Finite Group Databasebeta

← #13166604

Subgroups — #13166604

Order 1728.

712 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 84 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×#88994C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
4C2×Dic3×#157C211yesmaximal, characteristic, complemented
5C3⋊D3×#217C211yesmaximal, characteristic
6#11722722C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593747—31nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×Dic3×C3⋊D3C2211yescharacteristic
16#88994—22no—
17C3×C22⋊#157—22no—
18Dic3⋊#157—22no—
19Dic3×#157—22no—
20#88864C411yescharacteristic, complemented
21C32×#217C411yescharacteristic, complemented
22#88803C411yescharacteristic, complemented
23#88803C411yescharacteristic, complemented
24C2×C22⋊#157—31no—
25C2×C22⋊#157D311yescharacteristic
26C2×C4⋊#157—31no—
27C2×C4×#157—31no—
28C22⋊C4×C3⋊D3—31no—
29D3×#217—62no—
30#63781—31no—
31#63781—31no—
32C23×C33C2×C411yescharacteristic
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3—21no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C22×C3×C3⋊D3C2311yescharacteristic
41C2×C3⋊#157—21no—
42C2×C3⋊#157—21no—
43C2×C3⋊#157—21no—
44C2×C3⋊#157—21no—
45C2×C3⋊#157—21no—
46C2×C3⋊#157D412yes—
47C2×C3×#157—21no—
48C2×C3×#157—21no—
49C2×C3×#157—21no—
50C2×C3×#157—21no—
51C2×C3×#157—21no—
52C2×C3×#157D412yes—
53C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
55C2×C32×Dic3C2×C411yescharacteristic
56C2×C32×Dic3C2×C411yescharacteristic
57Dic3×C3⋊D3—21no—
58Dic3×C3⋊D3—21no—
59Dic3×C3⋊D3—21no—
60Dic3×C3⋊D3—21no—
61#4944—91nomaximal
62C23×C3⋊D3C2×D311yescharacteristic
63C23×C3×D3—62no—
64C22×#157—31no—
65C22×#157—31no—
66C22×#157C2×D311yescharacteristic
67C22×#157C2×D311yescharacteristic
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—31no—
70C2×D3×Dic3—62no—
71C2×D3×Dic3—62no—
72C22⋊#157—22no—
73C22⋊#157—62no—
74C4⋊#157—62no—
75C4×#157—62no—
76C32×C22⋊C4—31no—
77#3757—31no—
78#3757—62no—
79#3757—122no—
80#3752—31no—
81#3752—31no—
82C3×#217—22no—
83(C2×D3)⋊Dic3—62no—
84(C2×D3)⋊Dic3—62no—
85C22×C33—21no—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33C22×C411yescharacteristic
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3C2×D411yescharacteristic
97C2×C3×C3⋊D3C2×D411yescharacteristic
98C2×C3×C3⋊D3#4111yescharacteristic
99C2×C3×C3⋊D3#4111yescharacteristic
100C2×C3×C3⋊D3#4111yescharacteristic
101C2×C3×C3⋊D3#4111yescharacteristic
102C3⋊#157—22no—
103C3⋊#157—41no—
104C3×#157—22no—
105C3×#157—41no—
106C3⋊(C3⋊Dic3)—21no—
107C3⋊(C3⋊Dic3)—21no—
108C32×Dic3—21no—
109C32×Dic3—21no—
110C2×C3×C22⋊C4—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×Dic3⋊C4—91no—
114C2×C4×Dic3—91no—
115D3×C22⋊C4—182no—
116C3⋊#95—91no—
117C3⋊#95—91no—
118C23×C32—22no—
119C23×C32—42no—
120C23×C32C4×D311yescharacteristic
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3C4×D311yescharacteristic
128C22×C3⋊D3C22×D311yescharacteristic
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—64no—
132C22×C3×D3—64no—
133C22×C3×D3—122no—
134C22×C3×D3—122no—
135C22×C3×D3—122no—
136C22×C3×D3—122no—
137C2×#157—21no—
138C2×#157—21no—
139C2×#157—21no—
140C2×#157—21no—
141C2×#157—21no—
142C2×#157—32no—
143C2×#157—61no—
144C2×#157—61no—
145C2×#157—61no—
146C2×#157—61no—
147C2×#157—61no—
148C2×#157#6712yes—
149C2×C32×C4—31no—
150C2×C32×C4—31no—
151C2×C3⋊Dic3—31no—
152C2×C3⋊Dic3—31no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3—122no—
157C4×C3⋊D3—61no—
158C4×C3⋊D3—61no—
159C4×C3⋊D3—61no—
160C4×C3⋊D3—61no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163D3×Dic3—122no—
164D3×Dic3—122no—
165D3×Dic3—122no—
166D3×Dic3—122no—
167#388—271noSylow
168C2×C33—41no—
169C2×C33#11711yescharacteristic
170C2×C33C4×D411yescharacteristic
171C2×C33C4×D411yescharacteristic
172C3×C3⋊D3—41no—
173C3×C3⋊D3#12111yescharacteristic
174C3×C3⋊D3#12311yescharacteristic
175C3×C3⋊D3#12411yescharacteristic
176C3×C3⋊D3#12611yescharacteristic
177C24×C3—91no—
178C23×D3—62no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×Dic3—91no—
182C22×Dic3—91no—
183C22×Dic3—91no—
184C22×Dic3—91no—
185C2×C4×D3—182no—
186C2×C4×D3—182no—
187C3×C22⋊C4—62no—
188C3×C22⋊C4—182no—
189#217—91no—
190#217—91no—
191#217—91no—
192#217—182no—
193#217—182no—
194#217—362no—
195#217#15711yescharacteristic, complemented
196(C2×D3)⋊C4—182no—
197(C2×D3)⋊C4—182no—
198Dic3⋊C4—182no—
199C4×Dic3—182no—
200C22×C32—21no—
201C22×C32—24no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—44no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—82no—
210C22×C32—82no—
211C22×C32C2×C4×D311yescharacteristic
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3#23511yescharacteristic
220C2×C3⋊D3D3×D411yescharacteristic
221C2×C3⋊D3#23711yescharacteristic
222C2×C3⋊D3#23711yescharacteristic
223C2×C3⋊D3#23711yescharacteristic
224C2×C3⋊D3C2×#6711yescharacteristic
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—64no—
232C2×C3×D3—122no—
233C2×C3×D3—122no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236C2×C3×D3—124no—
237C2×C3×D3—124no—
238#157—22no—
239#157—41no—
240#157—62no—
241#157—121no—
242C32×C4—61no—
243C32×C4—61no—
244C3⋊Dic3—61no—
245C3⋊Dic3—61no—
246C3⋊Dic3—122no—
247C3⋊Dic3—122no—
248C3⋊Dic3—242no—
249C3⋊Dic3—242no—
250C3×Dic3—42no—
251C3×Dic3—42no—
252C2×C4⋊C4—271no—
253C2×C22⋊C4—91no—
254C2×C22⋊C4—271no—
255C2×C22⋊C4—271no—
256C2×C42—271no—
257#95—271no—
258#95—271no—
259C33#38811yescharacteristic, complemented, Sylow
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—91no—
263C23×C3—91no—
264C23×C3—181no—
265C23×C3—181no—
266C23×C3—181no—
267C23×C3—181no—
268C23×C3—181no—
269C23×C3—181no—
270C23×C3C2×#15711yescharacteristic
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—64no—
274C22×D3—64no—
275C22×D3—122no—
276C22×D3—122no—
277C22×D3—122no—
278C22×D3—122no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—92no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×Dic3—92no—
288C2×Dic3—91no—
289C2×Dic3—91no—
290C2×Dic3—182no—
291C2×Dic3—182no—
292C2×Dic3—181no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295C2×Dic3—181no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C2×Dic3—181no—
301C2×Dic3—362no—
302C2×Dic3—362no—
303C2×Dic3C2×#15711yescharacteristic
304C2×Dic3C2×#15711yescharacteristic
305C4×D3—362no—
306C4×D3—362no—
307C4×D3—362no—
308C4×D3—362no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—42no—
313C2×C32—42no—
314C2×C32—42no—
315C2×C32—41no—
316C2×C32—44no—
317C2×C32—84no—
318C2×C32#87711yescharacteristic
319C2×C32#67⋊C411yescharacteristic
320C2×C32C4×#6711yescharacteristic
321C3⋊D3—41no—
322C3⋊D3#87911yescharacteristic
323C3⋊D3#88511yescharacteristic
324C3⋊D3#93511yescharacteristic
325C3⋊D3#93811yescharacteristic
326C3×D3—62no—
327C3×D3—62no—
328C3×D3—62no—
329C3×D3—62no—
330C3×D3—124no—
331C24—91no—
332C22×C4—91no—
333C22×C4—91no—
334C22×C4—271no—
335C22×C4—271no—
336C22×C4—271no—
337C22×C4—271no—
338C4⋊C4—542no—
339C22⋊C4—31no—
340C22⋊C4—182no—
341C22⋊C4—271no—
342C22⋊C4—271no—
343C22⋊C4—271no—
344C22⋊C4—542no—
345C42—542no—
346C22×C3—24no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—91no—
358C22×C3—91no—
359C22×C3—91no—
360C22×C3—91no—
361C22×C3—91no—
362C22×C3—91no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3—181no—
366C22×C3—361no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3C22×#15711yescharacteristic
371C2×D3—62no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—62no—
377C2×D3—64no—
378C2×D3—122no—
379C2×D3—122no—
380C2×D3—124no—
381C2×D3—124no—
382C2×D3—124no—
383C2×D3—124no—
384C3×C4—122no—
385C3×C4—122no—
386C3×C4—182no—
387C3×C4—361no—
388Dic3—21no—
389Dic3—21no—
390Dic3—182no—
391Dic3—181no—
392Dic3—181no—
393Dic3—362no—
394Dic3—362no—
395Dic3—361no—
396Dic3—722no—
397Dic3—722no—
398C32—22no—
399C32—42no—
400C32#494411yescharacteristic, complemented
401C23—91no—
402C23—91no—
403C23—181no—
404C23—181no—
405C23—181no—
406C23—181no—
407C23—181no—
408C23—181no—
409C23D3×#15711yescharacteristic
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—92no—
413C2×C4—181no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—272no—
419C2×C4—271no—
420C2×C4—271no—
421C2×C4—541no—
422C2×C4—541no—
423C2×C4—541no—
424C2×C4—541no—
425C2×C4—541no—
426C2×C4—541no—
427C2×C4—541no—
428C2×C4—541no—
429C2×C4—541no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—44no—
434C2×C3—41no—
435C2×C3—42no—
436C2×C3—42no—
437C2×C3—42no—
438C2×C3—84no—
439C2×C3—91no—
440C2×C3—91no—
441C2×C3—91no—
442C2×C3—91no—
443C2×C3—361no—
444C2×C3#6428611yescharacteristic
445C2×C3D4×#15711yescharacteristic
446C2×C3D4×#15711yescharacteristic
447D3—62no—
448D3—62no—
449D3—62no—
450D3—62no—
451D3—124no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—91no—
456C22—91no—
457C22—91no—
458C22—91no—
459C22—91no—
460C22—91no—
461C22—181no—
462C22—181no—
463C22—181no—
464C22—361no—
465C22—361no—
466C22—361no—
467C22—361no—
468C22C2×D3×#15711yescharacteristic
469C4—61no—
470C4—61no—
471C4—182no—
472C4—361no—
473C4—542no—
474C4—541no—
475C4—541no—
476C4—1081no—
477C3—22no—
478C3—42no—
479C3#1059374711yescharacteristic, complemented
480C2—41no—
481C2—91no—
482C2—91no—
483C2—91no—
484C2—91no—
485C2—361no—
486C2#1172426111yescharacteristic
487C2#67×#15711yescharacteristic
488C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC23D3×#157
hypocenterC33#388
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×Dic3⋊#157C2semidirect
4. C2×Dic3×#157C2semidirect
6. #11722722C2semidirect
7. #11722722C2semidirect
20. #88864C4semidirect
21. C32×#217C4semidirect
22. #88803C4semidirect
23. #88803C4semidirect
195. #217#157semidirect
259. C33#388semidirect
400. C32#4944semidirect
479. C3#10593747semidirect

Maximal subgroups

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

subgroupindex
C2×#889942
C2×C3×C22⋊#1572
C2×Dic3⋊#1572
C2×Dic3×#1572
C3⋊D3×#2172
#117227222
#105937473
#49449