Finite Group Databasebeta

← #13166356

Subgroups — #13166356

Order 1728.

720 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 74 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×#89014C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4#11722727C214yesmaximal, complemented
5#10593818—31nomaximal
6C2×#6308C411yescharacteristic, complemented
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12C411yescharacteristic, complemented
9C2×#6290C2212yes—
10C2×D3×C3⋊Dic3C412yescomplemented
11#89014—21no—
12#89014—21no—
13#89014—21no—
14#89014—21no—
15#88994C2212yes—
16C4⋊(C3⋊#157)—21no—
17C4×C3⋊#157—21no—
18C3×C22⋊#157C2212yes—
19C3×C4⋊#157—21no—
20C3×C4×#157—21no—
21C2×#3830—31no—
22C2×#3795—62no—
23#64286—31no—
24#64286D311yescharacteristic, complemented
25#63786—34no—
26C22×C3×C3⋊D3C2311yescharacteristic
27C22×C32×D3C2×C412yes—
28C2×C3⋊#157—22no—
29C2×C3×#157—22no—
30C2×C33×C4C2×C411yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C412yes—
32#6308—21no—
33#6308—21no—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C32×D12—21no—
40C32×D12—21no—
41#6290—22no—
42#6290—22no—
43D3×C3⋊Dic3—22no—
44D3×C3⋊Dic3—22no—
45#5006—91nomaximal
46C2×C32×D4—31no—
47C2×#638—32no—
48C22×C3⋊Dic3—32no—
49#3830—61no—
50#3830—61no—
51#3830—61no—
52#3830—61no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—62no—
56C2×#634—122no—
57C2×C3×D12—22no—
58C2×C3×C4×D3—62no—
59C2×#625—64no—
60C2×D3×Dic3—64no—
61#3795—64no—
62#3795—64no—
63#3795—64no—
64#3795—64no—
65C22⋊#157—32no—
66C22⋊#157C2×D312yes—
67C4⋊#157—21no—
68C4⋊#157—61no—
69C4×#157—21no—
70C4×#157—61no—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—41no—
73C2×C3×C3⋊D3C2×D411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—42no—
77C3⋊#157—42no—
78C3×#157—42no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22⋊C411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C3×C3⋊Dic3C22⋊C411yescharacteristic
84C2×#237—182no—
85C2×#235—91no—
86C3×#117—91no—
87#926—91no—
88#806—94no—
89C23×C32—32no—
90C22×C3⋊D3C22×D311yescharacteristic
91C22×C3×D3—24no—
92C22×C3×D3—62no—
93C2×#157—22no—
94C2×#157—62no—
95C32×D4—61no—
96C32×D4—61no—
97C2×C32×C4—22no—
98C2×C32×C4—42no—
99C2×C32×C4C4×D311yescharacteristic
100#638—62no—
101#638—62no—
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—64no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—124no—
107C2×C3⋊Dic3C4×D311yescharacteristic
108C4×C3⋊D3D1211yescharacteristic
109C4×C3⋊D3D1211yescharacteristic
110C4×C3⋊D3#6711yescharacteristic
111C4×C3⋊D3#6711yescharacteristic
112#634—61no—
113#634—61no—
114#634—64no—
115#634—64no—
116#634—124no—
117#634—124no—
118C2×C3×Dic3—62no—
119C3×D12—24no—
120C3×D12—24no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125#625—68no—
126#625—68no—
127D3×Dic3—68no—
128D3×Dic3—68no—
129#410—271noSylow
130C2×C33—21no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3×C3⋊D3—21no—
133C3×C3⋊D3—21no—
134C32×D3—42no—
135C2×C3×D4—62no—
136C22×C3×C4—91no—
137C2×#67—92no—
138C2×#67—184no—
139C22×Dic3—184no—
140#237—184no—
141#237—184no—
142#237—184no—
143#237—184no—
144#235—181no—
145#235—181no—
146#235—181no—
147#235—181no—
148C2×D12#15711yescharacteristic, complemented
149C2×C4×D3—62no—
150C2×C4×D3—92no—
151C2×Dic6—91no—
152C2×Dic6—182no—
153C2×Dic6—362no—
154C3×C4⋊C4—181no—
155C3×C22⋊C4—92no—
156C3×C42—181no—
157#217—92no—
158C4⋊Dic3—181no—
159C4×Dic3—181no—
160C22×C32—22no—
161C22×C32—42no—
162C22×C32—62no—
163C22×C32—122no—
164C22×C32C2×C4×D311yescharacteristic
165C2×C3⋊D3—41no—
166C2×C3⋊D3C2×D1211yescharacteristic
167C2×C3⋊D3C2×#6711yescharacteristic
168C2×C3×D3—28no—
169C2×C3×D3—48no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—124no—
173#157—42no—
174#157—122no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—42no—
178C32×C4—42no—
179C32×C4(C2×D3)⋊C411yescharacteristic
180C32×C4(C2×D3)⋊C411yescharacteristic
181C3⋊Dic3—62no—
182C3⋊Dic3—68no—
183C3⋊Dic3—128no—
184C3⋊Dic3(C2×D3)⋊C411yescharacteristic
185C3⋊Dic3(C2×D3)⋊C411yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—62no—
188C2×#41—271no—
189#117—91no—
190#117—271no—
191#99—274no—
192C33#41011yescharacteristic, complemented, Sylow
193C23×C3—64no—
194C23×C3—91no—
195C22×D3—62no—
196C22×D3C2×#15712yes—
197C3×D4—64no—
198C3×D4—64no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—182no—
207C2×C3×C4C2×#15711yescharacteristic
208#67—182no—
209#67—182no—
210#67—188no—
211#67—188no—
212C2×Dic3—62no—
213C2×Dic3—92no—
214C2×Dic3—184no—
215C2×Dic3—188no—
216C2×Dic3—188no—
217C2×Dic3—182no—
218C2×Dic3—364no—
219D12—21no—
220D12—21no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—182no—
226C4×D3—182no—
227Dic6—184no—
228Dic6—184no—
229Dic6—181no—
230Dic6—181no—
231Dic6—364no—
232Dic6—364no—
233C2×C32—21no—
234C2×C32—24no—
235C2×C32—22no—
236C2×C32—44no—
237C2×C32—42no—
238C2×C32—122no—
239C2×C32C2×(C2×D3)⋊C411yescharacteristic
240C3⋊D3—21no—
241C3⋊D3—21no—
242C3×D3—48no—
243C3×D3—122no—
244C3×D3—122no—
245#41—541no—
246#41—541no—
247#41—541no—
248#41—541no—
249C2×Q8—271no—
250C2×D4—31no—
251C2×D4—272no—
252C22×C4—91no—
253C22×C4—272no—
254C4⋊C4—181no—
255C4⋊C4—541no—
256C22⋊C4—92no—
257C22⋊C4—272no—
258C42—181no—
259C42—541no—
260C22×C3—22no—
261C22×C3—42no—
262C22×C3—68no—
263C22×C3—91no—
264C22×C3—91no—
265C22×C3—128no—
266C22×C3—361no—
267C22×C3C22×#15711yescharacteristic
268C2×D3—22no—
269C2×D3—42no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—124no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—42no—
276C3×C4—42no—
277C3×C4—91no—
278C3×C4—91no—
279C3×C4—362no—
280C3×C4C22⋊#15711yescharacteristic
281C3×C4C22⋊#15711yescharacteristic
282Dic3—62no—
283Dic3—62no—
284Dic3—188no—
285Dic3—182no—
286Dic3—368no—
287Dic3—362no—
288C32—22no—
289C32—42no—
290C32#500611yescharacteristic, complemented
291C23—32no—
292C23—91no—
293Q8—541no—
294Q8—541no—
295D4—61no—
296D4—61no—
297D4—542no—
298D4—542no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—272no—
306C2×C4—542no—
307C2×C4—542no—
308C2×C4—542no—
309C2×C4D3×#15711yescharacteristic
310C2×C3—24no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—128no—
316C2×C3—181no—
317C2×C3—181no—
318C2×C3C2×C22⋊#15711yescharacteristic
319D3—42no—
320D3—122no—
321D3—122no—
322C22—62no—
323C22—91no—
324C22—91no—
325C22—122no—
326C22—361no—
327C22C2×D3×#15711yescharacteristic
328C4—91no—
329C4—91no—
330C4—362no—
331C4—542no—
332C4—1082no—
333C4(C2×D3)⋊#15711yescharacteristic
334C4(C2×D3)⋊#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3#1059381811yescharacteristic, complemented
338C2—21no—
339C2—122no—
340C2—181no—
341C2—181no—
342C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89014C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C3×#64286C2semidirect
4. #11722727C2semidirect
6. C2×#6308C4semidirect
8. C2×C32×D12C4semidirect
10. C2×D3×C3⋊Dic3C4semidirect
24. #64286D3semidirect
148. C2×D12#157semidirect
192. C33#410semidirect
290. C32#5006semidirect
337. C3#10593818semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
#117245142
C3×#642862
#117227272
#105938183
#50069