Finite Group Databasebeta

← #13166355

Subgroups — #13166355

Order 1728.

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

normal subgroups only

This group has 752 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×#89020C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4#11722728C212yesmaximal, complemented
5#11722727C212yesmaximal, complemented
6#10593792—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×C4×D3C411yescharacteristic, complemented
10C2×#6293C411yescharacteristic, complemented
11C2×#6292C2211yescharacteristic
12C2×#6291C411yescharacteristic, complemented
13C2×#6290C2211yescharacteristic
14#89020—21no—
15#89020—21no—
16#89020—21no—
17#89020—21no—
18#88994C2212yes—
19C4⋊(C3⋊#157)—21no—
20C4×C3⋊#157—21no—
21C3×C22⋊#157C2212yes—
22C3×C4⋊#157—21no—
23C3×C4×#157—21no—
24C2×#3828—31no—
25C2×#3798—62no—
26#64286—31no—
27#64286D311yescharacteristic, complemented
28#63786—32no—
29#63774—32no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3C2×C411yescharacteristic
33C2×C3⋊#157—22no—
34C2×C3×#157—22no—
35C2×C33×C4C2×C411yescharacteristic
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C4×C3⋊(C3⋊D3)—21no—
38C4×C3⋊(C3⋊D3)—21no—
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3D411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47#6293—21no—
48#6293—21no—
49#6292—21no—
50#6292—21no—
51#6291—21no—
52#6291—21no—
53#6290—21no—
54#6290—21no—
55#5006—91nomaximal
56C22×C32×C4—31no—
57C2×#638—31no—
58C2×#638—31no—
59#3828—61no—
60#3828—61no—
61#3828—61no—
62#3828—61no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—62no—
66C2×C4×C3⋊D3—122no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634—31no—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3—62no—
71C2×#627—62no—
72C2×#626—62no—
73C2×#626—62no—
74C2×#625—62no—
75#3798—64no—
76#3798—64no—
77#3798—64no—
78#3798—64no—
79C22⋊#157—32no—
80C22⋊#157C2×D312yes—
81C4⋊#157—21no—
82C4⋊#157—61no—
83C4×#157—21no—
84C4×#157—61no—
85C22×C33C22×C411yescharacteristic
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3C2×D411yescharacteristic
90C2×C3×C3⋊D3C2×D411yescharacteristic
91C2×C32×D3—21no—
92C2×C32×D3—41no—
93C3⋊#157—42no—
94C3×#157—42no—
95C33×C4C22⋊C411yescharacteristic
96C33×C4C22⋊C411yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3×C3⋊Dic3C22⋊C411yescharacteristic
99C3×C3⋊Dic3C22⋊C411yescharacteristic
100C32×Dic3—21no—
101C2×#235—91no—
102C2×#235—182no—
103C3×#117—91no—
104#926—91no—
105#806—92no—
106#806—92no—
107C23×C32—31no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—62no—
110C22×C3⋊D3—122no—
111C22×C3⋊D3C22×D311yescharacteristic
112C22×C3×D3—22no—
113C22×C3×D3—62no—
114C2×#157—22no—
115C2×#157—62no—
116C2×C32×C4—22no—
117C2×C32×C4—31no—
118C2×C32×C4—42no—
119C2×C32×C4—61no—
120C2×C32×C4—61no—
121C2×C32×C4C4×D311yescharacteristic
122#638—61no—
123#638—61no—
124#638—61no—
125#638—61no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3C4×D311yescharacteristic
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—61no—
132C4×C3⋊D3—61no—
133C4×C3⋊D3—61no—
134C4×C3⋊D3—64no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—124no—
137C4×C3⋊D3—124no—
138C4×C3⋊D3D1211yescharacteristic
139C4×C3⋊D3D1211yescharacteristic
140C4×C3⋊D3#6711yescharacteristic
141C4×C3⋊D3#6711yescharacteristic
142#634—61no—
143#634—61no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—62no—
146C3×C4×D3—24no—
147C3×C4×D3—24no—
148C3×C4×D3—62no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152#627—64no—
153#627—64no—
154#626—64no—
155#626—64no—
156#626—64no—
157#626—64no—
158#625—64no—
159#625—64no—
160#410—271noSylow
161C2×C33—21no—
162C2×C33C2×C22⋊C411yescharacteristic
163C3⋊(C3⋊D3)—41no—
164C3×C3⋊D3—21no—
165C3×C3⋊D3—21no—
166C32×D3—41no—
167C22×C3×C4—62no—
168C22×C3×C4—91no—
169C2×#67—91no—
170C2×#67—91no—
171C2×#67—182no—
172C2×#67—182no—
173#235—181no—
174#235—181no—
175#235—181no—
176#235—181no—
177#235—184no—
178#235—184no—
179#235—184no—
180#235—184no—
181C2×D12—91no—
182C2×D12—182no—
183C2×C4×D3—62no—
184C2×C4×D3—91no—
185C2×C4×D3—182no—
186C2×C4×D3—362no—
187C2×C4×D3#15711yescharacteristic, complemented
188C2×Dic6—91no—
189C2×Dic6—182no—
190C3×C4⋊C4—181no—
191C3×C22⋊C4—92no—
192C3×C42—181no—
193#217—92no—
194C4⋊Dic3—181no—
195C4×Dic3—181no—
196C22×C32—22no—
197C22×C32—42no—
198C22×C32—61no—
199C22×C32—121no—
200C22×C32C2×C4×D311yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—64no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—128no—
207C2×C3⋊D3—121no—
208C2×C3⋊D3C2×D1211yescharacteristic
209C2×C3⋊D3C2×#6711yescharacteristic
210C2×C3×D3—24no—
211C2×C3×D3—44no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C2×C3×D3—124no—
215#157—42no—
216#157—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—61no—
222C32×C4(C2×D3)⋊C411yescharacteristic
223C32×C4(C2×D3)⋊C411yescharacteristic
224C3⋊Dic3—61no—
225C3⋊Dic3—64no—
226C3⋊Dic3—124no—
227C3⋊Dic3(C2×D3)⋊C411yescharacteristic
228C3⋊Dic3(C2×D3)⋊C411yescharacteristic
229C3×Dic3—24no—
230C3×Dic3—62no—
231C3×Dic3—62no—
232C2×#41—271no—
233#117—91no—
234#117—271no—
235#99—272no—
236#99—272no—
237C33#41011yescharacteristic, complemented, Sylow
238C23×C3—62no—
239C23×C3—91no—
240C22×D3—62no—
241C22×D3—91no—
242C22×D3—182no—
243C22×D3—362no—
244C22×D3C2×#15711yescharacteristic
245C2×C3×C4—22no—
246C2×C3×C4—42no—
247C2×C3×C4—62no—
248C2×C3×C4—64no—
249C2×C3×C4—64no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—182no—
256C2×C3×C4C2×#15711yescharacteristic
257#67—181no—
258#67—181no—
259#67—184no—
260#67—184no—
261#67—181no—
262#67—181no—
263#67—184no—
264#67—184no—
265C2×Dic3—62no—
266C2×Dic3—91no—
267C2×Dic3—182no—
268C2×Dic3—182no—
269C2×Dic3—362no—
270C2×Dic3C2×#15711yescharacteristic
271D12—184no—
272D12—184no—
273D12—181no—
274D12—181no—
275C4×D3—21no—
276C4×D3—21no—
277C4×D3—62no—
278C4×D3—62no—
279C4×D3—62no—
280C4×D3—62no—
281C4×D3—184no—
282C4×D3—184no—
283C4×D3—181no—
284C4×D3—181no—
285C4×D3—364no—
286C4×D3—364no—
287Dic6—181no—
288Dic6—181no—
289Dic6—184no—
290Dic6—184no—
291C2×C32—21no—
292C2×C32—24no—
293C2×C32—22no—
294C2×C32—44no—
295C2×C32—42no—
296C2×C32—121no—
297C2×C32C2×(C2×D3)⋊C411yescharacteristic
298C3⋊D3—21no—
299C3⋊D3—21no—
300C3⋊D3—124no—
301C3⋊D3—128no—
302C3⋊D3—121no—
303C3×D3—44no—
304C3×D3—122no—
305C3×D3—122no—
306#41—541no—
307#41—541no—
308#41—541no—
309#41—541no—
310C2×Q8—271no—
311C2×D4—271no—
312C2×D4—271no—
313C2×D4—271no—
314C22×C4—31no—
315C22×C4—91no—
316C22×C4—271no—
317C4⋊C4—181no—
318C4⋊C4—541no—
319C22⋊C4—92no—
320C22⋊C4—272no—
321C42—181no—
322C42—541no—
323C22×C3—22no—
324C22×C3—42no—
325C22×C3—64no—
326C22×C3—91no—
327C22×C3—91no—
328C22×C3—124no—
329C22×C3—361no—
330C22×C3C22×#15711yescharacteristic
331C2×D3—21no—
332C2×D3—41no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—124no—
336C2×D3—184no—
337C2×D3—181no—
338C2×D3—364no—
339C2×D3—364no—
340C2×D3—368no—
341C2×D3—361no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—64no—
347C3×C4—91no—
348C3×C4—91no—
349C3×C4—362no—
350C3×C4C22⋊#15711yescharacteristic
351C3×C4C22⋊#15711yescharacteristic
352Dic3—21no—
353Dic3—62no—
354Dic3—62no—
355Dic3—184no—
356Dic3—181no—
357Dic3—364no—
358Dic3—362no—
359C32—22no—
360C32—42no—
361C32#500611yescharacteristic, complemented
362C23—31no—
363C23—91no—
364C23—271no—
365Q8—541no—
366Q8—541no—
367D4—541no—
368D4—541no—
369D4—541no—
370D4—541no—
371D4—541no—
372D4—541no—
373C2×C4—31no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—91no—
377C2×C4—91no—
378C2×C4—91no—
379C2×C4—91no—
380C2×C4—91no—
381C2×C4—182no—
382C2×C4—271no—
383C2×C4—541no—
384C2×C4—541no—
385C2×C4—542no—
386C2×C4D3×#15711yescharacteristic
387C2×C3—24no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—44no—
391C2×C3—42no—
392C2×C3—124no—
393C2×C3—181no—
394C2×C3—181no—
395C2×C3C2×C22⋊#15711yescharacteristic
396D3—41no—
397D3—122no—
398D3—122no—
399D3—364no—
400D3—368no—
401D3—361no—
402C22—61no—
403C22—91no—
404C22—91no—
405C22—121no—
406C22—361no—
407C22—541no—
408C22—1081no—
409C22C2×D3×#15711yescharacteristic
410C4—61no—
411C4—91no—
412C4—91no—
413C4—362no—
414C4—541no—
415C4—1082no—
416C4(C2×D3)⋊#15711yescharacteristic
417C4(C2×D3)⋊#15711yescharacteristic
418C3—22no—
419C3—42no—
420C3#1059379211yescharacteristic, complemented
421C2—21no—
422C2—121no—
423C2—181no—
424C2—181no—
425C2—1081no—
426C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC4(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89020C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C3×#64286C2semidirect
4. #11722728C2semidirect
5. #11722727C2semidirect
7. C2×C4×C3⋊(C3⋊D3)C4semidirect
9. C2×C32×C4×D3C4semidirect
10. C2×#6293C4semidirect
12. C2×#6291C4semidirect
27. #64286D3semidirect
187. C2×C4×D3#157semidirect
237. C33#410semidirect
361. C32#5006semidirect
420. C3#10593792semidirect

Maximal subgroups

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

subgroupindex
C2×#890202
#117245142
C3×#642862
#117227282
#117227272
#105937923
#50069