
291Dic3—62no—
292Dic3—62no—
293Dic3—122no—
294Dic3—121no—
295Dic3—121no—
296Dic3—241no—
297C32—21no—
298C32—21no—
299C32—41no—
300C32—41no—
301C32#484811yescharacteristic, complemented
302C23—182no—
303Q8—272no—
304Q8—541no—
305D4—362no—
306D4—542no—
307D4—542no—
308D4—1082no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4#157⋊D311yescharacteristic
314C8—182no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—722no—
328C2×C3#6422711yescharacteristic
329C2×C3#6423011yescharacteristic
330C2×C3#6423611yescharacteristic
331D3—242no—
332D3—244no—
333D3—242no—
334C22—362no—
335C22—362no—
336C22—362no—
337C22C2×#157⋊D311yescharacteristic
338C4—21no—
339C4—92no—
340C4—91no—
341C4—91no—
342C4—182no—
343C4—361no—
344C3—21no—
345C3—21no—
346C3—41no—
347C3—41no—
348C3#1059335511yescharacteristic, complemented
349C2—722no—
350C2#1172433911yescharacteristic
351C2#1172434311yescharacteristic
352C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical#11723697C2
hypercenterC2×C4#157⋊D3
hypocenterC33#460
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
1. C2×#88942C2semidirect
3. #11723665C2semidirect
4. C3×#63779C2semidirect
5. #11722793C2semidirect
22. #63779D3semidirect
32. #6296D4semidirect
69. C3×C3⋊Dic3SD16semidirect
170. C3⋊Dic3#204semidirect
190. C33#460semidirect
301. C32#4848semidirect
348. C3#10593355semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
#117236972
#117236652
C3×#637792
#117227932
#105933553
#48489