Finite Group Databasebeta

← #13140114

Subgroups — #13140114

Order 1728.

545 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88448C211yesmaximal, characteristic
2C2×#88448C211yesmaximal, characteristic
3#11720574C211yesmaximal, characteristic
4#63658⋊C3C211yesmaximal, characteristic
5#63658⋊C3C211yesmaximal, characteristic
6#11720334C211yesmaximal, characteristic
7#11720334C211yesmaximal, characteristic
8#10591211—31nomaximal
9#10591209—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#88448—22no—
18#88448—22no—
19C2×#3724—31no—
20C2×#3724—31no—
21C2×Dic3⋊Dic3—31no—
22C2×Dic3⋊Dic3—31no—
23#63658D311yescharacteristic
24#63658D311yescharacteristic
25C3×C3⋊#95—31no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C3×C3⋊#95—61no—
30#63579—31no—
31#63579—31no—
32#63579—31no—
33#63579—31no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084—21no—
43C2×#1084—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—
54C2×(C3⋊Dic3)⋊C3D412yes—
55C2×(C3⋊Dic3)⋊C3Q812yes—
56#4660—91no—
57#4659—91no—
58#4658—91no—
59C22×C32×C4—21no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C3⋊Dic3C2×D311yescharacteristic
63C22×C3⋊Dic3C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D311yescharacteristic
65C22×C3⋊Dic3C2×D311yescharacteristic
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—31no—
74C22×C3×Dic3—61no—
75C22×C3×Dic3—61no—
76#3724—62no—
77#3724—62no—
78Dic3⋊Dic3—62no—
79Dic3⋊Dic3—62no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C22×H(3,𝔽3)C2×Q811yescharacteristic
82C22×H(3,𝔽3)#4111yescharacteristic
83C22×H(3,𝔽3)#4111yescharacteristic
84C22×H(3,𝔽3)#4111yescharacteristic
85C22×H(3,𝔽3)#4111yescharacteristic
86C22×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)—41no—
88#1084—41no—
89#1084—41no—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—22no—
92(C3⋊Dic3)⋊C3—41no—
93(C3⋊Dic3)⋊C3—41no—
94C2×C4⋊Dic3—91no—
95C2×C4⋊Dic3—91no—
96C2×Dic3⋊C4—91no—
97C2×Dic3⋊C4—91no—
98C2×Dic3⋊C4—91no—
99C2×Dic3⋊C4—91no—
100C3×#95—91no—
101C3×#95—91no—
102C3×#95—91no—
103C3⋊#95—31no—
104C3⋊#95—31no—
105C3⋊#95—31no—
106C3⋊#95—31no—
107C3⋊#95—61no—
108C3⋊#95—91no—
109C3⋊#95—91no—
110C3⋊#95—91no—
111C3⋊#95—91no—
112C3⋊#95—91no—
113C3⋊#95—91no—
114C3⋊#95—91no—
115C3⋊#95—91no—
116C23×C32—21no—
117C23×C32C22×D311yescharacteristic
118C23×C32C22×D311yescharacteristic
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4—41no—
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⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3Dic612yes—
139C2×C3⋊Dic3D1212yes—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
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—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—121no—
170#400—271noSylow
171C2×H(3,𝔽3)C22⋊Q811yescharacteristic
172C2×H(3,𝔽3)#12311yescharacteristic
173C2×H(3,𝔽3)#12411yescharacteristic
174C2×H(3,𝔽3)#12511yescharacteristic
175C2×H(3,𝔽3)#12511yescharacteristic
176C2×H(3,𝔽3)#12611yescharacteristic
177C2×H(3,𝔽3)#12611yescharacteristic
178C22×C3×C4—31no—
179C22×C3×C4—31no—
180C22×C3×C4—61no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C22×C3×C4—91no—
184C22×C3×C4—91no—
185C22×C3×C4—91no—
186C22×C3×C4—91no—
187C22×C3×C4D3211yescharacteristic
188C22×Dic3—31no—
189C22×Dic3—31no—
190C22×Dic3—31no—
191C22×Dic3—31no—
192C22×Dic3—31no—
193C22×Dic3—31no—
194C22×Dic3—31no—
195C22×Dic3—31no—
196C22×Dic3—61no—
197C22×Dic3—61no—
198C22×Dic3—91no—
199C22×Dic3—91no—
200C22×Dic3—91no—
201C22×Dic3—91no—
202C4⋊Dic3—182no—
203C4⋊Dic3—182no—
204Dic3⋊C4—182no—
205Dic3⋊C4—182no—
206Dic3⋊C4—182no—
207Dic3⋊C4—182no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32C2×Dic611yescharacteristic
216C22×C32C2×D1211yescharacteristic
217C22×C32#23511yescharacteristic
218C22×C32#23511yescharacteristic
219C22×C32#23511yescharacteristic
220C22×C32#23511yescharacteristic
221C22×C32D3×D411yescharacteristic
222C22×C32#23711yescharacteristic
223C22×C32#23711yescharacteristic
224C22×C32#23711yescharacteristic
225C22×C32#23711yescharacteristic
226C22×C32#23711yescharacteristic
227C22×C32D3×Q811yescharacteristic
228C22×C32#23911yescharacteristic
229C32×C4—42no—
230C32×C4—41no—
231C32×C4—41no—
232C3⋊Dic3—22no—
233C3⋊Dic3—22no—
234C3⋊Dic3—41no—
235C3⋊Dic3—41no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—122no—
239C3×Dic3—122no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C2×C4⋊C4—271no—
247C2×C4⋊C4—271no—
248#95—91no—
249#95—91no—
250#95—91no—
251#95—271no—
252#95—271no—
253H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—61no—
257C23×C3C2×D3211yescharacteristic
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—92no—
269C2×C3×C4—92no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
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×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×Dic3—32no—
289C2×Dic3—32no—
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—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—62no—
307C2×Dic3—62no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—92no—
315C2×Dic3—92no—
316C2×Dic3—121no—
317C2×Dic3—121no—
318C2×Dic3—121no—
319C2×Dic3—121no—
320C2×Dic3—181no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—181no—
324C2×Dic3—181no—
325C2×Dic3—181no—
326C2×Dic3—181no—
327C2×Dic3—181no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32#87011yescharacteristic
338C2×C32#87511yescharacteristic
339C2×C32#87811yescharacteristic
340C2×C32#87911yescharacteristic
341C2×C32#87911yescharacteristic
342C2×C32#87911yescharacteristic
343C2×C32#88511yescharacteristic
344C2×C32#88611yescharacteristic
345C2×C32#88911yescharacteristic
346C2×C32#88911yescharacteristic
347C2×C32#89011yescharacteristic
348C2×C32#89411yescharacteristic
349C2×C32#89711yescharacteristic
350C2×C32#89811yescharacteristic
351C22×C4—91no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C4⋊C4—542no—
359C4⋊C4—542no—
360C22×C3—31no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3D3×Dic611yescharacteristic
382C22×C3#379611yescharacteristic
383C22×C3#379811yescharacteristic
384C22×C3D3×D1211yescharacteristic
385C22×C3#380411yescharacteristic
386C22×C3#380411yescharacteristic
387C22×C3#380511yescharacteristic
388C3×C4—41no—
389C3×C4—121no—
390C3×C4—121no—
391C3×C4—122no—
392C3×C4—182no—
393C3×C4—182no—
394C3×C4—361no—
395C3×C4—361no—
396C3×C4—361no—
397C3×C4—361no—
398Dic3—62no—
399Dic3—62no—
400Dic3—121no—
401Dic3—121no—
402Dic3—121no—
403Dic3—121no—
404Dic3—122no—
405Dic3—122no—
406Dic3—121no—
407Dic3—121no—
408Dic3—182no—
409Dic3—182no—
410Dic3—361no—
411Dic3—361no—
412C32—21no—
413C32#465911yescharacteristic, complemented
414C32#466011yescharacteristic, complemented
415C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—92no—
420C2×C4—92no—
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×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4—181no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—61no—
456C2×C3—61no—
457C2×C3—61no—
458C2×C3#6384611yescharacteristic
459C2×C3#6385211yescharacteristic
460C2×C3#6386211yescharacteristic
461C2×C3#6386711yescharacteristic
462C2×C3#6389011yescharacteristic
463C2×C3#6389311yescharacteristic
464C2×C3#6395611yescharacteristic
465C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
466C22#8864911yescharacteristic
467C22#8865011yescharacteristic
468C22#8865411yescharacteristic
469C22#8867111yescharacteristic
470C22#8867111yescharacteristic
471C22#8867211yescharacteristic
472C4—41no—
473C4—182no—
474C4—182no—
475C4—361no—
476C4—361no—
477C4—361no—
478C4—361no—
479C3—31no—
480C3—31no—
481C3—61no—
482C3#1059120711yescharacteristic
483C2#1172108711yescharacteristic
484C2#1172109111yescharacteristic
485C2#1172110311yescharacteristic
486C2#1172111211yescharacteristic
487C2#1172113011yescharacteristic
488C2#1172113311yescharacteristic
489C2#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)#400
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
253. H(3,𝔽3)#400semidirect
413. C32#4659semidirect
414. C32#4660semidirect

Maximal subgroups

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

subgroupindex
C2×#884482
#117205742
#63658⋊C32
#117203342
#105912113
#105912093