Finite Group Databasebeta

← #13140060

Subgroups — #13140060

Order 1728.

559 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88447C211yesmaximal, characteristic
2#11720574C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#63658⋊C3C211yesmaximal, characteristic
5#11720334C211yesmaximal, characteristic
6#11720334C211yesmaximal, characteristic
7#11720334C211yesmaximal, characteristic
8#10591248—31nomaximal
9#10591248—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#88447C414yescomplemented
18C2×Dic32—31no—
19C2×Dic32—31no—
20#63658D311yescharacteristic
21#63658D311yescharacteristic
22C3×C3⋊#95—31no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—31no—
25C3×C3⋊#95—31no—
26C3×C3⋊#95—61no—
27#63579—31no—
28#63579—31no—
29#63579—31no—
30#63579—31no—
31#63579—31no—
32#63579—31no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×#1084—21no—
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084C2×C412yes—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3C2×C412yes—
54C2×(C3⋊Dic3)⋊C3C2×C412yes—
55#4651—91no—
56#4651—91no—
57#4651—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C3⋊Dic3C2×D311yescharacteristic
62C22×C3⋊Dic3C2×D311yescharacteristic
63C22×C3⋊Dic3C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D311yescharacteristic
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—61no—
75Dic32—34no—
76Dic32—34no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4111yescharacteristic
80C22×H(3,𝔽3)#4111yescharacteristic
81C22×H(3,𝔽3)#4111yescharacteristic
82C22×H(3,𝔽3)#4111yescharacteristic
83C22×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)—41no—
85#1084—22no—
86#1084—41no—
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C2×C4×Dic3—91no—
92C2×C4×Dic3—91no—
93C2×C4×Dic3—91no—
94C3×#95—91no—
95C3×#95—91no—
96C3×#95—91no—
97C3⋊#95—31no—
98C3⋊#95—31no—
99C3⋊#95—31no—
100C3⋊#95—31no—
101C3⋊#95—61no—
102C3⋊#95—91no—
103C3⋊#95—91no—
104C3⋊#95—91no—
105C3⋊#95—91no—
106C3⋊#95—91no—
107C3⋊#95—91no—
108C3⋊#95—91no—
109C3⋊#95—91no—
110C3⋊#95—91no—
111C3⋊#95—91no—
112C3⋊#95—91no—
113C23×C32—21no—
114C23×C32C22×D311yescharacteristic
115C23×C32C22×D311yescharacteristic
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C4×D312yes—
136C2×C3⋊Dic3C4×D312yes—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—121no—
164C2×C3×Dic3—121no—
165C2×C3×Dic3—121no—
166C2×C3×Dic3—121no—
167#383—271noSylow
168C2×H(3,𝔽3)#11711yescharacteristic
169C2×H(3,𝔽3)#11711yescharacteristic
170C2×H(3,𝔽3)#11711yescharacteristic
171C2×H(3,𝔽3)#12611yescharacteristic
172C2×H(3,𝔽3)#12611yescharacteristic
173C2×H(3,𝔽3)#12611yescharacteristic
174C2×H(3,𝔽3)#12611yescharacteristic
175C22×C3×C4—31no—
176C22×C3×C4—31no—
177C22×C3×C4—61no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4—91no—
184C22×C3×C4D3211yescharacteristic
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—31no—
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—31no—
192C22×Dic3—31no—
193C22×Dic3—61no—
194C22×Dic3—61no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—91no—
199C4×Dic3—94no—
200C4×Dic3—94no—
201C4×Dic3—94no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32C2×C4×D311yescharacteristic
210C22×C32C2×C4×D311yescharacteristic
211C22×C32#23511yescharacteristic
212C22×C32#23511yescharacteristic
213C22×C32#23511yescharacteristic
214C22×C32#23511yescharacteristic
215C22×C32#23711yescharacteristic
216C22×C32#23711yescharacteristic
217C22×C32#23711yescharacteristic
218C22×C32#23711yescharacteristic
219C22×C32#23711yescharacteristic
220C22×C32#23711yescharacteristic
221C22×C32#23911yescharacteristic
222C22×C32#23911yescharacteristic
223C32×C4—42no—
224C32×C4—41no—
225C32×C4—41no—
226C3⋊Dic3—22no—
227C3⋊Dic3—22no—
228C3⋊Dic3—41no—
229C3⋊Dic3—41no—
230C3×Dic3—62no—
231C3×Dic3—62no—
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—122no—
235C3×Dic3—122no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C2×C42—271no—
241#95—91no—
242#95—91no—
243#95—91no—
244#95—271no—
245#95—271no—
246#95—271no—
247H(3,𝔽3)#38311yescharacteristic, complemented, Sylow
248C23×C3—31no—
249C23×C3—31no—
250C23×C3—61no—
251C23×C3C2×D3211yescharacteristic
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—62no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4—92no—
265C2×C3×C4—121no—
266C2×C3×C4—121no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—32no—
286C2×Dic3—62no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—62no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—92no—
309C2×Dic3—92no—
310C2×Dic3—121no—
311C2×Dic3—121no—
312C2×Dic3—121no—
313C2×Dic3—121no—
314C2×Dic3—181no—
315C2×Dic3—181no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318C2×Dic3—181no—
319C2×Dic3—181no—
320C2×Dic3—181no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—181no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32#87211yescharacteristic
332C2×C32#87211yescharacteristic
333C2×C32#87711yescharacteristic
334C2×C32#87711yescharacteristic
335C2×C32#87911yescharacteristic
336C2×C32#87911yescharacteristic
337C2×C32#87911yescharacteristic
338C2×C32#87911yescharacteristic
339C2×C32#89211yescharacteristic
340C2×C32#89211yescharacteristic
341C2×C32#89811yescharacteristic
342C2×C32#89811yescharacteristic
343C2×C32#89811yescharacteristic
344C2×C32#89811yescharacteristic
345C22×C4—91no—
346C22×C4—91no—
347C22×C4—91no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—91no—
351C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C42—274no—
353C22×C3—31no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3#379511yescharacteristic
375C22×C3#379611yescharacteristic
376C22×C3#379911yescharacteristic
377C22×C3C4×D3211yescharacteristic
378C22×C3#380411yescharacteristic
379C22×C3#380411yescharacteristic
380C22×C3#380511yescharacteristic
381C3×C4—41no—
382C3×C4—121no—
383C3×C4—121no—
384C3×C4—122no—
385C3×C4—182no—
386C3×C4—182no—
387C3×C4—182no—
388C3×C4—361no—
389C3×C4—361no—
390C3×C4—361no—
391Dic3—62no—
392Dic3—62no—
393Dic3—62no—
394Dic3—62no—
395Dic3—122no—
396Dic3—121no—
397Dic3—121no—
398Dic3—122no—
399Dic3—121no—
400Dic3—121no—
401Dic3—182no—
402Dic3—182no—
403Dic3—361no—
404Dic3—361no—
405C32—21no—
406C32#465111yescharacteristic, complemented
407C32#465111yescharacteristic, complemented
408C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—92no—
415C2×C4—181no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—61no—
449C2×C3—61no—
450C2×C3—61no—
451C2×C3#6383611yescharacteristic
452C2×C3#6384811yescharacteristic
453C2×C3#6385811yescharacteristic
454C2×C3#6386111yescharacteristic
455C2×C3#6387711yescharacteristic
456C2×C3#6387811yescharacteristic
457C2×C3#6395611yescharacteristic
458C22#8864811yescharacteristic
459C22#8864911yescharacteristic
460C22#8865211yescharacteristic
461C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
462C22#8867111yescharacteristic
463C22#8867111yescharacteristic
464C22#8867211yescharacteristic
465C4—41no—
466C4—182no—
467C4—182no—
468C4—182no—
469C4—361no—
470C4—361no—
471C4—361no—
472C3—31no—
473C3—31no—
474C3—61no—
475C3#1059124811yescharacteristic
476C2#1172107611yescharacteristic
477C2#1172108811yescharacteristic
478C2#1172110811yescharacteristic
479C2#1172111111yescharacteristic
480C2#1172112011yescharacteristic
481C2#1172112111yescharacteristic
482C2#1172129711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#383
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
17. #88447C4semidirect
247. H(3,𝔽3)#383semidirect
406. C32#4651semidirect
407. C32#4651semidirect

Maximal subgroups

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

subgroupindex
C2×#884472
#117205742
#63658⋊C32
#117203342
#105912483