Finite Group Databasebeta

← #13166634

Subgroups — #13166634

Order 1728.

736 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 64 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×#89029C211yesmaximal, characteristic
2D3×#3792C212yesmaximal, complemented
3#11723113C211yesmaximal, characteristic, complemented
4C3×#63788C211yesmaximal, characteristic, complemented
5#11722730C212yesmaximal, complemented
6#10593824—31nomaximal
7C22×C3⋊(C3⋊Dic3)C411yescharacteristic
8C2×C3×#638C2211yescharacteristic
9C2×C32×#67C411yescharacteristic
10C2×#6293C2211yescharacteristic
11C2×#6290C411yescharacteristic
12C2×Dic3×C3⋊D3C411yescharacteristic
13#89029—22no—
14#89029—22no—
15C2×D3×C3⋊Dic3C2211yescharacteristic
16#88993C2212yes—
17C3×#3792C2212yes—
18#88925—22no—
19D3×#622—22no—
20C2×#3830—31no—
21C2×#3805—62no—
22C2×#3792—32no—
23#63788—31no—
24#63788D311yescharacteristic, complemented
25#63780—32no—
26C23×C33C2×C411yescharacteristic
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C22×C32×D3C2×C411yescharacteristic
29C2×C3⋊(C3⋊Dic3)—21no—
30C2×C3⋊(C3⋊Dic3)—21no—
31C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
33C3×#638—22no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C32×#67—22no—
36C2×C32×Dic3C2×C411yescharacteristic
37#6293—22no—
38#6290—22no—
39Dic3×C3⋊D3—21no—
40Dic3×C3⋊D3—21no—
41D3×C3⋊Dic3D412yes—
42D3×C3⋊Dic3D412yes—
43C3⋊#622—22no—
44C3×#622—22no—
45#4745—91nomaximal
46C2×C32×D4—31no—
47C2×#638—31no—
48C2×#638C2×D311yescharacteristic
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—62no—
52C22×C3⋊Dic3—122no—
53#3830—62no—
54#3830—62no—
55C2×C4×C3⋊D3—31no—
56C2×#634—31no—
57C2×C3×#67—22no—
58C2×C3×#67—62no—
59C2×#627—62no—
60C2×#625—62no—
61#3805—68no—
62#3805—68no—
63C2×D3×Dic3—62no—
64C2×D3×Dic3—62no—
65#3792—32no—
66#3792—62no—
67#3792C2×D312yes—
68C2×#622—62no—
69C22×C33—21no—
70C22×C33—41no—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—41no—
74C2×C32×D3—41no—
75C2×C32×D3C22⋊C412yes—
76C3⋊(C3⋊Dic3)—21no—
77C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
78C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
79C3×C3⋊Dic3C2×D412yes—
80C32×Dic3—21no—
81C2×#237—91no—
82C2×#237—182no—
83D3×M16—92no—
84C3×#100—91no—
85#833—91no—
86#824—92no—
87C23×C32—22no—
88C23×C32—31no—
89C23×C32—42no—
90C23×C32C4×D311yescharacteristic
91C22×C3⋊D3C4×D311yescharacteristic
92C22×C3×D3—22no—
93C22×C3×D3—62no—
94C32×D4—62no—
95C2×C32×C4—31no—
96#638—22no—
97#638—62no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—32no—
100C2×C3⋊Dic3—32no—
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—64no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3—124no—
109C2×C3⋊Dic3—124no—
110C2×C3⋊Dic3—122no—
111C2×C3⋊Dic3—122no—
112C2×C3⋊Dic3C22×D311yescharacteristic
113C4×C3⋊D3—61no—
114C4×C3⋊D3—61no—
115#634—62no—
116C3×#67—28no—
117C3×#67—68no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—62no—
120#627—68no—
121#625—68no—
122D3×Dic3—64no—
123D3×Dic3—64no—
124D3×Dic3—64no—
125D3×Dic3—64no—
126#622—22no—
127#622—62no—
128#413—271noSylow
129C2×C33—21no—
130C2×C33—41no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3×C3⋊D3—41no—
133C32×D3—22no—
134C2×C3×D4—62no—
135C2×C3×D4—91no—
136C2×#67—62no—
137C2×#67—91no—
138C2×#67—182no—
139C2×#67#15711yescharacteristic
140C22×Dic3—91no—
141C22×Dic3—91no—
142C22×Dic3—182no—
143C22×Dic3—182no—
144C22×Dic3—362no—
145#237—182no—
146#237—182no—
147#237—188no—
148#237—188no—
149C2×C4×D3—91no—
150C2×C4×D3—182no—
151C2×Dic6—91no—
152C2×Dic6—182no—
153C3×M16—92no—
154#208—92no—
155ΓL(1,𝔽25)—182no—
156D3×C8—182no—
157C22×C32—21no—
158C22×C32—24no—
159C22×C32—22no—
160C22×C32—32no—
161C22×C32—44no—
162C22×C32—42no—
163C22×C32—41no—
164C22×C32—44no—
165C22×C32—48no—
166C22×C32—121no—
167C22×C32C2×C4×D311yescharacteristic
168C2×C3⋊D3—21no—
169C2×C3⋊D3—41no—
170C2×C3×D3—24no—
171C2×C3×D3—44no—
172C2×C3×D3—64no—
173C2×C3×D3—124no—
174C32×C4—61no—
175C3⋊Dic3—31no—
176C3⋊Dic3—31no—
177C3⋊Dic3—62no—
178C3⋊Dic3—62no—
179C3⋊Dic3—61no—
180C3⋊Dic3—64no—
181C3⋊Dic3—124no—
182C3⋊Dic3—122no—
183C3⋊Dic3—122no—
184C3⋊Dic3D3×D412yes—
185C3×Dic3—24no—
186C3×Dic3—64no—
187C2×#41—271no—
188C2×M16—272no—
189#101—272no—
190#100—91no—
191#100—271no—
192C33#41311yescharacteristic, complemented, Sylow
193C23×C3—22no—
194C23×C3—42no—
195C23×C3—62no—
196C23×C3—91no—
197C23×C3C2×#15711yescharacteristic
198C22×D3—62no—
199C22×D3C2×#15711yescharacteristic
200C3×D4—68no—
201C3×D4—182no—
202C2×C3×C4—62no—
203C2×C3×C4—91no—
204#67—22no—
205#67—68no—
206#67—188no—
207#67—182no—
208C2×Dic3—62no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—184no—
212C2×Dic3—184no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3—181no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—184no—
220C2×Dic3—184no—
221C2×Dic3—362no—
222C2×Dic3—362no—
223C2×Dic3—364no—
224C2×Dic3—364no—
225C2×Dic3C2×#15711yescharacteristic
226C4×D3—92no—
227C4×D3—92no—
228C4×D3—184no—
229C4×D3—184no—
230Dic6—182no—
231Dic6—188no—
232C3×C8—182no—
233C3⋊C8—182no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—22no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32—44no—
240C2×C32—48no—
241C2×C32—41no—
242C2×C32—62no—
243C2×C32D3×C22⋊C411yescharacteristic
244C3⋊D3—41no—
245C3×D3—44no—
246C3×D3—124no—
247#41—542no—
248#41—542no—
249C2×Q8—271no—
250C2×D4—31no—
251C2×D4—91no—
252C2×D4—271no—
253C22×C4—271no—
254C22×C4—271no—
255C22×C4—271no—
256M16—92no—
257M16—272no—
258M16—542no—
259C2×C8—542no—
260C22×C3—24no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—44no—
264C22×C3—41no—
265C22×C3—44no—
266C22×C3—48no—
267C22×C3—42no—
268C22×C3—64no—
269C22×C3—124no—
270C22×C3—181no—
271C22×C3—361no—
272C22×C3C22×#15711yescharacteristic
273C2×D3—41no—
274C2×D3—64no—
275C2×D3—124no—
276C2×D3C22⋊#15712yes—
277C3×C4—64no—
278C3×C4—92no—
279Dic3—21no—
280Dic3—64no—
281Dic3—91no—
282Dic3—91no—
283Dic3—182no—
284Dic3—182no—
285Dic3—184no—
286Dic3—181no—
287Dic3—362no—
288Dic3—362no—
289Dic3—364no—
290C32—22no—
291C32—42no—
292C32#474511yescharacteristic, complemented
293C23—31no—
294C23—91no—
295C23D3×#15711yescharacteristic
296Q8—542no—
297D4—62no—
298D4—182no—
299D4—542no—
300C2×C4—31no—
301C2×C4—91no—
302C2×C4—272no—
303C2×C4—272no—
304C2×C4—271no—
305C2×C4—271no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4—541no—
309C2×C4—541no—
310C8—182no—
311C8—542no—
312C2×C3—24no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—44no—
316C2×C3—44no—
317C2×C3—48no—
318C2×C3—41no—
319C2×C3—42no—
320C2×C3—124no—
321C2×C3—361no—
322C2×C3C2×C22⋊#15711yescharacteristic
323D3—22no—
324D3—124no—
325C22—21no—
326C22—32no—
327C22—41no—
328C22—121no—
329C22—181no—
330C22—361no—
331C22C2×D3×#15711yescharacteristic
332C4—61no—
333C4—92no—
334C4—271no—
335C4—271no—
336C4—541no—
337C3—22no—
338C3—42no—
339C3#1059382411yescharacteristic, complemented
340C2—21no—
341C2—41no—
342C2—62no—
343C2—361no—
344C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC23×C33C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89029C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. D3×#3792C2semidirect
3. #11723113C2semidirect
4. C3×#63788C2semidirect
5. #11722730C2semidirect
24. #63788D3semidirect
192. C33#413semidirect
292. C32#4745semidirect
339. C3#10593824semidirect

Maximal subgroups

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

subgroupindex
C2×#890292
D3×#37922
#117231132
C3×#637882
#117227302
#105938243
#47459