Finite Group Databasebeta

← #13166635

Subgroups — #13166635

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×#89031C211yesmaximal, characteristic
2#11724269C212yesmaximal, complemented
3#11723113C211yesmaximal, characteristic, complemented
4C3×#63788C211yesmaximal, characteristic, complemented
5#11722730C212yesmaximal, complemented
6#10593808—31nomaximal
7C2×#6312C411yescharacteristic
8C2×C3×#638C2211yescharacteristic
9C22×C32×Dic3C411yescharacteristic
10C2×#6293C2211yescharacteristic
11C2×#6292C411yescharacteristic
12C2×#6289C2211yescharacteristic
13#89031—22no—
14#89031—22no—
15C2×Dic3×C3⋊D3C411yescharacteristic
16#88993C2212yes—
17C3×#3792C2212yes—
18#88926—22no—
19#88924—22no—
20C2×#3828—31no—
21C2×#3804—62no—
22#64283—32no—
23#63788—31no—
24#63788D311yescharacteristic, complemented
25#63780—32no—
26C23×C33C2×C411yescharacteristic
27C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
28C22×C3×C3⋊D3C2×C411yescharacteristic
29#6312—22no—
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C3×#638—22no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3C2×C411yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37#6293—22no—
38#6292—22no—
39#6289D412yes—
40#6289D412yes—
41Dic3×C3⋊D3—21no—
42Dic3×C3⋊D3—21no—
43C3⋊#622—22no—
44C3×#622—22no—
45#4745—91nomaximal
46C22×C32×C4—31no—
47C2×#638—31no—
48C2×#638—62no—
49C2×#638—122no—
50C2×#638C2×D311yescharacteristic
51#3828—62no—
52#3828—62no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×#634—31no—
57C2×C3×#67—62no—
58C22×C3×Dic3—22no—
59C2×#627—62no—
60C2×#626—62no—
61C2×#624—62no—
62#3804—68no—
63#3804—68no—
64C2×D3×Dic3—62no—
65#3792—32no—
66#3792C2×D312yes—
67#3788—62no—
68#3787—62no—
69C22×C33—21no—
70C22×C33—41no—
71C22×C33C22×C411yescharacteristic
72C2×C3⋊(C3⋊D3)—41no—
73C2×C3⋊(C3⋊D3)C22⋊C412yes—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3—41no—
76C3⋊(C3⋊Dic3)—21no—
77C3×C3⋊Dic3C2×D412yes—
78C32×Dic3—21no—
79C32×Dic3C22⋊C411yescharacteristic
80C32×Dic3C22⋊C411yescharacteristic
81C2×#237—91no—
82C2×#235—182no—
83D3×M16—92no—
84C3×#100—91no—
85#833—91no—
86#824—92no—
87C23×C32—22no—
88C23×C32—42no—
89C23×C32C4×D311yescharacteristic
90C22×C3⋊D3—31no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—122no—
93C22×C3⋊D3C4×D311yescharacteristic
94C22×C3×D3—62no—
95C2×C32×C4—31no—
96C2×C32×C4—31no—
97C2×C32×C4—61no—
98C2×C32×C4—61no—
99#638—22no—
100#638—62no—
101#638—68no—
102#638—128no—
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4⋊(C3⋊D3)—62no—
108C4×C3⋊D3—32no—
109C4×C3⋊D3—32no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112#634—62no—
113C3×#67—68no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—62no—
119#627—68no—
120#626—68no—
121#624—64no—
122#624—64no—
123D3×Dic3—64no—
124D3×Dic3—64no—
125#622—22no—
126#622—62no—
127#413—271noSylow
128C2×C33—21no—
129C2×C33—41no—
130C2×C33C2×C22⋊C411yescharacteristic
131C3⋊(C3⋊D3)—22no—
132C3×C3⋊D3—41no—
133C2×C3×D4—91no—
134C22×C3×C4—62no—
135C2×#67—62no—
136C2×#67—91no—
137C2×#67—91no—
138C2×#67—182no—
139C2×#67—362no—
140C22×Dic3—91no—
141C22×Dic3#15711yescharacteristic
142#237—182no—
143#237—182no—
144#235—188no—
145#235—188no—
146C2×D12—182no—
147C2×C4×D3—91no—
148C2×C4×D3—182no—
149C2×C4×D3—182no—
150C2×Dic6—91no—
151C2×Dic6—182no—
152C3×M16—92no—
153#208—92no—
154ΓL(1,𝔽25)—182no—
155D3×C8—182no—
156C22×C32—21no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32—44no—
160C22×C32—42no—
161C22×C32—41no—
162C22×C32—44no—
163C22×C32—84no—
164C22×C32C2×C4×D311yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—32no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—64no—
169C2×C3⋊D3—124no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—124no—
172C2×C3⋊D3—244no—
173C2×C3×D3—64no—
174C2×C3×D3—124no—
175C32×C4—31no—
176C32×C4—31no—
177C32×C4—61no—
178C3⋊Dic3—61no—
179C3⋊Dic3—64no—
180C3⋊Dic3—124no—
181C3⋊Dic3D3×D412yes—
182C3×Dic3—24no—
183C3×Dic3—22no—
184C3×Dic3—22no—
185C3×Dic3—64no—
186C2×#41—271no—
187C2×M16—272no—
188#101—272no—
189#100—91no—
190#100—271no—
191C33#41311yescharacteristic, complemented, Sylow
192C23×C3—22no—
193C23×C3—42no—
194C23×C3—91no—
195C23×C3C2×#15711yescharacteristic
196C22×D3—62no—
197C22×D3—91no—
198C22×D3—182no—
199C22×D3—362no—
200C3×D4—182no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—91no—
206#67—68no—
207#67—182no—
208#67—182no—
209#67—188no—
210#67—368no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—62no—
214C2×Dic3—91no—
215C2×Dic3—182no—
216C2×Dic3—181no—
217C2×Dic3—181no—
218C2×Dic3—362no—
219C2×Dic3C2×#15711yescharacteristic
220C2×Dic3C2×#15711yescharacteristic
221D12—188no—
222C4×D3—92no—
223C4×D3—92no—
224C4×D3—184no—
225C4×D3—184no—
226C4×D3—184no—
227C4×D3—184no—
228Dic6—182no—
229Dic6—188no—
230C3×C8—182no—
231C3⋊C8—182no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—44no—
236C2×C32—42no—
237C2×C32—41no—
238C2×C32—44no—
239C2×C32—84no—
240C2×C32D3×C22⋊C411yescharacteristic
241C3⋊D3—41no—
242C3⋊D3—62no—
243C3⋊D3—124no—
244C3⋊D3—244no—
245C3×D3—124no—
246#41—542no—
247#41—542no—
248C2×Q8—271no—
249C2×D4—91no—
250C2×D4—271no—
251C2×D4—271no—
252C22×C4—31no—
253C22×C4—271no—
254C22×C4—271no—
255M16—92no—
256M16—272no—
257M16—542no—
258C2×C8—542no—
259C22×C3—24no—
260C22×C3—21no—
261C22×C3—22no—
262C22×C3—44no—
263C22×C3—41no—
264C22×C3—44no—
265C22×C3—42no—
266C22×C3—84no—
267C22×C3—181no—
268C22×C3—361no—
269C22×C3C22×#15711yescharacteristic
270C2×D3—64no—
271C2×D3—92no—
272C2×D3—124no—
273C2×D3—184no—
274C2×D3—364no—
275C2×D3—361no—
276C2×D3—364no—
277C2×D3—724no—
278C3×C4—62no—
279C3×C4—62no—
280C3×C4—64no—
281C3×C4—92no—
282Dic3—21no—
283Dic3—64no—
284Dic3—184no—
285Dic3—181no—
286Dic3—364no—
287Dic3C22⋊#15711yescharacteristic
288Dic3C22⋊#15711yescharacteristic
289C32—22no—
290C32—42no—
291C32#474511yescharacteristic, complemented
292C23—91no—
293C23—271no—
294C23D3×#15711yescharacteristic
295Q8—542no—
296D4—182no—
297D4—542no—
298D4—542no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—91no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—271no—
307C2×C4—541no—
308C2×C4—541no—
309C8—182no—
310C8—542no—
311C2×C3—24no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—44no—
315C2×C3—44no—
316C2×C3—41no—
317C2×C3—42no—
318C2×C3—84no—
319C2×C3—361no—
320C2×C3C2×C22⋊#15711yescharacteristic
321D3—124no—
322D3—182no—
323D3—364no—
324D3—724no—
325C22—21no—
326C22—41no—
327C22—181no—
328C22—272no—
329C22—361no—
330C22—1081no—
331C22C2×D3×#15711yescharacteristic
332C4—31no—
333C4—31no—
334C4—61no—
335C4—92no—
336C4—541no—
337C3—22no—
338C3—42no—
339C3#1059380811yescharacteristic, complemented
340C2—21no—
341C2—41no—
342C2—361no—
343C2—542no—
344C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC23×C33C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89031C2
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. #11724269C2semidirect
3. #11723113C2semidirect
4. C3×#63788C2semidirect
5. #11722730C2semidirect
24. #63788D3semidirect
191. C33#413semidirect
291. C32#4745semidirect
339. C3#10593808semidirect

Maximal subgroups

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

subgroupindex
C2×#890312
#117242692
#117231132
C3×#637882
#117227302
#105938083
#47459