Finite Group Databasebeta

← #13173423

Subgroups — #13173423

Order 1728.

1152 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C23×C3⋊#157C211yesmaximal, characteristic, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
3C2×#88994C211yesmaximal, characteristic, complemented
4C2×#88994C212yesmaximal, complemented
5#11723106C212yesmaximal, complemented
6#10593826—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3⋊#157—22no—
10C22×C3⋊#157—24no—
11C22×C3⋊#157C2212yes—
12C22×C3⋊#157C2212yes—
13C2×C33×D4C411yescharacteristic, complemented
14C2×C3×#638C411yescharacteristic, complemented
15C2×C3×#638C411yescharacteristic, complemented
16C3×D4×C3⋊D3—22no—
17C3×D4×C3⋊D3—22no—
18C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20#88994—22no—
21#88994—24no—
22C23×#157—31no—
23C2×D4×C3⋊D3D311yescharacteristic
24C2×C3×D3×D4—62no—
25C2×C22⋊#157—32no—
26C2×C22⋊#157—31no—
27#63781—32no—
28C23×C33C2×C411yescharacteristic
29C23×C33C2×C411yescharacteristic, complemented
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3—22no—
36C22×C3×C3⋊D3C2×C411yescharacteristic
37C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
38C22×C3×C3⋊D3D412yes—
39C22×C3×C3⋊D3D412yes—
40C22×C3×C3⋊D3C2311yescharacteristic
41C2×C3⋊#157—28no—
42C2×C3⋊#157—22no—
43C2×C3⋊#157—22no—
44C2×C3⋊#157—22no—
45C2×C3⋊#157—22no—
46C2×C3⋊#157—22no—
47C2×C3⋊#157D414yescomplemented
48C33×D4—22no—
49C2×C33×C4C2×C411yescharacteristic, complemented
50C3×#638—22no—
51C3×#638—22no—
52C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56#5220—91nomaximal
57C23×C3⋊D3C2×D311yescharacteristic
58C23×C3⋊D3C2×D311yescharacteristic
59C23×C3×D3—62no—
60C23×C3×D3—62no—
61C22×#157—32no—
62C22×#157—32no—
63C22×#157—64no—
64C22×#157—62no—
65C2×C32×D4—22no—
66C2×C32×D4—42no—
67C2×C32×D4Dic311yescharacteristic, complemented
68C2×#638Dic311yescharacteristic, complemented
69C2×#638Dic311yescharacteristic, complemented
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C3×#67—62no—
75C2×C3×#67—62no—
76C3×D3×D4—68no—
77C3×D3×D4—68no—
78C2×C3×D12—62no—
79C2×C3×C4×D3—62no—
80C22⋊#157—64no—
81C22⋊#157—62no—
82C22×C33—22no—
83C22×C33—21no—
84C22×C33—22no—
85C22×C33C22⋊C412yescomplemented
86C22×C33C22×C411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—24no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—42no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3C22⋊C412yescomplemented
97C2×C3×C3⋊D3C2×D412yes—
98C2×C3×C3⋊D3C2×D411yescharacteristic
99C2×C3×C3⋊D3C2×D411yescharacteristic
100C2×C3×C3⋊D3#4112yes—
101C3⋊#157—24no—
102C3⋊#157—42no—
103C33×C4—21no—
104C3×C3⋊Dic3—21no—
105C22×C3×D4—91no—
106C23×Dic3—91no—
107C2×D3×D4—62no—
108C2×#217—92no—
109C2×#217—91no—
110C3⋊#95—92no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—42no—
114C23×C32—42no—
115C23×C32C2×Dic311yescharacteristic
116C23×C32C2×Dic311yescharacteristic, complemented
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—22no—
120C22×C3⋊D3—22no—
121C22×C3⋊D3—22no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3C2×Dic311yescharacteristic
124C22×C3⋊D3C2×Dic311yescharacteristic, complemented
125C22×C3⋊D3#6712yes—
126C22×C3⋊D3#6712yes—
127C22×C3⋊D3C22×D311yescharacteristic
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—124no—
136C22×C3×D3—124no—
137C22×C3×D3—124no—
138C22×C3×D3—124no—
139C2×#157—34no—
140C2×#157—68no—
141C2×#157—62no—
142C2×#157—62no—
143C2×#157—62no—
144C2×#157—62no—
145C2×#157—62no—
146C32×D4—22no—
147C32×D4—28no—
148C32×D4—48no—
149C2×C32×C4—22no—
150C2×C32×C4—42no—
151C2×C32×C4C2×Dic311yescharacteristic, complemented
152#638—22no—
153#638—22no—
154C2×C3⋊Dic3C2×Dic311yescharacteristic, complemented
155C4⋊(C3⋊D3)—22no—
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158C3×#67—68no—
159C3×#67—68no—
160C2×C3×Dic3—62no—
161C3×D12—68no—
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164#386—271noSylow
165C2×C33—22no—
166C2×C33—41no—
167C2×C33C2×C22⋊C411yescharacteristic
168C2×C33C4×D412yes—
169C3×C3⋊D3—22no—
170C3×C3⋊D3—41no—
171C3×C3⋊D3#12011yescharacteristic
172C3×C3⋊D3#12112yes—
173C3×C3⋊D3#12311yescharacteristic
174C24×C3—91no—
175C24×C3—91no—
176C23×D3—62no—
177C23×D3—62no—
178C2×C3×D4—22no—
179C2×C3×D4—42no—
180C2×C3×D4—91no—
181C2×C3×D4—91no—
182C2×C3×D4—91no—
183C2×C3×D4—182no—
184C2×C3×D4—182no—
185C2×C3×D4#15711yescharacteristic, complemented
186C22×C3×C4—91no—
187C2×#67—62no—
188C2×#67—62no—
189C22×Dic3—92no—
190C22×Dic3—92no—
191C22×Dic3—184no—
192C22×Dic3—182no—
193D3×D4—68no—
194D3×D4—68no—
195C2×D12—62no—
196C2×C4×D3—62no—
197#217—184no—
198#217—182no—
199C22×C32—21no—
200C22×C32—24no—
201C22×C32—22no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—44no—
206C22×C32—42no—
207C22×C32—44no—
208C22×C32—44no—
209C22×C32—44no—
210C22×C32—84no—
211C22×C32—84no—
212C22×C32#21712yescomplemented
213C22×C32C22×Dic311yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—24no—
219C2×C3⋊D3—22no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—42no—
223C2×C3⋊D3#21712yescomplemented
224C2×C3⋊D3D3×D412yes—
225C2×C3⋊D3#23712yes—
226C2×C3⋊D3C2×#6711yescharacteristic
227C2×C3⋊D3C2×#6711yescharacteristic
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—64no—
233C2×C3×D3—64no—
234C2×C3×D3—124no—
235C2×C3×D3—128no—
236C2×C3×D3—124no—
237C2×C3×D3—124no—
238C2×C3×D3—124no—
239C2×C3×D3—124no—
240C2×C3×D3—128no—
241C2×C3×D3—124no—
242#157—64no—
243#157—122no—
244C32×C4—21no—
245C32×C4—24no—
246C32×C4—44no—
247C3⋊Dic3—21no—
248C3×Dic3—64no—
249C22×D4—91no—
250C23×C4—271no—
251C2×C22⋊C4—272no—
252C2×C22⋊C4—271no—
253#95—272no—
254C33#38611yescharacteristic, complemented, Sylow
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—91no—
260C23×C3—91no—
261C23×C3—92no—
262C23×C3—92no—
263C23×C3—91no—
264C23×C3—182no—
265C23×C3—182no—
266C23×C3—181no—
267C23×C3—181no—
268C23×C3—182no—
269C23×C3—182no—
270C23×C3C2×#15711yescharacteristic
271C23×C3C2×#15711yescharacteristic, complemented
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—62no—
275C22×D3—64no—
276C22×D3—64no—
277C22×D3—64no—
278C22×D3—64no—
279C22×D3—124no—
280C22×D3—124no—
281C22×D3—124no—
282C22×D3—124no—
283C3×D4—28no—
284C3×D4—22no—
285C3×D4—48no—
286C3×D4—182no—
287C3×D4—182no—
288C3×D4—182no—
289C2×C3×C4—22no—
290C2×C3×C4—42no—
291C2×C3×C4—91no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C2×#15711yescharacteristic, complemented
295#67—68no—
296#67—68no—
297C2×Dic3—62no—
298C2×Dic3—94no—
299C2×Dic3—182no—
300C2×Dic3—182no—
301C2×Dic3—188no—
302C2×Dic3—182no—
303C2×Dic3—182no—
304C2×Dic3—182no—
305D12—68no—
306C4×D3—64no—
307C4×D3—64no—
308C2×C32—24no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—42no—
312C2×C32—44no—
313C2×C32—44no—
314C2×C32—41no—
315C2×C32—44no—
316C2×C32—84no—
317C2×C32—84no—
318C2×C32D4×Dic312yes—
319C2×C32C2×#21711yescharacteristic
320C3⋊D3—22no—
321C3⋊D3—41no—
322C3⋊D3#93511yescharacteristic
323C3⋊D3#93711yescharacteristic
324C3⋊D3#93912yes—
325C3×D3—62no—
326C3×D3—64no—
327C3×D3—62no—
328C3×D3—124no—
329C3×D3—124no—
330C24—91no—
331C24—91no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—182no—
336C2×D4—182no—
337C2×D4C3⋊#15711yescharacteristic, complemented
338C22×C4—91no—
339C22×C4—272no—
340C22×C4—272no—
341C22×C4—544no—
342C22×C4—542no—
343C22⋊C4—544no—
344C22⋊C4—542no—
345C22×C3—22no—
346C22×C3—24no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—24no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—44no—
356C22×C3—84no—
357C22×C3—84no—
358C22×C3—92no—
359C22×C3—92no—
360C22×C3—91no—
361C22×C3—91no—
362C22×C3—92no—
363C22×C3—182no—
364C22×C3—184no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3—182no—
368C22×C3—181no—
369C22×C3—361no—
370C22×C3—362no—
371C22×C3—361no—
372C22×C3C22⋊#15712yescomplemented
373C22×C3C22×#15711yescharacteristic
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—124no—
381C2×D3—124no—
382C2×D3—128no—
383C2×D3—124no—
384C2×D3—124no—
385C2×D3—124no—
386C2×D3—124no—
387C2×D3—128no—
388C3×C4—24no—
389C3×C4—21no—
390C3×C4—44no—
391C3×C4—181no—
392Dic3—64no—
393Dic3—184no—
394Dic3—362no—
395C32—22no—
396C32—42no—
397C32#522011yescharacteristic, complemented
398C23—91no—
399C23—91no—
400C23—92no—
401C23—92no—
402C23—91no—
403C23—182no—
404C23—182no—
405C23—182no—
406C23—181no—
407C23—181no—
408C23—182no—
409C23C2×C3⋊#15711yescharacteristic
410C23C2×C3⋊#15711yescharacteristic, complemented
411D4—22no—
412D4—182no—
413D4—182no—
414D4—182no—
415C2×C4—91no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—274no—
419C2×C4—542no—
420C2×C4—542no—
421C2×C4—548no—
422C2×C4—542no—
423C2×C4—542no—
424C2×C4—542no—
425C2×C4C2×C3⋊#15711yescharacteristic, complemented
426C2×C3—22no—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—44no—
430C2×C3—44no—
431C2×C3—41no—
432C2×C3—44no—
433C2×C3—42no—
434C2×C3—84no—
435C2×C3—84no—
436C2×C3—91no—
437C2×C3—92no—
438C2×C3—91no—
439C2×C3—182no—
440C2×C3—361no—
441C2×C3D4×#15712yes—
442C2×C3C2×C22⋊#15711yescharacteristic
443D3—62no—
444D3—62no—
445D3—64no—
446D3—124no—
447D3—124no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—92no—
452C22—92no—
453C22—91no—
454C22—91no—
455C22—92no—
456C22—182no—
457C22—184no—
458C22—182no—
459C22—182no—
460C22—182no—
461C22—181no—
462C22—361no—
463C22—362no—
464C22—361no—
465C22#8899412yescomplemented
466C22C22×C3⋊#15711yescharacteristic
467C4—21no—
468C4—181no—
469C4—544no—
470C4—1082no—
471C3—22no—
472C3—42no—
473C3#1059382611yescharacteristic, complemented
474C2—22no—
475C2—41no—
476C2—91no—
477C2—92no—
478C2—91no—
479C2—182no—
480C2—361no—
481C2D4×C3⋊#15712yes—
482C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC2×C33×D4C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊#157C2semidirect
3. C2×#88994C2semidirect
4. C2×#88994C2semidirect
5. #11723106C2semidirect
13. C2×C33×D4C4semidirect
14. C2×C3×#638C4semidirect
15. C2×C3×#638C4semidirect
18. C2×C3×C4⋊(C3⋊D3)C4semidirect
29. C23×C33C2×C4semidirect
37. C22×C3×C3⋊D3C2×C4semidirect
47. C2×C3⋊#157D4semidirect
49. C2×C33×C4C2×C4semidirect
52. C2×C3×C3⋊Dic3C2×C4semidirect
67. C2×C32×D4Dic3semidirect
68. C2×#638Dic3semidirect
69. C2×#638Dic3semidirect
72. C2×C4⋊(C3⋊D3)Dic3semidirect
85. C22×C33C22⋊C4semidirect
96. C2×C3×C3⋊D3C22⋊C4semidirect
116. C23×C32C2×Dic3semidirect
124. C22×C3⋊D3C2×Dic3semidirect
151. C2×C32×C4C2×Dic3semidirect
154. C2×C3⋊Dic3C2×Dic3semidirect
185. C2×C3×D4#157semidirect
212. C22×C32#217semidirect
223. C2×C3⋊D3#217semidirect
254. C33#386semidirect
271. C23×C3C2×#157semidirect
294. C2×C3×C4C2×#157semidirect
337. C2×D4C3⋊#157semidirect
372. C22×C3C22⋊#157semidirect
397. C32#5220semidirect
410. C23C2×C3⋊#157semidirect
425. C2×C4C2×C3⋊#157semidirect
465. C22#88994semidirect
473. C3#10593826semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C2×C3×D4×C3⋊D32
C2×#889942
#117231062
#105938263
#52209