Finite Group Databasebeta

← #13140100

Subgroups — #13140100

Order 1728.

601 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3750⋊C3C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3C2×#88448C211yesmaximal, characteristic
4C2×#88448C211yesmaximal, characteristic
5#11720574C211yesmaximal, characteristic
6#63658⋊C3C211yesmaximal, characteristic
7#11720334C211yesmaximal, characteristic
8#10591356—31nomaximal
9#10591339—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#3750⋊C3—22no—
18#88449—22no—
19#88448—22no—
20#88448—22no—
21C2×#3750D311yescharacteristic
22C2×C3×Dic3⋊C4—31no—
23C2×#3724—31no—
24C2×#3724—31no—
25C2×Dic3⋊Dic3—31no—
26C2×Dic3⋊Dic3—31no—
27C2×Dic3⋊Dic3—31no—
28C2×Dic3⋊Dic3—31no—
29#63658D311yescharacteristic
30C3×C3⋊#95—31no—
31C3×C3⋊#95—31no—
32C3×C3⋊#95—31no—
33C3×C3⋊#95—61no—
34#63579—31no—
35#63579—31no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084—21no—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084Q812yes—
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—
54C2×(C3⋊Dic3)⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3D412yes—
56C2×(C3⋊Dic3)⋊C3D412yes—
57C2×(C3⋊Dic3)⋊C3Q812yes—
58#4981—91no—
59#4657—91no—
60#4657—91no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D311yescharacteristic
65C22×C3⋊Dic3C2×D311yescharacteristic
66C22×C3⋊Dic3C2×D311yescharacteristic
67C22×C3⋊Dic3C2×D311yescharacteristic
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—61no—
78#3750—22no—
79C3×Dic3⋊C4—62no—
80#3724—62no—
81#3724—62no—
82Dic3⋊Dic3—62no—
83Dic3⋊Dic3—62no—
84Dic3⋊Dic3—62no—
85Dic3⋊Dic3—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)C2×Q811yescharacteristic
89C22×H(3,𝔽3)C2×Q811yescharacteristic
90C22×H(3,𝔽3)#4111yescharacteristic
91C22×H(3,𝔽3)#4111yescharacteristic
92C22×H(3,𝔽3)#4111yescharacteristic
93C4×H(3,𝔽3)—41no—
94#1084—22no—
95#1084—41no—
96(C3⋊Dic3)⋊C3—22no—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×C4⋊C4—91no—
101C2×C4⋊Dic3—91no—
102C2×C4⋊Dic3—91no—
103C2×C4⋊Dic3—91no—
104C2×Dic3⋊C4—31no—
105C2×Dic3⋊C4—91no—
106C2×Dic3⋊C4—91no—
107C2×Dic3⋊C4—91no—
108C2×Dic3⋊C4—91no—
109C2×Dic3⋊C4—91no—
110C2×Dic3⋊C4—91no—
111C2×Dic3⋊C4—91no—
112C3×#95—91no—
113C3×#95—91no—
114C3⋊#95—31no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C3⋊#95—61no—
118C3⋊#95—91no—
119C3⋊#95—91no—
120C3⋊#95—91no—
121C3⋊#95—91no—
122C23×C32—21no—
123C23×C32C22×D311yescharacteristic
124C23×C32C22×D311yescharacteristic
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3Dic612yes—
144C2×C3⋊Dic3D1212yes—
145C2×C3⋊Dic3#6712yes—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—62no—
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—62no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—121no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—121no—
176#398—271noSylow
177C2×H(3,𝔽3)#12111yescharacteristic
178C2×H(3,𝔽3)C22⋊Q811yescharacteristic
179C2×H(3,𝔽3)C22⋊Q811yescharacteristic
180C2×H(3,𝔽3)#12311yescharacteristic
181C2×H(3,𝔽3)#12511yescharacteristic
182C2×H(3,𝔽3)#12511yescharacteristic
183C2×H(3,𝔽3)C4⋊Q811yescharacteristic
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—61no—
187C22×C3×C4—91no—
188C22×C3×C4—91no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4D3211yescharacteristic
194C22×Dic3—31no—
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—61no—
203C22×Dic3—61no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C22×Dic3—91no—
207C22×Dic3—91no—
208C3×C4⋊C4—182no—
209C4⋊Dic3—182no—
210C4⋊Dic3—182no—
211C4⋊Dic3—182no—
212Dic3⋊C4—62no—
213Dic3⋊C4—182no—
214Dic3⋊C4—182no—
215Dic3⋊C4—182no—
216Dic3⋊C4—182no—
217Dic3⋊C4—182no—
218Dic3⋊C4—182no—
219Dic3⋊C4—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32C2×Dic611yescharacteristic
228C22×C32C2×D1211yescharacteristic
229C22×C32#23511yescharacteristic
230C22×C32#23511yescharacteristic
231C22×C32#23511yescharacteristic
232C22×C32D3×D411yescharacteristic
233C22×C32D3×D411yescharacteristic
234C22×C32#23711yescharacteristic
235C22×C32#23711yescharacteristic
236C22×C32#23711yescharacteristic
237C22×C32D3×Q811yescharacteristic
238C22×C32D3×Q811yescharacteristic
239C22×C32D3×Q811yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C32×C4—42no—
242C32×C4—41no—
243C32×C4—41no—
244C3⋊Dic3—22no—
245C3⋊Dic3—22no—
246C3⋊Dic3—22no—
247C3⋊Dic3—41no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—122no—
254C3×Dic3—122no—
255C3×Dic3—121no—
256C3×Dic3—121no—
257C3×Dic3—121no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—271no—
260C2×C4⋊C4—271no—
261C2×C4⋊C4—271no—
262#95—91no—
263#95—91no—
264#95—271no—
265H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
266C23×C3—31no—
267C23×C3—31no—
268C23×C3—61no—
269C23×C3C2×D3211yescharacteristic
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
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×C4—181no—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—62no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—62no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—181no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32#87011yescharacteristic
350C2×C32#87811yescharacteristic
351C2×C32#88311yescharacteristic
352C2×C32#88411yescharacteristic
353C2×C32#88811yescharacteristic
354C2×C32#88911yescharacteristic
355C2×C32#88911yescharacteristic
356C2×C32#88911yescharacteristic
357C2×C32#89611yescharacteristic
358C2×C32#89711yescharacteristic
359C2×C32#89711yescharacteristic
360C2×C32#93011yescharacteristic
361C2×C32#93511yescharacteristic
362C2×C32#94411yescharacteristic
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4—91no—
367C22×C4—91no—
368C22×C4—91no—
369C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
370C4⋊C4—182no—
371C4⋊C4—542no—
372C4⋊C4—542no—
373C4⋊C4—542no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3D3×Dic611yescharacteristic
396C22×C3#379711yescharacteristic
397C22×C3#379811yescharacteristic
398C22×C3D3×D1211yescharacteristic
399C22×C3#380411yescharacteristic
400C22×C3#380511yescharacteristic
401C22×C3D3×#6711yescharacteristic
402C3×C4—41no—
403C3×C4—121no—
404C3×C4—121no—
405C3×C4—122no—
406C3×C4—182no—
407C3×C4—182no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4—361no—
411C3×C4—361no—
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—62no—
416Dic3—62no—
417Dic3—122no—
418Dic3—121no—
419Dic3—121no—
420Dic3—122no—
421Dic3—121no—
422Dic3—182no—
423Dic3—182no—
424Dic3—182no—
425Dic3—361no—
426C32—21no—
427C32#465711yescharacteristic, complemented
428C32#498111yescharacteristic, complemented
429C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—92no—
436C2×C4—92no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3#6383911yescharacteristic
473C2×C3#6386711yescharacteristic
474C2×C3#6387011yescharacteristic
475C2×C3#6389311yescharacteristic
476C2×C3#6389511yescharacteristic
477C2×C3#6390611yescharacteristic
478C2×C3#6396911yescharacteristic
479C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
480C22#8865011yescharacteristic
481C22#8865111yescharacteristic
482C22#8865411yescharacteristic
483C22#8867111yescharacteristic
484C22#8867211yescharacteristic
485C22#8867611yescharacteristic
486C4—41no—
487C4—182no—
488C4—182no—
489C4—182no—
490C4—182no—
491C4—361no—
492C4—361no—
493C3—31no—
494C3—31no—
495C3—61no—
496C3#1059133911yescharacteristic
497C2#1172107911yescharacteristic
498C2#1172109111yescharacteristic
499C2#1172109811yescharacteristic
500C2#1172113311yescharacteristic
501C2#1172114011yescharacteristic
502C2#1172115011yescharacteristic
503C2#1172131111yescharacteristic
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)#398
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
265. H(3,𝔽3)#398semidirect
427. C32#4657semidirect
428. C32#4981semidirect

Maximal subgroups

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

subgroupindex
C2×#3750⋊C32
C2×#884492
C2×#884482
#117205742
#63658⋊C32
#117203342
#105913563
#105913393