Finite Group Databasebeta

← #13166522

Subgroups — #13166522

Order 1728.

800 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C22×D3×#157C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#157C211yesmaximal, characteristic
5C3⋊D3×(C2×D3)⋊C4C211yesmaximal, characteristic
6C3×#63781C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593746—31nomaximal
9C22×D3×C3⋊D3C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C2×D3×#157—22no—
15C2×D3×#157—22no—
16C2×D3×#157—22no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C4⋊(C3⋊#157)—22no—
20Dic3⋊#157—22no—
21(C2×D3)⋊#157—22no—
22#88859C411yescharacteristic, complemented
23C32×(C2×D3)⋊C4C411yescharacteristic, complemented
24#88804C411yescharacteristic, complemented
25(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic, complemented
26C23×#157—31no—
27C2×C22⋊#157—31no—
28C2×C4⋊#157—31no—
29C2×C4⋊#157—31no—
30C22⋊C4×C3⋊D3—31no—
31D3×(C2×D3)⋊C4—62no—
32#63781—31no—
33#63781D311yescharacteristic, complemented
34C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
35C22×C3×C3⋊D3C2311yescharacteristic
36C22×C32×D3C2×C411yescharacteristic
37C2×D3×C3⋊D3—21no—
38C2×D3×C3⋊D3—21no—
39C2×D3×C3⋊D3—21no—
40C2×D3×C3⋊D3—21no—
41C2×D3×C3⋊D3D412yes—
42C2×D3×C3⋊D3Q812yes—
43C2×C3⋊#157—21no—
44C2×C3⋊#157—21no—
45C2×C3⋊#157—21no—
46C2×C3⋊#157—21no—
47C2×C3⋊#157—21no—
48C2×C3⋊#157D412yes—
49C2×C3×#157—21no—
50C2×C3×#157—21no—
51C2×C3×#157—21no—
52C2×C3×#157—21no—
53C2×C3×#157—21no—
54C2×C3×#157D412yes—
55D3×#157—28no—
56C2×C33×C4C2×C411yescharacteristic, complemented
57C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
58C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
59C3×C4×C3⋊D3—21no—
60C3×C4×C3⋊D3—21no—
61C2×C32×Dic3C2×C411yescharacteristic, complemented
62Dic3×C3⋊D3—21no—
63Dic3×C3⋊D3—21no—
64#4668—91nomaximal
65C23×C3⋊D3—31no—
66C22×D32—62no—
67C22×#157—31no—
68C22×#157—31no—
69C22×#157—62no—
70C22×#157—62no—
71C22×#157—62no—
72C22×#157C2×D311yescharacteristic
73C22×#157C2×D311yescharacteristic
74C2×C4×C3⋊D3—31no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×C4×D3—62no—
77C2×D3×Dic3—62no—
78C22⋊#157—62no—
79C4⋊#157—62no—
80C4⋊#157—62no—
81C32×C22⋊C4—31no—
82#3757—31no—
83#3752—31no—
84#3752—31no—
85#3752—62no—
86#3752—122no—
87C3×(C2×D3)⋊C4—22no—
88#3722—62no—
89(C2×D3)⋊Dic3—62no—
90C22×C33C22×C411yescharacteristic
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)C4⋊C412yescomplemented
94C2×C3×C3⋊D3C2×D411yescharacteristic
95C2×C3×C3⋊D3C2×D411yescharacteristic
96C2×C3×C3⋊D3C2×D411yescharacteristic
97C2×C3×C3⋊D3C2×Q811yescharacteristic
98C2×C3×C3⋊D3#4111yescharacteristic
99C2×C3×C3⋊D3#4111yescharacteristic
100C2×C32×D3—21no—
101C2×C32×D3—21no—
102C2×C32×D3C4⋊C412yescomplemented
103D3×C3⋊D3—22no—
104D3×C3⋊D3—22no—
105D3×C3⋊D3—22no—
106D3×C3⋊D3—22no—
107C3⋊#157—22no—
108C3⋊#157—41no—
109C3×#157—22no—
110C3×#157—41no—
111C33×C4—21no—
112C3⋊(C3⋊Dic3)—21no—
113C3×C3⋊Dic3—21no—
114C32×Dic3—21no—
115C22×C4×D3—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×C4⋊Dic3—91no—
119C2×Dic3⋊C4—91no—
120D3×C22⋊C4—182no—
121C3×#95—91no—
122C3⋊#95—91no—
123C23×C32—31no—
124C22×C3⋊D3—31no—
125C22×C3⋊D3—32no—
126C22×C3⋊D3—32no—
127C22×C3⋊D3—62no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—61no—
132C22×C3⋊D3—122no—
133C22×C3⋊D3C22×D311yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—62no—
136C2×D32—64no—
137C2×D32—64no—
138C2×D32—122no—
139C2×D32—122no—
140C2×D32—122no—
141C2×D32—122no—
142C2×#157—21no—
143C2×#157—21no—
144C2×#157—21no—
145C2×#157—21no—
146C2×#157—21no—
147C2×#157—32no—
148C2×#157—68no—
149C2×#157—61no—
150C2×#157—61no—
151C2×#157—61no—
152C2×#157—61no—
153C2×#157—61no—
154C2×#157D1212yes—
155C2×C32×C4—22no—
156C2×C32×C4—31no—
157C2×C32×C4—42no—
158C2×C32×C4C4×D311yescharacteristic, complemented
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—122no—
162C2×C3⋊Dic3C4×D311yescharacteristic, complemented
163C4×C3⋊D3—21no—
164C4×C3⋊D3—21no—
165C4×C3⋊D3—61no—
166C4×C3⋊D3—61no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—62no—
169C3×C4×D3—122no—
170C3×C4×D3—122no—
171D3×Dic3—122no—
172D3×Dic3—122no—
173#385—271noSylow
174C2×C33C2×C4⋊C411yescharacteristic
175C2×C33C4×D411yescharacteristic
176C2×C33C4×D411yescharacteristic
177C3⋊(C3⋊D3)—22no—
178C3×C3⋊D3#12011yescharacteristic
179C3×C3⋊D3C22⋊Q811yescharacteristic
180C3×C3⋊D3C22⋊Q811yescharacteristic
181C3×C3⋊D3#12311yescharacteristic
182C32×D3—22no—
183C23×D3—91no—
184C23×D3—182no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×Dic3—91no—
189C22×Dic3—91no—
190C22×Dic3—91no—
191C2×C4×D3—62no—
192C2×C4×D3—182no—
193C2×C4×D3—182no—
194C2×C4×D3—182no—
195C2×C4×D3—182no—
196C3×C22⋊C4—62no—
197#217—182no—
198(C2×D3)⋊C4—91no—
199(C2×D3)⋊C4—91no—
200(C2×D3)⋊C4—91no—
201(C2×D3)⋊C4—182no—
202(C2×D3)⋊C4—182no—
203(C2×D3)⋊C4—182no—
204(C2×D3)⋊C4—362no—
205(C2×D3)⋊C4#15711yescharacteristic, complemented
206C4⋊Dic3—182no—
207Dic3⋊C4—182no—
208C22×C32—22no—
209C22×C32—32no—
210C22×C32—42no—
211C22×C32—61no—
212C22×C32—61no—
213C22×C32C2×C4×D311yescharacteristic
214C2×C3⋊D3—32no—
215C2×C3⋊D3—64no—
216C2×C3⋊D3—61no—
217C2×C3⋊D3—61no—
218C2×C3⋊D3—62no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—62no—
221C2×C3⋊D3—62no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—122no—
224C2×C3⋊D3—122no—
225C2×C3⋊D3—242no—
226C2×C3⋊D3—242no—
227C2×C3⋊D3C2×D1211yescharacteristic
228C2×C3⋊D3#23511yescharacteristic
229C2×C3⋊D3D3×D411yescharacteristic
230C2×C3⋊D3D3×D411yescharacteristic
231C2×C3⋊D3#23711yescharacteristic
232C2×C3⋊D3D3×Q811yescharacteristic
233C2×C3×D3—24no—
234C2×C3×D3—42no—
235C2×C3×D3—42no—
236C2×C3×D3—62no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242D32—124no—
243D32—124no—
244D32—124no—
245D32—124no—
246#157—22no—
247#157—41no—
248#157—62no—
249#157—121no—
250C32×C4—21no—
251C32×C4—42no—
252C32×C4—61no—
253C32×C4—82no—
254C3⋊Dic3—21no—
255C3⋊Dic3—61no—
256C3⋊Dic3—122no—
257C3⋊Dic3—242no—
258C3×Dic3—42no—
259C3×Dic3—122no—
260C23×C4—271no—
261C2×C4⋊C4—271no—
262C2×C4⋊C4—271no—
263C2×C22⋊C4—271no—
264C2×C22⋊C4—271no—
265#95—91no—
266#95—271no—
267C33#38511yescharacteristic, complemented, Sylow
268C23×C3—62no—
269C23×C3—91no—
270C22×D3—62no—
271C22×D3—91no—
272C22×D3—92no—
273C22×D3—92no—
274C22×D3—184no—
275C22×D3—184no—
276C22×D3—182no—
277C22×D3—181no—
278C22×D3—181no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—362no—
282C22×D3—362no—
283C22×D3—362no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3C2×#15711yescharacteristic
287C2×C3×C4—22no—
288C2×C3×C4—42no—
289C2×C3×C4—62no—
290C2×C3×C4—92no—
291C2×C3×C4—91no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4—181no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4C2×#15711yescharacteristic, complemented
300C2×Dic3—62no—
301C2×Dic3—92no—
302C2×Dic3—91no—
303C2×Dic3—182no—
304C2×Dic3—181no—
305C2×Dic3—181no—
306C2×Dic3—181no—
307C2×Dic3—181no—
308C2×Dic3—181no—
309C2×Dic3—181no—
310C2×Dic3—181no—
311C2×Dic3—362no—
312C2×Dic3C2×#15711yescharacteristic, complemented
313C4×D3—122no—
314C4×D3—122no—
315C4×D3—188no—
316C4×D3—362no—
317C4×D3—362no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—42no—
322C2×C32—42no—
323C2×C32—42no—
324C2×C32—62no—
325C2×C32C4×D1211yescharacteristic
326C2×C32#67⋊C411yescharacteristic
327C2×C32D3×C4⋊C411yescharacteristic
328C3⋊D3—62no—
329C3⋊D3—124no—
330C3⋊D3—244no—
331C3⋊D3#88211yescharacteristic
332C3⋊D3#88311yescharacteristic
333C3⋊D3#89611yescharacteristic
334C3⋊D3#89711yescharacteristic
335C3×D3—44no—
336C3×D3—62no—
337C3×D3—62no—
338C3×D3—62no—
339C3×D3—62no—
340C24—271no—
341C22×C4—91no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—271no—
345C22×C4—271no—
346C22×C4—271no—
347C22×C4—542no—
348C22×C4—542no—
349C22×C4—542no—
350C4⋊C4—542no—
351C4⋊C4—542no—
352C22⋊C4—31no—
353C22⋊C4—271no—
354C22⋊C4—271no—
355C22⋊C4—271no—
356C22⋊C4—542no—
357C22×C3—22no—
358C22×C3—42no—
359C22×C3—64no—
360C22×C3—91no—
361C22×C3—91no—
362C22×C3—91no—
363C22×C3—91no—
364C22×C3—91no—
365C22×C3—91no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3C22×#15711yescharacteristic
369C2×D3—21no—
370C2×D3—21no—
371C2×D3—62no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—62no—
377C2×D3—92no—
378C2×D3—184no—
379C2×D3—182no—
380C2×D3—182no—
381C2×D3—182no—
382C2×D3—182no—
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—364no—
386C2×D3—364no—
387C2×D3—364no—
388C2×D3—364no—
389C2×D3—364no—
390C2×D3—362no—
391C2×D3—362no—
392C2×D3—722no—
393C2×D3—722no—
394C2×D3C4⋊#15712yescomplemented
395C3×C4—21no—
396C3×C4—42no—
397C3×C4—82no—
398C3×C4—122no—
399C3×C4—182no—
400C3×C4—181no—
401C3×C4—361no—
402Dic3—21no—
403Dic3—122no—
404Dic3—182no—
405Dic3—181no—
406Dic3—362no—
407Dic3—361no—
408Dic3—722no—
409C32—22no—
410C32—42no—
411C32#466811yescharacteristic, complemented
412C23—31no—
413C23—91no—
414C23—272no—
415C23—272no—
416C23—271no—
417C23—541no—
418C23—541no—
419C23—541no—
420C23—541no—
421C2×C4—31no—
422C2×C4—92no—
423C2×C4—91no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—272no—
432C2×C4—271no—
433C2×C4—541no—
434C2×C4—541no—
435C2×C4—548no—
436C2×C4—541no—
437C2×C4—541no—
438C2×C4—541no—
439C2×C4—541no—
440C2×C4—541no—
441C2×C4D3×#15711yescharacteristic, complemented
442C2×C3—22no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—42no—
446C2×C3—42no—
447C2×C3—42no—
448C2×C3—91no—
449C2×C3—91no—
450C2×C3—91no—
451C2×C3—91no—
452C2×C3—124no—
453C2×C3C2×C4⋊#15711yescharacteristic
454C2×C3D4×#15711yescharacteristic
455C2×C3D4×#15711yescharacteristic
456D3—22no—
457D3—62no—
458D3—62no—
459D3—62no—
460D3—62no—
461D3—182no—
462D3—364no—
463D3—724no—
464C22—32no—
465C22—61no—
466C22—61no—
467C22—91no—
468C22—91no—
469C22—91no—
470C22—91no—
471C22—91no—
472C22—91no—
473C22—272no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22—542no—
478C22—541no—
479C22—541no—
480C22C2×D3×#15711yescharacteristic
481C4—21no—
482C4—61no—
483C4—182no—
484C4—181no—
485C4—361no—
486C4—542no—
487C4—541no—
488C4—1081no—
489C3—22no—
490C3—42no—
491C3#1059374611yescharacteristic, complemented
492C2—62no—
493C2—91no—
494C2—91no—
495C2—91no—
496C2—91no—
497C2—542no—
498C2D3×C4⋊#15711yescharacteristic
499C2D12×#15711yescharacteristic
500C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC22×D3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#385
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. C2×Dic3⋊#157C2semidirect
6. C3×#63781C2semidirect
7. #11722722C2semidirect
22. #88859C4semidirect
23. C32×(C2×D3)⋊C4C4semidirect
24. #88804C4semidirect
25. (C2×D3)⋊(C3⋊Dic3)C4semidirect
33. #63781D3semidirect
56. C2×C33×C4C2×C4semidirect
57. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
58. C2×C3×C3⋊Dic3C2×C4semidirect
61. C2×C32×Dic3C2×C4semidirect
93. C2×C3⋊(C3⋊D3)C4⋊C4semidirect
102. C2×C32×D3C4⋊C4semidirect
158. C2×C32×C4C4×D3semidirect
162. C2×C3⋊Dic3C4×D3semidirect
205. (C2×D3)⋊C4#157semidirect
267. C33#385semidirect
299. C2×C3×C4C2×#157semidirect
312. C2×Dic3C2×#157semidirect
394. C2×D3C4⋊#157semidirect
411. C32#4668semidirect
441. C2×C4D3×#157semidirect
491. C3#10593746semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×C4⋊(C3⋊#157)2
C2×Dic3⋊#1572
C2×(C2×D3)⋊#1572
C3⋊D3×(C2×D3)⋊C42
C3×#637812
#117227222
#105937463
#46689