Finite Group Databasebeta

← #13159217

Subgroups — #13159217

Order 1728.

992 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 144 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⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723449C211yesmaximal, characteristic
3C3×#63637C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic
8#10592561—31nomaximal
9#10590529—34nomaximal
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21#64106—31no—
22C4⋊C4×C3⋊D3—31no—
23C4⋊C4×C3⋊D3—34no—
24C4⋊C4×C3⋊D3—64no—
25#63871—34no—
26C32×#106—31no—
27#63647—31no—
28#63643—31no—
29#63637D311yescharacteristic, complemented
30#63635—31no—
31C3×#810—34no—
32C3×#808D314yescomplemented
33#63573—34no—
34#63566—34no—
35#63557—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4×C3⋊(C3⋊D3)—21no—
42C4×C3⋊(C3⋊D3)—21no—
43C4×C3⋊(C3⋊D3)—42no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C2×C32×Dic3—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8—21no—
50#4862—94no—
51#4805—91no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—34no—
54C2×C4×C3⋊D3—64no—
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3—64no—
57C2×C4×C3⋊D3—68no—
58C2×#634C2×D311yescharacteristic
59C2×C3×Dic6—34no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)—64no—
70#3750—61no—
71#3750—64no—
72#3750—68no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8C2×D314yes—
77Dic3⋊Dic3—64no—
78#3722—64no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)SD1612yes—
83C33×C4—41no—
84C33×C4#4111yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90#946—91no—
91#897—94no—
92D3×C4⋊C4—94no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—184no—
95C3×#106—34no—
96C3×#103—91no—
97#835—91no—
98(C2×D3)⋊C8—94no—
99(C2×D3)⋊C8—91no—
100#817—91no—
101#816—94no—
102#810—34no—
103#808—94no—
104#808C3⋊D311yescharacteristic, complemented
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—61no—
122C4×C3⋊D3—68no—
123C4×C3⋊D3—68no—
124C4×C3⋊D3—64no—
125C4×C3⋊D3—61no—
126C4×C3⋊D3—61no—
127C4×C3⋊D3—64no—
128C4×C3⋊D3—122no—
129C4×C3⋊D3—128no—
130C4×C3⋊D3—1216no—
131#634—21no—
132#634—21no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—64no—
135C3×Dic6—64no—
136C3×Dic6—64no—
137C32×C8—61no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—24no—
140C3×C3⋊C8—64no—
141#483—271noSylow
142C2×C33#12311yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33#13711yescharacteristic
145C3⋊(C3⋊D3)—22no—
146C2×C3×Q8—91no—
147C2×C4×D3—94no—
148C2×C4×D3—91no—
149C2×C4×D3—184no—
150C2×C4×D3—184no—
151C2×C4×D3—181no—
152C2×C4×D3—188no—
153C2×Dic6—34no—
154C2×C3×C8—34no—
155C2×C3×C8—91no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4D3214yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—184no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3—184no—
166C4⋊Dic3C2×C3⋊D311yescharacteristic
167Dic3⋊C4—181no—
168Dic3⋊C4—184no—
169Dic3⋊C4—181no—
170Dic3⋊C4—188no—
171C2×C3⋊C8—34no—
172C2×C3⋊C8C2×C3⋊D311yescharacteristic
173C22×C32—24no—
174C22×C32D3×D414yes—
175C22×C32D3×D411yescharacteristic
176C2×C3⋊D3—32no—
177C2×C3⋊D3—38no—
178C2×C3⋊D3—68no—
179C2×C3⋊D3—68no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—64no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—61no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—48no—
188C32×C4—44no—
189C32×C4—41no—
190C32×C4—121no—
191C32×C4#23514yes—
192C32×C4#23511yescharacteristic
193C32×C4#23711yescharacteristic
194C32×C4#23914yes—
195C3⋊Dic3—41no—
196C3⋊Dic3—68no—
197C3⋊Dic3—61no—
198C3⋊Dic3—64no—
199C3⋊Dic3—128no—
200C3⋊Dic3—124no—
201C3⋊Dic3—121no—
202C3×Dic3—44no—
203C3×Dic3—124no—
204C22⋊Q8—271no—
205C2×C4⋊C4—271no—
206#106—31no—
207#106—271no—
208#103—91no—
209#103—271no—
210C22⋊C8—271no—
211C33#48311yescharacteristic, complemented, Sylow
212C22×D3—94no—
213C22×D3—91no—
214C22×D3—184no—
215C3×Q8—181no—
216C3×Q8—181no—
217C2×C3×C4—28no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—64no—
222C2×C3×C4—181no—
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—64no—
227C2×Dic3—94no—
228C2×Dic3—91no—
229C2×Dic3—184no—
230C2×Dic3—188no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C4×D3—188no—
234C4×D3—188no—
235C4×D3—184no—
236C4×D3—184no—
237C4×D3—181no—
238C4×D3—181no—
239C4×D3—362no—
240C4×D3—368no—
241C4×D3—3616no—
242Dic6—64no—
243Dic6—64no—
244C3×C8—64no—
245C3×C8—181no—
246C3⋊C8—21no—
247C3⋊C8—64no—
248C2×C32—24no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32#88311yescharacteristic
252C2×C32#89414yes—
253C2×C32D3×SD1614yes—
254C2×C32D3×SD1611yescharacteristic
255C2×C32#91514yes—
256C2×C32#91811yescharacteristic
257C3⋊D3—616no—
258C3⋊D3—62no—
259C3⋊D3—68no—
260C2×Q8—91no—
261C22×C4—271no—
262C22×C4—541no—
263C2×C8—31no—
264C2×C8—91no—
265C4⋊C4—31no—
266C4⋊C4—271no—
267C4⋊C4—541no—
268C4⋊C4—541no—
269C4⋊C4D3×C3⋊D311yescharacteristic
270C22⋊C4—541no—
271C22×C3—24no—
272C22×C3#381114yes—
273C22×C3D4×C3⋊D311yescharacteristic
274C2×D3—92no—
275C2×D3—98no—
276C2×D3—188no—
277C2×D3—188no—
278C2×D3—188no—
279C2×D3—181no—
280C2×D3—184no—
281C2×D3—181no—
282C2×D3—184no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—48no—
286C3×C4—41no—
287C3×C4—44no—
288C3×C4—124no—
289C3×C4—361no—
290C3×C4#379614yes—
291C3×C4#379814yes—
292C3×C4#382811yescharacteristic
293C3×C4#383211yescharacteristic
294Dic3—41no—
295Dic3—124no—
296Dic3—188no—
297Dic3—181no—
298Dic3—184no—
299Dic3—368no—
300Dic3—361no—
301Dic3—364no—
302C32—24no—
303C32#480511yescharacteristic, complemented
304C32#486214yescomplemented
305C23—271no—
306Q8—181no—
307Q8—181no—
308C2×C4—21no—
309C2×C4—61no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—541no—
313C2×C4—541no—
314C2×C4—541no—
315C2×C4—1082no—
316C2×C4C2×D3×C3⋊D311yescharacteristic
317C8—61no—
318C8—181no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3#6385314yes—
323C2×C3#6393014yes—
324C2×C3#6395114yes—
325C2×C3#6410311yescharacteristic
326C2×C3SD16×C3⋊D311yescharacteristic
327C2×C3#6412411yescharacteristic
328D3—1816no—
329D3—182no—
330D3—188no—
331C22—272no—
332C22—541no—
333C22—541no—
334C22#8903911yescharacteristic
335C4—41no—
336C4—121no—
337C4—361no—
338C4—541no—
339C4—1081no—
340C4#8901711yescharacteristic
341C4#8902011yescharacteristic
342C3—24no—
343C3#1059052914yescomplemented
344C3#1059256111yescharacteristic, complemented
345C2—542no—
346C2#1172342111yescharacteristic
347C2#1172355311yescharacteristic
348C2#1172358811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#63637C2semidirect
4. C32×#808C2semidirect
5. #11722054C2semidirect
6. #11722040C2semidirect
29. #63637D3semidirect
32. C3×#808D3semidirect
104. #808C3⋊D3semidirect
211. C33#483semidirect
303. C32#4805semidirect
304. C32#4862semidirect
343. C3#10590529semidirect
344. C3#10592561semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊(C3⋊D3)2
#117234492
C3×#636372
C32×#8082
#117220542
#117220402
#117220242
#105925613
#105905293