Finite Group Databasebeta

← #13167398

Subgroups — #13167398

Order 1728.

470 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 61 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
2#11723678C211yesmaximal, characteristic, complemented
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4#11722797C212yesmaximal, complemented
5#11722793C212yesmaximal
6#10593347—31nomaximal
7C2×C3×C4×C3⋊D3—21no—
8C2×C3×#634—21no—
9C2×#6295C2211yescharacteristic
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#88815—21no—
13#88815C2212yes—
14#88815C2211yescharacteristic
15#88814—21no—
16C4×#626—61no—
17C4×#625—31no—
18#63873—31no—
19#63866—61no—
20C4⋊#622D311yescharacteristic, complemented
21#63743—32no—
22#63739—32no—
23C22×C3×C3⋊D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3×C4×C3⋊D3—42no—
30C3×#634—42no—
31#6295—21no—
32#6295Q812yescomplemented
33C3×#622—22no—
34#4846—91nomaximal
35C2×C4×C3⋊D3—21no—
36C2×#634—21no—
37C2×C3×C4×D3—61no—
38C2×C3×C4×D3—61no—
39C2×C3×C4×D3—121no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—61no—
42C2×C3×Dic6—121no—
43C2×#626—61no—
44C2×#625—31no—
45C2×#622C2×D312yes—
46C4×C3⋊Dic3C2×D311yescharacteristic
47C3×C4×Dic3—61no—
48C3×C4×Dic3—61no—
49#3724—32no—
50#3724—31no—
51#3724—61no—
52Dic3⋊Dic3—62no—
53Dic3⋊Dic3—61no—
54Dic3⋊Dic3—62no—
55#3722—61no—
56(C2×D3)⋊Dic3—61no—
57(C2×D3)⋊Dic3—61no—
58C22×C33C2×D411yescharacteristic
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3—41no—
61C2×C3×C3⋊D3—41no—
62C33×C4D812yes—
63C3×C3⋊Dic3—21no—
64C3×C3⋊Dic3—42no—
65C3×C3⋊Dic3—41no—
66C3×C3⋊Dic3C2×Q811yescharacteristic
67C3×C3⋊Dic3#4111yescharacteristic
68#944—181no—
69C4×#67—181no—
70C4×D12—91no—
71#869—91no—
72C3×C4⋊C8—91no—
73#821—92no—
74#818—92no—
75C22×C3⋊D3—21no—
76C22×C3×D3—61no—
77C22×C3×D3—61no—
78C22×C3×D3—121no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4D1211yescharacteristic
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3D1211yescharacteristic
87C2×C3⋊Dic3C22×D311yescharacteristic
88C4×C3⋊D3—42no—
89#634—42no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—121no—
100C3×C4×D3—122no—
101C3×C4×D3—124no—
102C3×C4×D3—122no—
103C3×Dic6—122no—
104C3×Dic6—124no—
105C3×Dic6—122no—
106#626—62no—
107#626—62no—
108#625—32no—
109#625—61no—
110#622—22no—
111#474—271noSylow
112C2×C33C22⋊Q811yescharacteristic
113C2×C33C2×D811yescharacteristic
114C2×C33#13711yescharacteristic
115C3×C3⋊D3—42no—
116C2×C3×Q8—181no—
117C22×C3×C4—181no—
118C2×#67—181no—
119C2×D12—91no—
120C2×C4×D3—61no—
121C2×C4×D3—61no—
122C2×C4×D3—121no—
123C2×Dic6—61no—
124C2×Dic6—61no—
125C2×Dic6—121no—
126C2×C3×C8—92no—
127C3×C42—91no—
128#217—181no—
129(C2×D3)⋊C4—181no—
130(C2×D3)⋊C4—181no—
131C4⋊Dic3—91no—
132C4⋊Dic3—92no—
133C4⋊Dic3—181no—
134Dic3⋊C4—181no—
135Dic3⋊C4—182no—
136Dic3⋊C4—182no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32C2×D1211yescharacteristic
144C2×C3⋊D3—22no—
145C2×C3⋊D3—41no—
146C2×C3⋊D3—41no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—122no—
150C2×C3×D3—121no—
151C2×C3×D3—121no—
152C2×C3×D3—122no—
153C2×C3×D3—122no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—42no—
159C32×C4—42no—
160C32×C4D2412yes—
161C3⋊Dic3—21no—
162C3⋊Dic3—42no—
163C3⋊Dic3—41no—
164C3⋊Dic3#23711yescharacteristic
165C3⋊Dic3D3×Q811yescharacteristic
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—62no—
171C3×Dic3—121no—
172C3×Dic3—122no—
173C3×Dic3—121no—
174C3×Dic3—124no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C3×Dic3—121no—
178C4⋊Q8—271no—
179C4×D4—271no—
180#107—272no—
181C4⋊C8—91no—
182#102—272no—
183C33#47411yescharacteristic, complemented, Sylow
184C23×C3—181no—
185C22×D3—61no—
186C22×D3—61no—
187C22×D3—121no—
188C3×Q8—362no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—182no—
196C2×C3×C4—181no—
197C2×C3×C4—362no—
198C2×C3×C4GO+(4,𝔽2)11yescharacteristic
199#67—182no—
200#67—182no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—62no—
206C2×Dic3—61no—
207C2×Dic3—62no—
208C2×Dic3—61no—
209C2×Dic3—122no—
210C2×Dic3—121no—
211D12—92no—
212D12—181no—
213C4×D3—122no—
214C4×D3—124no—
215C4×D3—122no—
216Dic6—122no—
217Dic6—124no—
218Dic6—122no—
219C3×C8—182no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32#89711yescharacteristic
233C2×C32C2×D2411yescharacteristic
234C2×C32#90911yescharacteristic
235C3⋊D3—42no—
236C3×D3—122no—
237C3×D3—124no—
238C3×D3—122no—
239C2×Q8—181no—
240C2×D4—271no—
241C22×C4—181no—
242C2×C8—92no—
243C4⋊C4—272no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C22⋊C4—541no—
247C42—91no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—182no—
253C22×C3—361no—
254C22×C3—361no—
255C22×C3C2×GO+(4,𝔽2)11yescharacteristic
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—122no—
259C2×D3—121no—
260C2×D3—121no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—121no—
264C2×D3—121no—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—42no—
268C3×C4—42no—
269C3×C4—91no—
270C3×C4—91no—
271C3×C4—181no—
272C3×C4—361no—
273C3×C4—362no—
274C3×C4#377412yes—
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—61no—
279Dic3—62no—
280Dic3—121no—
281Dic3—122no—
282Dic3—121no—
283Dic3—124no—
284Dic3—122no—
285Dic3—122no—
286Dic3—121no—
287C32—21no—
288C32—21no—
289C32—41no—
290C32—41no—
291C32#484611yescharacteristic, complemented
292C23—181no—
293Q8—362no—
294D4—272no—
295D4—541no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—182no—
300C2×C4—362no—
301C2×C4#157⋊D311yescharacteristic
302C8—182no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—362no—
316C2×C3D32⋊Q811yescharacteristic
317C2×C3#6422611yescharacteristic
318C2×C3C2×#377411yescharacteristic
319D3—122no—
320D3—124no—
321D3—122no—
322C22—182no—
323C22—361no—
324C22—361no—
325C22C2×#157⋊D311yescharacteristic
326C4—91no—
327C4—91no—
328C4—181no—
329C4—361no—
330C4—362no—
331C4#8894112yes—
332C3—21no—
333C3—21no—
334C3—41no—
335C3—41no—
336C3#1059334711yescharacteristic, complemented
337C2—362no—
338C2#1172433311yescharacteristic
339C2#1172433811yescharacteristic
340C2C2×#8894111yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4#157⋊D3
hypocenterC33#474
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. #11723678C2semidirect
3. C3×C4⋊#622C2semidirect
4. #11722797C2semidirect
20. C4⋊#622D3semidirect
32. #6295Q8semidirect
183. C33#474semidirect
291. C32#4846semidirect
336. C3#10593347semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
#117236782
C3×C4⋊#6222
#117227972
#117227932
#105933473
#48469