Finite Group Databasebeta

← #13167589

Subgroups — #13167589

Order 1728.

672 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 71 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×C3×C22⋊#157C211yesmaximal, characteristic
2C2×#88940C211yesmaximal, characteristic, complemented
3#11723723C212yesmaximal
4#11722791C212yesmaximal, complemented
5#11722791C211yesmaximal, characteristic, complemented
6#10593486—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C22×C3×#157C2211yescharacteristic
10C2×#6294C2212yes—
11C2×#6294C2212yes—
12C3×C22⋊#157—22no—
13#88940—22no—
14C3×#3757—22no—
15#88814—22no—
16C2×C22⋊#157D311yescharacteristic
17C2×#3773—31no—
18#63974—32no—
19D3×#217—62no—
20#63737—32no—
21#63737—31no—
22C23×C33D411yescharacteristic
23C22×C3×C3⋊D3—21no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3×#157—22no—
30C2×C3×#157—21no—
31C2×C3×#157—22no—
32C2×C3×#157D412yes—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35#6294—22no—
36#6294—22no—
37#6294—22no—
38#6294—22no—
39#4953—91nomaximal
40C23×C3⋊D3C2×D311yescharacteristic
41C23×C3×D3—62no—
42C22×#157C2×D311yescharacteristic
43C22×#157C2×D311yescharacteristic
44C2×#624—32no—
45C2×#624—32no—
46C2×D3×Dic3—62no—
47C2×D3×Dic3—62no—
48C22⋊#157—22no—
49#3773—62no—
50#3757—22no—
51C3×#217—62no—
52C3×#217—62no—
53C3×#217—122no—
54#3722—62no—
55(C2×D3)⋊Dic3—62no—
56(C2×D3)⋊Dic3—62no—
57C22×C33—21no—
58C22×C33—22no—
59C22×C33C2×D411yescharacteristic
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—22no—
62C2×C3×C3⋊D3—41no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3C2×D411yescharacteristic
66C2×C3×C3⋊D3#4112yes—
67C2×C3×C3⋊D3#4112yes—
68C2×C3×C3⋊D3#4111yescharacteristic
69C3×#157—22no—
70C3×#157—41no—
71C3×C3⋊Dic3—42no—
72C3×C3⋊Dic3—42no—
73C2×C3×C22⋊C4—91no—
74C2×#217—92no—
75C2×C4⋊Dic3—91no—
76D3×C22⋊C4—182no—
77C3⋊#95—91no—
78C3⋊#95—92no—
79C23×C32—22no—
80C23×C32—42no—
81C23×C32D1211yescharacteristic
82C22×C3⋊D3—22no—
83C22×C3⋊D3—22no—
84C22×C3⋊D3—21no—
85C22×C3⋊D3—21no—
86C22×C3⋊D3D1211yescharacteristic
87C22×C3⋊D3C22×D311yescharacteristic
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C22×C3×D3—64no—
91C22×C3×D3—64no—
92C22×C3×D3—122no—
93C22×C3×D3—122no—
94C22×C3×D3—122no—
95C22×C3×D3—122no—
96C2×#157—22no—
97C2×#157—22no—
98C2×#157—21no—
99C2×#157#6712yes—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—122no—
107C2×C3×Dic3—122no—
108#624—62no—
109#624—62no—
110#624—62no—
111#624—62no—
112D3×Dic3—122no—
113D3×Dic3—122no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#397—271noSylow
117C2×C33—41no—
118C2×C33#12111yescharacteristic
119C2×C33#12312yes—
120C3×C3⋊D3—41no—
121C3×C3⋊D3#12311yescharacteristic
122C3×C3⋊D3#12411yescharacteristic
123C3×C3⋊D3#12612yes—
124C24×C3—91no—
125C23×D3—62no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×Dic3—92no—
129C22×Dic3—92no—
130C2×C4×D3—182no—
131C2×C4×D3—182no—
132C3×C22⋊C4—182no—
133C3×C22⋊C4—182no—
134#217—62no—
135#217—62no—
136#217—122no—
137#217—182no—
138(C2×D3)⋊C4—182no—
139(C2×D3)⋊C4—182no—
140C4⋊Dic3—182no—
141C22×C32—24no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—21no—
145C22×C32—42no—
146C22×C32—42no—
147C22×C32—42no—
148C22×C32—44no—
149C22×C32—82no—
150C22×C32—82no—
151C22×C32C2×D1211yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3⋊D3—21no—
154C2×C3⋊D3—42no—
155C2×C3⋊D3—41no—
156C2×C3⋊D3—41no—
157C2×C3⋊D3#23511yescharacteristic
158C2×C3⋊D3#23712yes—
159C2×C3⋊D3#23712yes—
160C2×C3⋊D3C2×#6711yescharacteristic
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—64no—
168C2×C3×D3—122no—
169C2×C3×D3—122no—
170C2×C3×D3—124no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C2×C3×D3—124no—
174#157—22no—
175#157—41no—
176C3⋊Dic3—42no—
177C3⋊Dic3—42no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—122no—
181C3×Dic3—122no—
182C3×Dic3—242no—
183C3×Dic3—242no—
184C2×C4⋊C4—271no—
185C2×C22⋊C4—91no—
186C2×C22⋊C4—272no—
187#95—271no—
188#95—272no—
189C33#39711yescharacteristic, complemented, Sylow
190C23×C3—22no—
191C23×C3—42no—
192C23×C3—91no—
193C23×C3—91no—
194C23×C3—182no—
195C23×C3—182no—
196C23×C3—181no—
197C23×C3—181no—
198C23×C3GO+(4,𝔽2)11yescharacteristic
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—64no—
202C22×D3—64no—
203C22×D3—122no—
204C22×D3—122no—
205C22×D3—122no—
206C22×D3—122no—
207C2×C3×C4—92no—
208C2×C3×C4—182no—
209C2×C3×C4—182no—
210C2×C3×C4—182no—
211C2×C3×C4—181no—
212C2×C3×C4—182no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—122no—
218C2×Dic3—122no—
219C2×Dic3—182no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—182no—
223C4×D3—362no—
224C4×D3—362no—
225C4×D3—362no—
226C4×D3—362no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—42no—
231C2×C32—42no—
232C2×C32—42no—
233C2×C32—41no—
234C2×C32—44no—
235C2×C32—84no—
236C2×C32#88612yes—
237C2×C32#93011yescharacteristic
238C3⋊D3—41no—
239C3⋊D3#87912yes—
240C3⋊D3#93511yescharacteristic
241C3⋊D3#93611yescharacteristic
242C3×D3—62no—
243C3×D3—62no—
244C3×D3—62no—
245C3×D3—62no—
246C3×D3—124no—
247C24—91no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4—272no—
251C22×C4—272no—
252C4⋊C4—542no—
253C22⋊C4—182no—
254C22⋊C4—182no—
255C22⋊C4—542no—
256C22×C3—24no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3—82no—
265C22×C3—82no—
266C22×C3—92no—
267C22×C3—92no—
268C22×C3—91no—
269C22×C3—91no—
270C22×C3—182no—
271C22×C3—181no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—362no—
275C22×C3C2×GO+(4,𝔽2)11yescharacteristic
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—64no—
283C2×D3—122no—
284C2×D3—122no—
285C2×D3—124no—
286C2×D3—124no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—182no—
290C3×C4—362no—
291C3×C4—362no—
292C3×C4—361no—
293Dic3—122no—
294Dic3—122no—
295Dic3—122no—
296Dic3—122no—
297Dic3—242no—
298Dic3—242no—
299C32—22no—
300C32—42no—
301C32#495311yescharacteristic, complemented
302C23—91no—
303C23—91no—
304C23—181no—
305C23—182no—
306C23—182no—
307C23—181no—
308C23#157⋊D311yescharacteristic
309C2×C4—92no—
310C2×C4—182no—
311C2×C4—181no—
312C2×C4—182no—
313C2×C4—182no—
314C2×C4—182no—
315C2×C4—542no—
316C2×C4—542no—
317C2×C4—542no—
318C2×C4—542no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—22no—
322C2×C3—44no—
323C2×C3—41no—
324C2×C3—42no—
325C2×C3—42no—
326C2×C3—42no—
327C2×C3—84no—
328C2×C3—91no—
329C2×C3—91no—
330C2×C3—92no—
331C2×C3—361no—
332C2×C3#6423312yes—
333C2×C3#6424211yescharacteristic
334D3—62no—
335D3—62no—
336D3—62no—
337D3—62no—
338D3—124no—
339C22—22no—
340C22—21no—
341C22—92no—
342C22—92no—
343C22—91no—
344C22—91no—
345C22—182no—
346C22—181no—
347C22—361no—
348C22—361no—
349C22—362no—
350C22C2×#157⋊D311yescharacteristic
351C4—182no—
352C4—361no—
353C4—362no—
354C4—362no—
355C3—22no—
356C3—42no—
357C3#1059348611yescharacteristic, complemented
358C2—41no—
359C2—91no—
360C2—91no—
361C2—92no—
362C2—361no—
363C2#1172434612yes—
364C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC22×C3×C3⋊D3C23
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×C3×C3⋊D3C22
hypercenterC23#157⋊D3
hypocenterC33#397
LCSG ▷ C22×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88940C2semidirect
4. #11722791C2semidirect
5. #11722791C2semidirect
189. C33#397semidirect
301. C32#4953semidirect
357. C3#10593486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊#1572
C2×#889402
#117237232
#117227912
#105934863
#49539