Finite Group Databasebeta

← #13140007

Subgroups — #13140007

Order 1728.

489 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11720574C211yesmaximal, characteristic
2#63658⋊C3C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#11720334C211yesmaximal, characteristic
5#11720334C211yesmaximal, characteristic
6#11720334C211yesmaximal, characteristic
7#11720334C211yesmaximal, characteristic
8#10591228—31nomaximal
9#10591228—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#63658D311yescharacteristic
18#63658D311yescharacteristic
19C3×C3⋊#95—31no—
20C3×C3⋊#95—31no—
21C3×C3⋊#95—31no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—61no—
24#63579—31no—
25#63579—31no—
26#63579—31no—
27#63579—31no—
28#63579—31no—
29#63579—31no—
30#63579—31no—
31#63579—31no—
32C23×H(3,𝔽3)C2311yescharacteristic
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×#1084—21no—
37C2×#1084—21no—
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
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)⋊C3—21no—
54C3⋊#401—91no—
55C3⋊#401—91no—
56C3⋊#401—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C22×C3⋊Dic3C2×D311yescharacteristic
61C22×C3⋊Dic3C2×D311yescharacteristic
62C22×C3⋊Dic3C2×D311yescharacteristic
63C22×C3⋊Dic3C2×D311yescharacteristic
64C22×C3×Dic3—31no—
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—61no—
73C22×C3×Dic3—61no—
74C22×H(3,𝔽3)#4111yescharacteristic
75C22×H(3,𝔽3)#4111yescharacteristic
76C22×H(3,𝔽3)#4111yescharacteristic
77C22×H(3,𝔽3)#4111yescharacteristic
78C22×H(3,𝔽3)#4111yescharacteristic
79C22×H(3,𝔽3)#4111yescharacteristic
80C22×H(3,𝔽3)#4111yescharacteristic
81C4×H(3,𝔽3)—41no—
82#1084—41no—
83#1084—41no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×#95—91no—
89C3×#95—91no—
90C3×#95—91no—
91C3⋊#95—31no—
92C3⋊#95—31no—
93C3⋊#95—31no—
94C3⋊#95—31no—
95C3⋊#95—61no—
96C3⋊#95—91no—
97C3⋊#95—91no—
98C3⋊#95—91no—
99C3⋊#95—91no—
100C3⋊#95—91no—
101C3⋊#95—91no—
102C3⋊#95—91no—
103C3⋊#95—91no—
104C3⋊#95—91no—
105C3⋊#95—91no—
106C3⋊#95—91no—
107C3⋊#95—91no—
108C3⋊#95—91no—
109C3⋊#95—91no—
110C23×C32—21no—
111C23×C32C22×D311yescharacteristic
112C23×C32C22×D311yescharacteristic
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
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—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
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—61no—
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—121no—
161C2×C3×Dic3—121no—
162C2×C3×Dic3—121no—
163C2×C3×Dic3—121no—
164#401—271noSylow
165C2×H(3,𝔽3)#12611yescharacteristic
166C2×H(3,𝔽3)#12611yescharacteristic
167C2×H(3,𝔽3)#12611yescharacteristic
168C2×H(3,𝔽3)#12611yescharacteristic
169C2×H(3,𝔽3)#12611yescharacteristic
170C2×H(3,𝔽3)#12611yescharacteristic
171C2×H(3,𝔽3)#12611yescharacteristic
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—61no—
175C22×C3×C4—91no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4D3211yescharacteristic
182C22×Dic3—31no—
183C22×Dic3—31no—
184C22×Dic3—31no—
185C22×Dic3—31no—
186C22×Dic3—31no—
187C22×Dic3—31no—
188C22×Dic3—31no—
189C22×Dic3—31no—
190C22×Dic3—61no—
191C22×Dic3—61no—
192C22×Dic3—91no—
193C22×Dic3—91no—
194C22×Dic3—91no—
195C22×Dic3—91no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32#23511yescharacteristic
204C22×C32#23511yescharacteristic
205C22×C32#23511yescharacteristic
206C22×C32#23511yescharacteristic
207C22×C32#23511yescharacteristic
208C22×C32#23511yescharacteristic
209C22×C32#23711yescharacteristic
210C22×C32#23711yescharacteristic
211C22×C32#23711yescharacteristic
212C22×C32#23711yescharacteristic
213C22×C32#23711yescharacteristic
214C22×C32#23711yescharacteristic
215C22×C32#23911yescharacteristic
216C22×C32#23911yescharacteristic
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—41no—
220C3⋊Dic3—41no—
221C3⋊Dic3—41no—
222C3⋊Dic3—41no—
223C3⋊Dic3—41no—
224C3×Dic3—122no—
225C3×Dic3—122no—
226C3×Dic3—121no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234#95—91no—
235#95—91no—
236#95—91no—
237#95—271no—
238#95—271no—
239#95—271no—
240#95—271no—
241H(3,𝔽3)#40111yescharacteristic, complemented, Sylow
242C23×C3—31no—
243C23×C3—31no—
244C23×C3—61no—
245C23×C3C2×D3211yescharacteristic
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—121no—
257C2×C3×C4—121no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
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×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—62no—
289C2×Dic3—62no—
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—61no—
302C2×Dic3—121no—
303C2×Dic3—121no—
304C2×Dic3—121no—
305C2×Dic3—121no—
306C2×Dic3—181no—
307C2×Dic3—181no—
308C2×Dic3—181no—
309C2×Dic3—181no—
310C2×Dic3—181no—
311C2×Dic3—181no—
312C2×Dic3—181no—
313C2×Dic3—181no—
314C2×Dic3—181no—
315C2×Dic3—181no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32#87611yescharacteristic
326C2×C32#87611yescharacteristic
327C2×C32#87911yescharacteristic
328C2×C32#87911yescharacteristic
329C2×C32#87911yescharacteristic
330C2×C32#87911yescharacteristic
331C2×C32#87911yescharacteristic
332C2×C32#87911yescharacteristic
333C2×C32#89811yescharacteristic
334C2×C32#89811yescharacteristic
335C2×C32#89811yescharacteristic
336C2×C32#89811yescharacteristic
337C2×C32#89811yescharacteristic
338C2×C32#89811yescharacteristic
339C22×C4—91no—
340C22×C4—91no—
341C22×C4—91no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—91no—
345C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3#379511yescharacteristic
368C22×C3#379611yescharacteristic
369C22×C3#379811yescharacteristic
370C22×C3#379911yescharacteristic
371C22×C3#380411yescharacteristic
372C22×C3#380411yescharacteristic
373C22×C3#380511yescharacteristic
374C3×C4—41no—
375C3×C4—121no—
376C3×C4—121no—
377C3×C4—122no—
378C3×C4—361no—
379C3×C4—361no—
380C3×C4—361no—
381C3×C4—361no—
382C3×C4—361no—
383C3×C4—361no—
384Dic3—121no—
385Dic3—121no—
386Dic3—121no—
387Dic3—121no—
388Dic3—122no—
389Dic3—122no—
390Dic3—121no—
391Dic3—121no—
392Dic3—121no—
393Dic3—121no—
394Dic3—361no—
395Dic3—361no—
396Dic3—361no—
397Dic3—361no—
398C32—21no—
399C32C3⋊#40111yescharacteristic, complemented
400C32C3⋊#40111yescharacteristic, complemented
401C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—181no—
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×C3—31no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—61no—
444C2×C3#6384811yescharacteristic
445C2×C3#6385811yescharacteristic
446C2×C3#6385911yescharacteristic
447C2×C3#6386111yescharacteristic
448C2×C3#6386211yescharacteristic
449C2×C3#6386811yescharacteristic
450C2×C3#6395611yescharacteristic
451C22#8864811yescharacteristic
452C22#8864911yescharacteristic
453C22#8865011yescharacteristic
454C22#8865211yescharacteristic
455C22#8867111yescharacteristic
456C22#8867111yescharacteristic
457C22#8867211yescharacteristic
458C4—41no—
459C4—361no—
460C4—361no—
461C4—361no—
462C4—361no—
463C4—361no—
464C4—361no—
465C3—31no—
466C3—31no—
467C3—61no—
468C3#1059122811yescharacteristic
469C2#1172108811yescharacteristic
470C2#1172109211yescharacteristic
471C2#1172110811yescharacteristic
472C2#1172110911yescharacteristic
473C2#1172111111yescharacteristic
474C2#1172111211yescharacteristic
475C2#1172129711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#401
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
241. H(3,𝔽3)#401semidirect
399. C32C3⋊#401semidirect
400. C32C3⋊#401semidirect

Maximal subgroups

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

subgroupindex
#117205742
#63658⋊C32
#117203342
#105912283