Finite Group Databasebeta

← #13150667

Subgroups — #13150667

Order 1728.

795 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721824C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C2×#3750⋊C3C211yesmaximal, characteristic
4#11721299C212yesmaximal
5#11721296C212yesmaximal
6#11721087C212yesmaximal, complemented
7#11721085C212yesmaximal, complemented
8#11721080C212yesmaximal, complemented
9#11721079C212yesmaximal, complemented
10#10594813—31nomaximal
11#10594812—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×#6196C2212yes—
18#88552C2212yes—
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21#3757⋊C3C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25#3750⋊C3C2212yes—
26#88449C2212yes—
27#88448C2212yes—
28#88448C2212yes—
29#88448C2212yes—
30#88448C2212yes—
31#88447C2212yes—
32#88447C2212yes—
33#64133D311yescharacteristic
34C2×#3750D311yescharacteristic
35C3×#924—31no—
36C3×#924—31no—
37C3×#924—31no—
38C3×#924—61no—
39C2×C3×Dic3⋊C4—31no—
40#63982—32no—
41#63964—32no—
42#63957—32no—
43#63955—32no—
44#63847—32no—
45#63847—32no—
46#63846—32no—
47#63845—32no—
48#63840—32no—
49#63839—32no—
50#63839—32no—
51#63839—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)—41no—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084C2312yes—
57C2×#1084C2312yes—
58#6229—41no—
59C2×(C3⋊Dic3)⋊C3—41no—
60C2×(C3⋊Dic3)⋊C3Q814yescomplemented
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65#634⋊C3—41no—
66#6196—42no—
67#5623—91no—
68#5623—91no—
69#5500—91no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×#627—32no—
83C2×#627—32no—
84#3757C2×D312yes—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86#3750C2×D312yes—
87#3750C2×D312yes—
88#3750C2×D312yes—
89C3×#217—32no—
90C3×#217—32no—
91C3×#217—32no—
92C3×#217—62no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—62no—
103#3724—32no—
104#3724—32no—
105#3724—32no—
106#3724—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113Dic32—32no—
114Dic32—32no—
115Dic32—32no—
116Dic32—32no—
117C22×H(3,𝔽3)—21no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)C2×Q812yes—
120C22×H(3,𝔽3)C2411yescharacteristic
121C4×H(3,𝔽3)—21no—
122C4×H(3,𝔽3)—21no—
123#1084—22no—
124#1084—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3C2×Q814yes—
129C3×C22⋊Q8—91no—
130C3×C22⋊Q8—91no—
131C2×C3×C4⋊C4—91no—
132#944—92no—
133#944—92no—
134#926—92no—
135#924—31no—
136#924—31no—
137#924—31no—
138#924—61no—
139#924—91no—
140#924—92no—
141C2×Dic3⋊C4—31no—
142C2×Dic3⋊C4—91no—
143#890—92no—
144#890—92no—
145#889—92no—
146#889—92no—
147#889—92no—
148#889—92no—
149#888—92no—
150#888—92no—
151#888—92no—
152#888—92no—
153#878—92no—
154#878—92no—
155#877—92no—
156#877—92no—
157C23×C32—21no—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—42no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—41no—
170C2×C3⋊Dic3Dic614yescomplemented
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175#634—41no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—121no—
188C3×Dic6—121no—
189C3×Dic6—121no—
190C3×Dic6—121no—
191C3×Dic6—122no—
192#627—122no—
193#627—122no—
194#544—271noSylow
195C2×H(3,𝔽3)—22no—
196C2×H(3,𝔽3)C22×Q811yescharacteristic
197C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
198C2×H(3,𝔽3)2−1+411yescharacteristic
199C2×C3×Q8—91no—
200C2×C3×Q8—91no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—61no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4D3211yescharacteristic
207C22×Dic3—31no—
208C22×Dic3—31no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C2×Dic6—31no—
212C2×Dic6—31no—
213C2×Dic6—31no—
214C2×Dic6—61no—
215C2×Dic6—91no—
216C2×Dic6—92no—
217C2×Dic6—92no—
218C2×Dic6—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—92no—
224C3×C4⋊C4—91no—
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227#217—32no—
228#217—32no—
229#217—32no—
230#217—62no—
231#217—92no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—61no—
236C4⋊Dic3—91no—
237C4⋊Dic3—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241C4⋊Dic3—92no—
242Dic3⋊C4—32no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—32no—
247Dic3⋊C4—62no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32C2×Dic612yes—
276C22×C32D3×Q812yes—
277C22×C32C23×D311yescharacteristic
278C22×C32C23×D311yescharacteristic
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—41no—
285C3⋊Dic3—22no—
286C3⋊Dic3—22no—
287C3⋊Dic3—22no—
288C3⋊Dic3C2×Dic614yes—
289C3×Dic3—34no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—122no—
298C3×Dic3—122no—
299C4⋊Q8—272no—
300C4⋊Q8—272no—
301#125—272no—
302#125—272no—
303C22⋊Q8—91no—
304C22⋊Q8—91no—
305C22⋊Q8—272no—
306#117—272no—
307C2×C4⋊C4—91no—
308H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—61no—
312C23×C3C2×D3211yescharacteristic
313C3×Q8—361no—
314C3×Q8—361no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—41no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—92no—
323C2×C3×C4—94no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—121no—
330C2×C3×C4—121no—
331C2×C3×C4—122no—
332C2×C3×C4—361no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—32no—
336C2×Dic3—34no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—94no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—121no—
352C2×Dic3—361no—
353Dic6—121no—
354Dic6—121no—
355Dic6—121no—
356Dic6—122no—
357Dic6—361no—
358Dic6—362no—
359Dic6—362no—
360Dic6—362no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—42no—
367C2×C32C22×Dic611yescharacteristic
368C2×C32#100411yescharacteristic
369C2×C32#100411yescharacteristic
370C2×C32C2×D3×Q811yescharacteristic
371C2×C32#100711yescharacteristic
372C2×C32#101011yescharacteristic
373C2×Q8—91no—
374C2×Q8—91no—
375C2×Q8—272no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4—91no—
382C4⋊C4—91no—
383C4⋊C4—92no—
384C4⋊C4—92no—
385C4⋊C4—272no—
386C4⋊C4—272no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C42—272no—
393C42—272no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—32no—
399C22×C3—32no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—61no—
405C22×C3—62no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3D3×Dic612yes—
409C22×C3C22×D3211yescharacteristic
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—61no—
416C3×C4—62no—
417C3×C4—94no—
418C3×C4—121no—
419C3×C4—182no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4—182no—
423C3×C4—182no—
424Dic3—34no—
425Dic3—62no—
426Dic3—62no—
427Dic3—62no—
428Dic3—62no—
429Dic3—62no—
430Dic3—62no—
431Dic3—62no—
432Dic3—94no—
433Dic3—122no—
434Dic3—122no—
435Dic3—182no—
436Dic3—182no—
437Dic3—182no—
438C32—21no—
439C32#550011yescharacteristic, complemented
440C32#562311yescharacteristic, complemented
441C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
442Q8—361no—
443Q8—361no—
444Q8—1082no—
445C2×C4—41no—
446C2×C4—92no—
447C2×C4—94no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—92no—
453C2×C4—361no—
454C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
456C2×C3—22no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—62no—
464C2×C3—62no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—122no—
469C2×C3C2×D3×Dic611yescharacteristic
470C2×C3#6429711yescharacteristic
471C2×C3#6433011yescharacteristic
472C22—21no—
473C22—21no—
474C22((C3⋊D3)⋊C3)⋊Q812yes—
475C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476C4—21no—
477C4—21no—
478C4—94no—
479C4—182no—
480C4—182no—
481C4—182no—
482C4—182no—
483C4—182no—
484C3—31no—
485C3—31no—
486C3—61no—
487C3#1059481211yescharacteristic
488C2—22no—
489C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
490C2#1172248211yescharacteristic
491C2#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 radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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
6. #11721087C2semidirect
7. #11721085C2semidirect
8. #11721080C2semidirect
9. #11721079C2semidirect
60. C2×(C3⋊Dic3)⋊C3Q8semidirect
170. C2×C3⋊Dic3Dic6semidirect
308. H(3,𝔽3)#544semidirect
439. C32#5500semidirect
440. C32#5623semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64133⋊C32
C2×#3750⋊C32
#117212992
#117212962
#117210872
#117210852
#117210802
#117210792
#105948133
#105948123