Finite Group Databasebeta

← #13150741

Subgroups — #13150741

Order 1728.

855 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88540C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721315C212yesmaximal, complemented
5#11721307C212yesmaximal, complemented
6#11721128C212yesmaximal, complemented
7#11721125C212yesmaximal, complemented
8#11721108C212yesmaximal, complemented
9#11721087C212yesmaximal, complemented
10#10594980—31nomaximal
11#10594973—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19#88540C2212yes—
20#88540C2212yes—
21#3757⋊C3C2212yes—
22#3752⋊C3C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26#88450C2212yes—
27#88450C2212yes—
28#88450C2212yes—
29#88448C2212yes—
30#88448C2212yes—
31#88448C2212yes—
32#88447C2212yes—
33#64139D311yescharacteristic
34#64133D311yescharacteristic, complemented
35C3×#930—31no—
36C3×#924—31no—
37C2×C3×Dic3⋊C4—31no—
38C2×C3×Dic3⋊C4—31no—
39C2×C3×Dic3⋊C4—61no—
40Dic3×#67—32no—
41#63966—32no—
42#63961—32no—
43#63952—32no—
44#63897—32no—
45#63896—32no—
46#63888—32no—
47#63884—32no—
48#63861—32no—
49#63848—32no—
50#63847—32no—
51#63845—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)—41no—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084—22no—
58C2×#1084—41no—
59C2×#1084C2312yes—
60C2×#1084C2312yes—
61#638⋊C3—24no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C4⋊((C3⋊D3)⋊C3)—41no—
66#634⋊C3—41no—
67((C3⋊D3)⋊C3)⋊C4—42no—
68#5645—91no—
69#5602—91no—
70#5515—91no—
71C22×C32×C4—21no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C2×#638C2×D312yes—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×#67—32no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—61no—
83C22×C3×Dic3—61no—
84C2×C3×D12—31no—
85C2×C3×Dic6—31no—
86C2×#624—32no—
87C2×D3×Dic3—32no—
88#3757C2×D312yes—
89#3752C2×D312yes—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic
92#3750C2×D312yes—
93C3×#217—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—62no—
103C3×Dic3⋊C4—62no—
104#3724—32no—
105#3724—32no—
106#3724—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—32no—
110#3722—32no—
111#3722—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3—32no—
116Dic32—32no—
117Dic32—32no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)—21no—
120C22×H(3,𝔽3)#4112yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C4×H(3,𝔽3)—21no—
126C4×H(3,𝔽3)—21no—
127#1084—22no—
128#1084—22no—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132C3×C22⋊Q8—91no—
133C3×#121—91no—
134C2×C3×C4⋊C4—91no—
135#946—92no—
136#935—92no—
137#935—92no—
138D4×Dic3—92no—
139#930—31no—
140#930—91no—
141#929—92no—
142C4×#67—92no—
143#924—31no—
144#924—91no—
145C2×Dic3⋊C4—31no—
146C2×Dic3⋊C4—31no—
147C2×Dic3⋊C4—61no—
148#898—92no—
149#898—92no—
150#896—92no—
151#894—92no—
152#894—92no—
153#890—92no—
154#889—92no—
155#889—92no—
156#886—92no—
157#67⋊C4—92no—
158#879—92no—
159#878—92no—
160C23×C32—21no—
161C23×C32C22×D311yescharacteristic
162C23×C32C22×D311yescharacteristic
163C22×C3⋊D3C22×D312yes—
164C22×C3×D3—32no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—42no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174#638—24no—
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D312yes—
177C2×C3⋊Dic3C22×D312yes—
178C4⋊(C3⋊D3)—41no—
179#634—41no—
180C3×#67—64no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—121no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—122no—
196C3×D12—121no—
197C3×Dic6—121no—
198#624—122no—
199D3×Dic3—122no—
200#539—271noSylow
201C2×H(3,𝔽3)—22no—
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
204C2×H(3,𝔽3)2−1+411yescharacteristic
205(C3⋊D3)⋊C3—42no—
206C2×C3×Q8—91no—
207C2×C3×D4—91no—
208C2×C3×D4—92no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—61no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4D3211yescharacteristic
215C2×#67—32no—
216C2×#67—92no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—61no—
222C22×Dic3—61no—
223C22×Dic3—92no—
224C2×D12—31no—
225C2×D12—91no—
226C2×C4×D3—92no—
227C2×C4×D3—92no—
228C2×Dic6—31no—
229C2×Dic6—91no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C22⋊C4—92no—
236C3×C22⋊C4—92no—
237#217—32no—
238#217—92no—
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262Dic3⋊C4—62no—
263Dic3⋊C4—62no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32#23712yes—
283C22×C32#23912yes—
284C22×C32C23×D311yescharacteristic
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—41no—
297C32×C4—41no—
298C3⋊Dic3—22no—
299C3⋊Dic3—22no—
300C3⋊Dic3—22no—
301C3×Dic3—64no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—62no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—122no—
310#126—272no—
311#125—272no—
312#123—272no—
313#123—272no—
314C22⋊Q8—91no—
315C22⋊Q8—272no—
316#121—91no—
317C4×D4—272no—
318C2×C4⋊C4—91no—
319H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C23×C3—92no—
324C23×C3C2×D3211yescharacteristic
325C22×D3—32no—
326C22×D3—92no—
327C3×Q8—361no—
328C3×D4—184no—
329C3×D4—361no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—41no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—92no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4—122no—
345C2×C3×C4—182no—
346C2×C3×C4—361no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—64no—
350#67—184no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—64no—
363C2×Dic3—92no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—121no—
367C2×Dic3—121no—
368C2×Dic3—122no—
369C2×Dic3—362no—
370D12—121no—
371D12—361no—
372C4×D3—362no—
373C4×D3—362no—
374Dic6—121no—
375Dic6—361no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32C2×#23711yescharacteristic
383C2×C32#100411yescharacteristic
384C2×C32#100411yescharacteristic
385C2×C32C2×#23911yescharacteristic
386C2×C32#101011yescharacteristic
387C2×C32#101011yescharacteristic
388C3⋊D3—42no—
389C3×D3—122no—
390C2×Q8—91no—
391C2×D4—92no—
392C2×D4—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—272no—
396C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
397C4⋊C4—92no—
398C4⋊C4—92no—
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—91no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C22⋊C4—92no—
406C22⋊C4—92no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C42—272no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—32no—
416C22×C3—32no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—61no—
424C22×C3—62no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3#379612yes—
429C22×C3C22×D3211yescharacteristic
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—182no—
445C3×C4—182no—
446C3×C4—182no—
447C3×C4—182no—
448C3×C4—182no—
449Dic3—62no—
450Dic3—62no—
451Dic3—62no—
452Dic3—62no—
453Dic3—64no—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—122no—
458Dic3—182no—
459Dic3—182no—
460Dic3—182no—
461C32—21no—
462C32#560211yescharacteristic, complemented
463C32#564511yescharacteristic, complemented
464C23—92no—
465C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
466Q8—361no—
467D4—184no—
468D4—361no—
469C2×C4—41no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—182no—
476C2×C4—361no—
477C2×C4—1082no—
478C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
479C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C2×C3—22no—
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—62no—
488C2×C3—62no—
489C2×C3—64no—
490C2×C3—61no—
491C2×C3—61no—
492C2×C3—61no—
493C2×C3—362no—
494C2×C3C2×#379611yescharacteristic
495C2×C3#6429811yescharacteristic
496C2×C3#6433011yescharacteristic
497D3—122no—
498D3—362no—
499C22—21no—
500C22—21no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22#8864912yes—
505C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
506C4—21no—
507C4—21no—
508C4—182no—
509C4—182no—
510C4—182no—
511C4—182no—
512C4—182no—
513C3—31no—
514C3—31no—
515C3—61no—
516C3#1059497411yescharacteristic
517C2—22no—
518C2—362no—
519C2C2×#8864911yescharacteristic
520C2#1172248311yescharacteristic
521C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#539
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88540C2semidirect
3. #64133⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721307C2semidirect
6. #11721128C2semidirect
7. #11721125C2semidirect
8. #11721108C2semidirect
9. #11721087C2semidirect
34. #64133D3semidirect
319. H(3,𝔽3)#539semidirect
462. C32#5602semidirect
463. C32#5645semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
#64139⋊C32
#64133⋊C32
#117213152
#117213072
#117211282
#117211252
#117211082
#117210872
#105949803
#105949733