Finite Group Databasebeta

← #13161976

Subgroups — #13161976

Order 1728.

1150 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 143 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
2#11723667C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C3×#63659C212yesmaximal, complemented
5#11722075C212yesmaximal, complemented
6#10591426—31nomaximal
7#10591421—32nomaximal
8C23×C3×C3⋊D3—21no—
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—21no—
11C3×D4×C3⋊D3—22no—
12C3×D4×C3⋊D3C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yescomplemented
16C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
17#88814—21no—
18#88813—21no—
19#88809—22no—
20C2×D4×C3⋊D3D311yescharacteristic
21C2×C3×D3×D4—32no—
22C2×C3×D3×D4—61no—
23#63849—31no—
24#63841—32no—
25C2×#3712—31no—
26C2×#3711—32no—
27#63659D312yescomplemented
28C3×#832—32no—
29C3×#832—32no—
30C3×#832—62no—
31#63564—32no—
32#63563—32no—
33#63562—32no—
34C23×C33—21no—
35C22×C3×C3⋊D3—21no—
36C22×C3×C3⋊D3—24no—
37C22×C3×C3⋊D3—44no—
38C22×C3×C3⋊D3D411yescharacteristic
39C33×D4—21no—
40C33×D4C2×C412yes—
41C2×C33×C4C2311yescharacteristic
42C3×#638—24no—
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yes—
47C3×C4×C3⋊D3C2×C412yes—
48C3×C4×C3⋊D3D412yes—
49C3×C3⋊(C3⋊C8)—22no—
50#5215—91no—
51#4770—92no—
52C23×C3⋊D3—21no—
53C23×C3×D3—62no—
54C23×C3×D3—62no—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4—41no—
58C2×C32×D4C2×D312yes—
59C2×C32×D4C2×D311yescharacteristic
60C2×#638—21no—
61D4×C3⋊D3—22no—
62D4×C3⋊D3Dic314yescomplemented
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×#67—62no—
66C2×C3×#67—62no—
67C3×D3×D4—38no—
68C3×D3×D4—64no—
69C3×D3×D4—64no—
70C3×D3×D4—64no—
71C2×C3×D12—32no—
72C2×C3×D12—61no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—61no—
75C4⋊(C3⋊Dic3)C2×D312yescomplemented
76C2×C3⋊(C3⋊C8)C2×D312yescomplemented
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—32no—
79C3×C4⋊Dic3—62no—
80C2×C3×C3⋊C8—32no—
81C2×C3×C3⋊C8—32no—
82C2×C3×C3⋊C8—62no—
83#3722—61no—
84(C2×D3)⋊Dic3—62no—
85Dic32—62no—
86Dic32—61no—
87#3712—62no—
88#3711—64no—
89C22×C33—22no—
90C22×C33—42no—
91C22×C33C2×D411yescharacteristic
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—48no—
95C2×C3×C3⋊D3—42no—
96C2×C3×C3⋊D3C22⋊C412yes—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—42no—
100C3×C3⋊Dic3C22⋊C412yes—
101C22×C3×D4—91no—
102C2×D3×D4—32no—
103C2×D3×D4—61no—
104#926—91no—
105C2×#208—91no—
106C2×ΓL(1,𝔽25)—92no—
107#892—92no—
108C3×#102—92no—
109#832—32no—
110#832—32no—
111#832—62no—
112#832—92no—
113#821—92no—
114#809—92no—
115C23×C32—21no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—42no—
119C23×C32—42no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3—24no—
122C22×C3⋊D3—44no—
123C22×C3⋊D3#6711yescharacteristic
124C22×C3×D3—32no—
125C22×C3×D3—61no—
126C22×C3×D3—68no—
127C22×C3×D3—62no—
128C22×C3×D3—68no—
129C22×C3×D3—62no—
130C22×C3×D3—128no—
131C22×C3×D3—124no—
132C22×C3×D3—124no—
133C32×D4—21no—
134C32×D4—22no—
135C32×D4—22no—
136C32×D4—24no—
137C32×D4—22no—
138C32×D4—42no—
139C32×D4—42no—
140C32×D4—42no—
141C32×D4C4×D314yes—
142C32×D4C2×Dic312yes—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4C22×D312yes—
147C2×C32×C4C22×D311yescharacteristic
148#638—24no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3#6711yescharacteristic
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)C2×Dic312yes—
153C4×C3⋊D3C2×Dic312yes—
154C4×C3⋊D3#6712yes—
155C3×#67—68no—
156C3×#67—68no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—122no—
162C3×D12—34no—
163C3×D12—62no—
164C3×D12—62no—
165C3×D12—62no—
166C3×C4×D3—34no—
167C3×C4×D3—34no—
168C3×C4×D3—62no—
169C3×C4×D3—62no—
170C3⋊(C3⋊C8)—22no—
171C3×C3⋊C8—62no—
172C3×C3⋊C8—62no—
173C3×C3⋊C8—122no—
174#418—271noSylow
175C2×C33—42no—
176C2×C33C2×C22⋊C411yescharacteristic
177C2×C33AGL(1,ℤ/8)12yes—
178C3×C3⋊D3—22no—
179C3×C3⋊D3—42no—
180C24×C3—181no—
181C23×D3—62no—
182C23×D3—62no—
183C2×C3×D4—21no—
184C2×C3×D4—22no—
185C2×C3×D4—41no—
186C2×C3×D4—94no—
187C2×C3×D4—91no—
188C2×C3×D4—182no—
189C2×C3×D4—181no—
190C2×C3×D4D3212yes—
191C2×C3×D4D3211yescharacteristic
192C22×C3×C4—91no—
193C2×#67—62no—
194C2×#67—62no—
195D3×D4—38no—
196D3×D4—64no—
197D3×D4—64no—
198D3×D4—64no—
199C2×D12—32no—
200C2×D12—61no—
201C2×C4×D3—32no—
202C2×C4×D3—61no—
203C2×C3×C8—92no—
204C3×C4⋊C4—92no—
205#217—181no—
206(C2×D3)⋊C4—182no—
207C4⋊Dic3—32no—
208C4⋊Dic3—32no—
209C4⋊Dic3—62no—
210C4×Dic3—182no—
211C4×Dic3—181no—
212#208—182no—
213C2×C3⋊C8—32no—
214C2×C3⋊C8—32no—
215C2×C3⋊C8—62no—
216ΓL(1,𝔽25)—184no—
217C22×C32—22no—
218C22×C32—24no—
219C22×C32—24no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—41no—
223C22×C32—44no—
224C22×C32—44no—
225C22×C32—42no—
226C22×C32—44no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—82no—
230C22×C32—82no—
231C22×C32—84no—
232C22×C32D3×D412yes—
233C22×C32C2×#6711yescharacteristic
234C2×C3⋊D3—22no—
235C2×C3⋊D3—22no—
236C2×C3⋊D3—48no—
237C2×C3⋊D3—42no—
238C2×C3⋊D3#21712yes—
239C2×C3×D3—34no—
240C2×C3×D3—62no—
241C2×C3×D3—64no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—1216no—
247C2×C3×D3—1216no—
248C2×C3×D3—124no—
249C2×C3×D3—122no—
250C2×C3×D3—122no—
251C32×C4—22no—
252C32×C4—21no—
253C32×C4—22no—
254C32×C4—21no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4C2×C4×D312yes—
258C32×C4D3×D412yes—
259C32×C4C22×Dic311yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C3⋊Dic3—42no—
262C3⋊Dic3#21712yes—
263C3×Dic3—34no—
264C3×Dic3—62no—
265C3×Dic3—122no—
266C3×Dic3—122no—
267C3×Dic3—242no—
268C22×D4—91no—
269C2×M16—271no—
270#117—271no—
271#102—92no—
272#102—272no—
273C33#41811yescharacteristic, complemented, Sylow
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—21no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—91no—
280C23×C3—184no—
281C23×C3—181no—
282C23×C3—364no—
283C22×D3—32no—
284C22×D3—61no—
285C22×D3—68no—
286C22×D3—62no—
287C22×D3—68no—
288C22×D3—62no—
289C22×D3—128no—
290C22×D3—124no—
291C22×D3—124no—
292C3×D4—22no—
293C3×D4—22no—
294C3×D4—21no—
295C3×D4—22no—
296C3×D4—24no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—92no—
301C3×D4—184no—
302C3×D4—181no—
303C3×D4D3×Dic314yes—
304C3×D4#62412yes—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—41no—
308C2×C3×C4—91no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—182no—
312C2×C3×C4C2×D3212yes—
313C2×C3×C4C2×D3211yescharacteristic
314#67—68no—
315#67—68no—
316C2×Dic3—32no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—122no—
321D12—34no—
322D12—62no—
323D12—62no—
324D12—62no—
325C4×D3—34no—
326C4×D3—34no—
327C4×D3—62no—
328C4×D3—62no—
329C3×C8—182no—
330C3⋊C8—62no—
331C3⋊C8—62no—
332C3⋊C8—122no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—21no—
338C2×C32—41no—
339C2×C32—42no—
340C2×C32—42no—
341C2×C32—44no—
342C2×C32—44no—
343C2×C32—84no—
344C2×C32—84no—
345C2×C32D3×C22⋊C412yes—
346C2×C32#91112yes—
347C2×C32#91412yes—
348C2×C32#93212yes—
349C2×C32C2×#21711yescharacteristic
350C3⋊D3—22no—
351C3⋊D3—42no—
352C3×D3—64no—
353C3×D3—64no—
354C3×D3—124no—
355C3×D3—124no—
356C24—181no—
357C2×D4—94no—
358C2×D4—91no—
359C2×D4—182no—
360C2×D4—181no—
361C2×D4#111311yescharacteristic
362C22×C4—91no—
363M16—542no—
364C2×C8—92no—
365C4⋊C4—92no—
366C22⋊C4—541no—
367C42—541no—
368C22×C3—24no—
369C22×C3—24no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—44no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—41no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—84no—
383C22×C3—92no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—368no—
387C22×C3—362no—
388C22×C3D3×#6712yes—
389C22×C3#381111yescharacteristic
390C2×D3—34no—
391C2×D3—62no—
392C2×D3—64no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—1216no—
398C2×D3—1216no—
399C2×D3—124no—
400C2×D3—122no—
401C2×D3—122no—
402C3×C4—21no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—92no—
409C3×C4—362no—
410C3×C4C2×D3×Dic312yes—
411C3×C4C2×#62411yescharacteristic
412C3×C4D3×#6712yes—
413C3×C4#381111yescharacteristic
414Dic3—34no—
415Dic3—62no—
416Dic3—122no—
417Dic3—122no—
418Dic3—242no—
419C32—22no—
420C32—21no—
421C32—41no—
422C32#477012yescomplemented
423C32#521511yescharacteristic, complemented
424C23—21no—
425C23—91no—
426C23—184no—
427C23—181no—
428C23—364no—
429D4—21no—
430D4—92no—
431D4—184no—
432D4—181no—
433D4#629412yes—
434C2×C4—91no—
435C2×C4—92no—
436C2×C4—92no—
437C2×C4—182no—
438C2×C4C2×#111311yescharacteristic
439C8—182no—
440C2×C3—22no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—44no—
446C2×C3—44no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3—84no—
451C2×C3—84no—
452C2×C3—182no—
453C2×C3—362no—
454C2×C3#6393212yes—
455C2×C3#6393412yes—
456C2×C3#6393712yes—
457C2×C3D3×#21712yes—
458C2×C3#6397411yescharacteristic
459D3—64no—
460D3—64no—
461D3—124no—
462D3—124no—
463C22—22no—
464C22—42no—
465C22—92no—
466C22—182no—
467C22—182no—
468C22—368no—
469C22—362no—
470C22#8904911yescharacteristic
471C4—92no—
472C4—362no—
473C4C2×#629411yescharacteristic
474C4#8904911yescharacteristic
475C3—21no—
476C3—22no—
477C3—41no—
478C3#1059142112yescomplemented
479C3#1059142611yescharacteristic, complemented
480C2—42no—
481C2—182no—
482C2—362no—
483C2#1172370612yes—
484C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenterC2×D4#1113
hypocenterC33#418
LCSG ▷ 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. #11723667C2semidirect
3. C2×#88809C2semidirect
4. C3×#63659C2semidirect
5. #11722075C2semidirect
12. C3×D4×C3⋊D3C4semidirect
15. C3×C4⋊(C3⋊Dic3)C22semidirect
16. C2×C3×C3⋊(C3⋊C8)C22semidirect
27. #63659D3semidirect
62. D4×C3⋊D3Dic3semidirect
75. C4⋊(C3⋊Dic3)C2×D3semidirect
76. C2×C3⋊(C3⋊C8)C2×D3semidirect
273. C33#418semidirect
422. C32#4770semidirect
423. C32#5215semidirect
478. C3#10591421semidirect
479. C3#10591426semidirect

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
#117236672
C2×#888092
C3×#636592
#117220752
#105914263
#105914213