Finite Group Databasebeta

← #13166366

Subgroups — #13166366

Order 1728.

724 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C3⋊D3×#235C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C2×C3×#3787C211yesmaximal, characteristic, complemented
4#11722709C212yesmaximal, complemented
5#11722707C212yesmaximal, complemented
6#10593841—31nomaximal
7#89076C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C32×#235C411yescharacteristic, complemented
10C3⋊D3×#67—21no—
11#89028—21no—
12C3⋊D3×D12C2211yescharacteristic
13C4×D3×C3⋊D3—21no—
14#89020—21no—
15#89018C411yescharacteristic, complemented
16C3⋊D3×Dic6C2211yescharacteristic
17#89014C411yescharacteristic, complemented
18#88994—21no—
19C4⋊(C3⋊#157)C2212yes—
20C4×C3⋊#157—21no—
21C2×C3×#622—21no—
22C3×#3787—21no—
23C3×#3787C2212yes—
24#41×C3⋊D3—31no—
25D3×#235—62no—
26#64286—31no—
27C2×#3787D311yescharacteristic, complemented
28#63784—32no—
29#63782—32no—
30C22×C3×C3⋊D3D411yescharacteristic
31C2×D3×C3⋊D3—21no—
32C2×C3⋊#157—22no—
33C2×C33×C4C2×C411yescharacteristic
34#6312—21no—
35C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
36C4×C3⋊(C3⋊D3)—21no—
37#6308C2×C411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3C2×C411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C32×#67—21no—
44C32×D12C2×C411yescharacteristic
45C32×C4×D3—21no—
46C32×Dic6C2×C411yescharacteristic
47#6293—21no—
48#6292—21no—
49#6291—21no—
50#6290—21no—
51#6289—21no—
52Dic3×C3⋊D3—21no—
53D3×C3⋊Dic3—21no—
54C3×#622—22no—
55#5113—91nomaximal
56C32×#41—31no—
57#3832—31no—
58Q8×C3⋊D3—31no—
59#3830—31no—
60#3830—61no—
61D4×C3⋊D3—31no—
62D4×C3⋊D3—61no—
63#3828—31no—
64#3828—62no—
65#3828—61no—
66#3828—122no—
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C3×#235—22no—
70C2×C3×C4×D3—62no—
71D3×#67—64no—
72#3804—64no—
73D3×D12—62no—
74C4×D32—64no—
75#3799—62no—
76#3798—64no—
77#3795—62no—
78D3×Dic6—62no—
79C22⋊#157—61no—
80C2×#622—21no—
81C4⋊#157—32no—
82C4×#157—61no—
83#3787—21no—
84#3787C2×D312yes—
85C22×C33C22⋊C411yescharacteristic
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3C22⋊C411yescharacteristic
89C2×C3×C3⋊D3C2×D411yescharacteristic
90C2×C32×D3—21no—
91D3×C3⋊D3—42no—
92C3⋊#157—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22×C411yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3C22⋊C411yescharacteristic
97C3×C3⋊Dic3C2×D411yescharacteristic
98C32×Dic3—21no—
99D3×#41—182no—
100C2×#235—91no—
101C22×C3×C8—91no—
102#926—91no—
103#821—92no—
104#816—92no—
105C22×C3⋊D3—61no—
106C22×C3⋊D3D1211yescharacteristic
107C22×C3×D3—62no—
108C2×D32—64no—
109C2×#157—62no—
110C32×Q8—31no—
111C32×D4—31no—
112C32×D4—61no—
113C2×C32×C4—22no—
114C2×C32×C4—42no—
115C2×C32×C4—61no—
116C2×C32×C4C4×D311yescharacteristic
117#638—61no—
118#638—61no—
119#638—61no—
120#638—64no—
121#638—124no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3#6711yescharacteristic
124C4⋊(C3⋊D3)—31no—
125C4⋊(C3⋊D3)—62no—
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—122no—
128C4×C3⋊D3—61no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—61no—
131C4×C3⋊D3—64no—
132C4×C3⋊D3—124no—
133C4×C3⋊D3C4×D311yescharacteristic
134C4×C3⋊D3D1211yescharacteristic
135C4×C3⋊D3#6711yescharacteristic
136C4×C3⋊D3C22×D311yescharacteristic
137#634—31no—
138#634—62no—
139#634—61no—
140#634—122no—
141C3×#67—24no—
142C2×C3×Dic3—62no—
143C3×D12—22no—
144C3×C4×D3—24no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148C3×C4×D3—62no—
149C3×Dic6—22no—
150#627—64no—
151#626—64no—
152#626—64no—
153#625—64no—
154#624—64no—
155D3×Dic3—64no—
156D3×Dic3—64no—
157#622—22no—
158#416—271noSylow
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3⋊(C3⋊D3)—41no—
162C3×C3⋊D3—21no—
163C3×C3⋊D3#13512yes—
164C32×D3—41no—
165C3×#41—62no—
166C22×C3×C4—91no—
167C2×#67—181no—
168#239—182no—
169D3×Q8—182no—
170#237—182no—
171#237—184no—
172D3×D4—182no—
173D3×D4—184no—
174#235—91no—
175#235—91no—
176#235—91no—
177#235—182no—
178#235—181no—
179#235—184no—
180#235—181no—
181#235—362no—
182#235#15711yescharacteristic, complemented
183C2×D12—91no—
184C2×C4×D3—62no—
185C2×C4×D3—181no—
186C2×C4×D3—184no—
187C2×Dic6—91no—
188C2×C3×C8—92no—
189C2×C3×C8—181no—
190C2×C3×C8—181no—
191#217—181no—
192C4⋊Dic3—92no—
193C4×Dic3—181no—
194C22×C32—22no—
195C22×C32—42no—
196C22×C32—61no—
197C22×C32(C2×D3)⋊C411yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—64no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—122no—
203C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
204C2×C3⋊D3C2×D1211yescharacteristic
205C2×C3×D3—24no—
206C2×C3×D3—62no—
207C2×C3×D3—62no—
208C2×C3×D3—124no—
209D32—128no—
210#157—122no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—42no—
214C32×C4—42no—
215C32×C4—61no—
216C32×C4(C2×D3)⋊C411yescharacteristic
217C32×C4C2×C4×D311yescharacteristic
218C3⋊Dic3—61no—
219C3⋊Dic3—64no—
220C3⋊Dic3—124no—
221C3⋊Dic3(C2×D3)⋊C411yescharacteristic
222C3⋊Dic3C2×#6711yescharacteristic
223C3×Dic3—24no—
224C3×Dic3—62no—
225C3×Dic3—62no—
226C2×#41—271no—
227C22×C8—91no—
228#117—271no—
229#103—272no—
230#102—272no—
231C33#41611yescharacteristic, complemented, Sylow
232C23×C3—91no—
233C22×D3—62no—
234C22×D3—181no—
235C22×D3—184no—
236C3×Q8—62no—
237C3×D4—62no—
238C3×D4—64no—
239C2×C3×C4—22no—
240C2×C3×C4—42no—
241C2×C3×C4—64no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4C2×#15711yescharacteristic
248#67—21no—
249#67—181no—
250#67—184no—
251#67—181no—
252#67—184no—
253#67—184no—
254#67—181no—
255#67—364no—
256C2×Dic3—62no—
257C2×Dic3—181no—
258C2×Dic3—184no—
259C2×Dic3—182no—
260D12—91no—
261D12—182no—
262D12—184no—
263D12—181no—
264D12—362no—
265D12C2×#15711yescharacteristic
266C4×D3—21no—
267C4×D3—62no—
268C4×D3—62no—
269C4×D3—62no—
270C4×D3—62no—
271C4×D3—181no—
272C4×D3—184no—
273C4×D3—184no—
274C4×D3—184no—
275C4×D3—181no—
276C4×D3—181no—
277C4×D3—364no—
278Dic6—91no—
279Dic6—182no—
280Dic6—181no—
281Dic6—184no—
282Dic6—362no—
283Dic6C2×#15711yescharacteristic
284C3×C8—182no—
285C2×C32—22no—
286C2×C32—21no—
287C2×C32—42no—
288C2×C32—42no—
289C2×C32—82no—
290C2×C32—121no—
291C2×C32C2×(C2×D3)⋊C411yescharacteristic
292C3⋊D3—21no—
293C3⋊D3—121no—
294C3⋊D3—124no—
295C3⋊D3—244no—
296C3⋊D3#90412yes—
297C3×D3—44no—
298C3×D3—64no—
299C3×D3—122no—
300#41—31no—
301#41—271no—
302#41—271no—
303#41—271no—
304#41—541no—
305#41—541no—
306C2×Q8—271no—
307C2×D4—271no—
308C2×D4—541no—
309C22×C4—91no—
310C22×C4—541no—
311C2×C8—92no—
312C2×C8—181no—
313C2×C8—181no—
314C4⋊C4—272no—
315C22⋊C4—541no—
316C42—541no—
317C22×C3—22no—
318C22×C3—42no—
319C22×C3—64no—
320C22×C3—91no—
321C22×C3—91no—
322C22×C3—182no—
323C22×C3C22⋊#15711yescharacteristic
324C2×D3—21no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—124no—
328C2×D3—184no—
329C2×D3—181no—
330C2×D3—364no—
331C2×D3—362no—
332C2×D3—368no—
333C3×C4—22no—
334C3×C4—22no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—64no—
338C3×C4—91no—
339C3×C4—91no—
340C3×C4C22⋊#15711yescharacteristic
341C3×C4C22×#15711yescharacteristic
342Dic3—21no—
343Dic3—62no—
344Dic3—62no—
345Dic3—184no—
346Dic3—181no—
347Dic3—364no—
348Dic3—362no—
349C32—22no—
350C32—42no—
351C32#511311yescharacteristic, complemented
352C23—91no—
353C23—541no—
354Q8—31no—
355Q8—271no—
356Q8—541no—
357D4—31no—
358D4—61no—
359D4—271no—
360D4—541no—
361D4—541no—
362D4—541no—
363D4—541no—
364C2×C4—61no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—91no—
369C2×C4—91no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4—541no—
373C2×C4—541no—
374C2×C4—542no—
375C2×C4D3×#15711yescharacteristic
376C8—182no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—42no—
380C2×C3—42no—
381C2×C3—82no—
382C2×C3—92no—
383C2×C3—124no—
384C2×C3—181no—
385C2×C3C2×C22⋊#15711yescharacteristic
386D3—41no—
387D3—64no—
388D3—122no—
389D3—364no—
390D3—361no—
391D3—724no—
392C22—61no—
393C22—91no—
394C22—91no—
395C22—182no—
396C22—541no—
397C22—1082no—
398C22(C2×D3)⋊#15711yescharacteristic
399C4—61no—
400C4—91no—
401C4—91no—
402C4—541no—
403C4—1082no—
404C4(C2×D3)⋊#15711yescharacteristic
405C4C2×D3×#15711yescharacteristic
406C3—22no—
407C3—42no—
408C3#1059384111yescharacteristic, complemented
409C2—21no—
410C2—92no—
411C2—121no—
412C2—181no—
413C2—1081no—
414C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC4(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC3⋊D3×#235C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C2×C3×#3787C2semidirect
4. #11722709C2semidirect
5. #11722707C2semidirect
7. #89076C4semidirect
9. C32×#235C4semidirect
15. #89018C4semidirect
17. #89014C4semidirect
27. C2×#3787D3semidirect
182. #235#157semidirect
231. C33#416semidirect
351. C32#5113semidirect
408. C3#10593841semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
#117245142
C2×C3×#37872
#117227092
#117227072
#105938413
#51139