Finite Group Databasebeta

← #13160966

Subgroups — #13160966

Order 1728.

1328 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 176 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×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10591458—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—22no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic, complemented
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C32×#214—22no—
21C2×C32×C3⋊C8C2211yescharacteristic, complemented
22#88806—21no—
23#88803—21no—
24#88793—22no—
25C2×D4×C3⋊D3D311yescharacteristic
26C2×C3×D3×D4—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29C2×#3755—31no—
30C2×C3×#214D314yescomplemented
31#63856—34no—
32C2×#3715—34no—
33#63659—31no—
34#63659—34no—
35#63659—64no—
36#63648—31no—
37#63647—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—42no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3D411yescharacteristic
46C33×D4—21no—
47C33×D4D412yes—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C3×#638—24no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)D412yes—
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8—21no—
61#5218—91no—
62#5169—94no—
63C23×C3⋊D3—21no—
64C23×C3×D3—64no—
65C2×C32×Q8—31no—
66C2×C32×D4—24no—
67C2×C32×D4C2×D314yes—
68C2×C32×D4C2×D311yescharacteristic
69C2×#638—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C3×#67—64no—
75C3×D3×D4—68no—
76C3×D3×D4—68no—
77C2×C3×D12—34no—
78C2×C3×C4×D3—34no—
79C2×C3×Dic6C2×D314yescomplemented
80C32×SD16—62no—
81C2×C32×C8—31no—
82#3755—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87#3750—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C3×#214—28no—
92C2×C3×C3⋊C8C2×D314yescomplemented
93#3724—64no—
94(C2×D3)⋊Dic3—64no—
95#3715—68no—
96C22×C33—22no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—48no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3SD1612yescomplemented
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)—41no—
110C3×C3⋊Dic3—21no—
111C32×Dic3—41no—
112C22×C3×D4—91no—
113C2×D3×D4—34no—
114C2×C3×SD16—34no—
115#946—94no—
116C2×#215—94no—
117C2×#214—91no—
118C2×#214C3⋊D311yescharacteristic, complemented
119#924—91no—
120#832—94no—
121#832—91no—
122#832—91no—
123#832—184no—
124#830—91no—
125#821—94no—
126(C2×D3)⋊C8—94no—
127C23×C32—21no—
128C23×C32—24no—
129C23×C32—44no—
130C22×C3⋊D3—24no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—42no—
133C22×C3⋊D3—42no—
134C22×C3⋊D3#6711yescharacteristic
135C22×C3×D3—34no—
136C22×C3×D3—616no—
137C22×C3×D3—64no—
138C22×C3×D3—128no—
139C22×C3×D3—128no—
140C32×Q8—61no—
141C32×Q8—61no—
142C32×D4—21no—
143C32×D4—28no—
144C32×D4—24no—
145C32×D4—44no—
146C32×D4#6718yes—
147C32×D4#6712yes—
148C2×C32×C4—24no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—24no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—64no—
155C2×C3⋊Dic3—124no—
156C2×C3⋊Dic3#6711yescharacteristic
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)#6712yes—
159C4×C3⋊D3—21no—
160C4×C3⋊D3—21no—
161C3×#67—616no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—34no—
164C3×D12—38no—
165C3×D12—64no—
166C3×C4×D3—64no—
167C3×C4×D3—64no—
168C3×Dic6—24no—
169C3×Dic6—24no—
170C32×C8—61no—
171C3⋊(C3⋊C8)—61no—
172C3⋊(C3⋊C8)—64no—
173C3⋊(C3⋊C8)—124no—
174C3×C3⋊C8—24no—
175#450—271noSylow
176C2×C33—42no—
177C2×C33#12011yescharacteristic
178C2×C33C2×SD1611yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C3×C3⋊D3—22no—
181C3×C3⋊D3—42no—
182C24×C3—181no—
183C23×D3—64no—
184C2×C3×Q8—34no—
185C2×C3×D4—24no—
186C2×C3×D4—91no—
187C2×C3×D4—181no—
188C2×C3×D4—182no—
189C2×C3×D4—182no—
190C2×C3×D4D3214yes—
191C2×C3×D4C2×C3⋊D311yescharacteristic
192C22×C3×C4—91no—
193C2×#67—64no—
194D3×D4—68no—
195D3×D4—68no—
196C2×D12—34no—
197C2×C4×D3—34no—
198C2×Dic6C2×C3⋊D311yescharacteristic, complemented
199C3×SD16—68no—
200C2×C3×C8—34no—
201#217—181no—
202#215—188no—
203#214—22no—
204#214—182no—
205(C2×D3)⋊C4—184no—
206C4⋊Dic3—94no—
207C4⋊Dic3—91no—
208C4⋊Dic3—184no—
209Dic3⋊C4—181no—
210Dic3⋊C4—184no—
211C2×C3⋊C8—94no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—184no—
214C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—28no—
218C22×C32—48no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—84no—
224C22×C32—84no—
225C22×C32D3×D414yes—
226C22×C32C2×#6711yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—48no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3#21412yescomplemented
234C2×C3×D3—38no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—68no—
238C2×C3×D3—1232no—
239C2×C3×D3—124no—
240C2×C3×D3—124no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—121no—
244C32×C4D3×D414yes—
245C32×C4C2×#6714yes—
246C32×C4C2×#6711yescharacteristic
247C32×C4C2×#6711yescharacteristic
248C3⋊Dic3—21no—
249C3⋊Dic3—121no—
250C3⋊Dic3—124no—
251C3⋊Dic3—244no—
252C3×Dic3—44no—
253C3×Dic3—64no—
254C22×D4—91no—
255C2×SD16—31no—
256C2×SD16—271no—
257C22⋊Q8—271no—
258#102—271no—
259#102—271no—
260C22⋊C8—271no—
261C33#45011yescharacteristic, complemented, Sylow
262C23×C3—24no—
263C23×C3—21no—
264C23×C3—44no—
265C23×C3—91no—
266C23×C3—184no—
267C23×C3—181no—
268C23×C3—362no—
269C23×C3—362no—
270C22×D3—34no—
271C22×D3—616no—
272C22×D3—64no—
273C22×D3—128no—
274C22×D3—128no—
275C3×Q8—64no—
276C3×Q8—64no—
277C3×D4—24no—
278C3×D4—21no—
279C3×D4—28no—
280C3×D4—44no—
281C3×D4—92no—
282C3×D4—184no—
283C3×D4—181no—
284C3×D4#62518yes—
285C3×D4#63812yes—
286C2×C3×C4—24no—
287C2×C3×C4—64no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4C2×D3214yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293#67—616no—
294C2×Dic3—21no—
295C2×Dic3—34no—
296C2×Dic3—184no—
297C2×Dic3—181no—
298C2×Dic3—364no—
299D12—38no—
300D12—64no—
301C4×D3—64no—
302C4×D3—64no—
303Dic6—21no—
304Dic6—21no—
305C3×C8—64no—
306C3⋊C8—21no—
307C3⋊C8—184no—
308C3⋊C8—181no—
309C3⋊C8—364no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—42no—
314C2×C32—48no—
315C2×C32—88no—
316C2×C32D3×SD1614yes—
317C2×C32#91414yes—
318C2×C32#93211yescharacteristic
319C2×C32C2×#21411yescharacteristic
320C2×C32#93714yes—
321C2×C32#95311yescharacteristic
322C3⋊D3—22no—
323C3⋊D3—42no—
324C3×D3—68no—
325C3×D3—128no—
326C24—181no—
327C2×Q8—31no—
328C2×D4—91no—
329C2×D4—181no—
330C2×D4—182no—
331C2×D4—182no—
332C2×D4D3×C3⋊D311yescharacteristic
333C22×C4—91no—
334SD16—62no—
335SD16—542no—
336C2×C8—31no—
337C2×C8—271no—
338C4⋊C4—271no—
339C4⋊C4—541no—
340C22⋊C4—541no—
341C22×C3—22no—
342C22×C3—28no—
343C22×C3—24no—
344C22×C3—48no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—84no—
350C22×C3—84no—
351C22×C3—92no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—182no—
355C22×C3—368no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3D3×#6714yes—
359C22×C3D4×C3⋊D311yescharacteristic
360C2×D3—38no—
361C2×D3—64no—
362C2×D3—64no—
363C2×D3—68no—
364C2×D3—1232no—
365C2×D3—124no—
366C2×D3—124no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—124no—
370C3×C4—181no—
371C3×C4C2×#62514yes—
372C3×C4D3×#6714yes—
373C3×C4D4×C3⋊D311yescharacteristic
374C3×C4C2×#63811yescharacteristic
375Dic3—41no—
376Dic3—64no—
377Dic3—361no—
378Dic3—364no—
379Dic3—724no—
380C32—24no—
381C32#516914yescomplemented
382C32#521811yescharacteristic, complemented
383C23—21no—
384C23—91no—
385C23—184no—
386C23—181no—
387C23—362no—
388C23—362no—
389Q8—61no—
390Q8—61no—
391D4—21no—
392D4—92no—
393D4—184no—
394D4—181no—
395D4#629012yes—
396C2×C4—61no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—541no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—541no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—42no—
408C2×C3—48no—
409C2×C3—88no—
410C2×C3—182no—
411C2×C3—362no—
412C2×C3D3×#21414yes—
413C2×C3#6393414yes—
414C2×C3#6397614yes—
415C2×C3SD16×C3⋊D311yescharacteristic
416C2×C3#6412311yescharacteristic
417C2×C3#6414611yescharacteristic
418D3—68no—
419D3—128no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—92no—
424C22—181no—
425C22—181no—
426C22—182no—
427C22—368no—
428C22—361no—
429C22—361no—
430C22C3⋊D3×#6711yescharacteristic
431C4—121no—
432C4—181no—
433C4—1081no—
434C4C2×#629011yescharacteristic
435C4C3⋊D3×#6711yescharacteristic
436C3—24no—
437C3#1059145814yescomplemented
438C3#1059286811yescharacteristic, complemented
439C2—42no—
440C2—182no—
441C2—362no—
442C2C3⋊D3×#21411yescharacteristic
443C2#1172356211yescharacteristic
444C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. #11723435C2semidirect
4. C2×#88793C2semidirect
5. #11722202C2semidirect
6. #11722034C2semidirect
7. #11722023C2semidirect
17. C2×C32×Dic6C22semidirect
18. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C2×C32×C3⋊C8C22semidirect
30. C2×C3×#214D3semidirect
79. C2×C3×Dic6C2×D3semidirect
92. C2×C3×C3⋊C8C2×D3semidirect
106. C2×C3×C3⋊D3SD16semidirect
118. C2×#214C3⋊D3semidirect
198. C2×Dic6C2×C3⋊D3semidirect
214. C2×C3⋊C8C2×C3⋊D3semidirect
233. C2×C3⋊D3#214semidirect
261. C33#450semidirect
381. C32#5169semidirect
382. C32#5218semidirect
437. C3#10591458semidirect
438. C3#10592868semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C32×#2142
#117234352
C2×#887932
#117222022
#117220342
#117220232
#105928683
#105914583