Finite Group Databasebeta

← #13156992

Subgroups — #13156992

Order 1728.

856 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64330C211yesmaximal, characteristic
2#11723605C211yesmaximal, characteristic, complemented
3C3×#63961C212yesmaximal, complemented
4#11722083C211yesmaximal, characteristic, complemented
5#11722062C212yesmaximal, complemented
6#10591533—32nomaximal
7#10591398—31nomaximal
8C2×C3×#638C2211yescharacteristic
9C2×C32×#67C2212yes—
10C22×C32×Dic3C2211yescharacteristic, complemented
11C3×#3811—21no—
12C3×D3×#67—22no—
13C3×#3805—21no—
14C3×#3804—22no—
15#88864C2211yescharacteristic, complemented
16C3×#3757C2212yescomplemented
17#88806—21no—
18#88803—21no—
19C3×#3724—22no—
20C3×(C2×D3)⋊Dic3—22no—
21C3×#1004—32no—
22#64330D311yescharacteristic
23#64138—31no—
24C3×#935—32no—
25C3×#929—32no—
26#63961—32no—
27#63961—61no—
28#63961D312yescomplemented
29#63661—32no—
30#63581—31no—
31#63580—32no—
32#63580—32no—
33#63580—32no—
34#63580—62no—
35C23×C33C2311yescharacteristic
36C22×C3×C3⋊D3D411yescharacteristic
37C22×C32×D3D412yes—
38C2×C3×D32—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C3×#638—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C32×#67—22no—
44C32×#67—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3D412yes—
48C3×#627—21no—
49C3×#626—22no—
50C3×#625—21no—
51C3×#624—21no—
52C3×D3×Dic3—22no—
53#5243—91no—
54#5053—92no—
55C2×C32×D4—32no—
56C22×C32×C4—31no—
57C2×#638C2×D311yescharacteristic
58C2×C3×#67—22no—
59C2×C3×#67—32no—
60C2×C3×#67—62no—
61C2×C3×#67—61no—
62C2×C3×#67C2×D312yes—
63C2×C3×#67C2×D312yes—
64C22×C3×Dic3—21no—
65C22×C3×Dic3C2×D312yescomplemented
66C3×#237—62no—
67C3×#237—62no—
68C3×D3×D4—62no—
69C3×D3×D4—62no—
70C3×#235—62no—
71#3811—21no—
72D3×#67—22no—
73#3805—21no—
74#3804—22no—
75#3757—31no—
76#3757—32no—
77#3757—61no—
78#3757—62no—
79#3757—121no—
80#3757C2×D312yescomplemented
81#3752—61no—
82#3750—61no—
83C3×#217—32no—
84C3×#217—32no—
85C3×#217—62no—
86C3×#217—62no—
87C3×(C2×D3)⋊C4—62no—
88C3×Dic3⋊C4—62no—
89C3×Dic3⋊C4—62no—
90#3724—22no—
91#3724—62no—
92#3724—62no—
93(C2×D3)⋊Dic3—22no—
94(C2×D3)⋊Dic3—62no—
95(C2×D3)⋊Dic3—62no—
96C22×C33—41no—
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D412yes—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—41no—
101C2×C32×D3—22no—
102C2×C32×D3—42no—
103C3×D32—81no—
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—42no—
107C32×Dic3—22no—
108C32×Dic3—41no—
109C3×H(5,𝔽2)—91no—
110#1004—32no—
111C3×#123—92no—
112#935—32no—
113#935—91no—
114#929—32no—
115#929—92no—
116#929—181no—
117#834—91no—
118#834—92no—
119#834—92no—
120#834—182no—
121#806—92no—
122#806—92no—
123C23×C32—21no—
124C23×C32—22no—
125C23×C32—32no—
126C23×C32—41no—
127C23×C32C22×D312yes—
128C23×C32C22×D311yescharacteristic
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—22no—
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C22×C3×D3—62no—
134C22×C3×D3#6712yes—
135C22×C3×D3#6712yes—
136C2×D32—21no—
137C32×D4—62no—
138C32×D4—62no—
139C2×C32×C4—32no—
140C2×C32×C4—61no—
141C2×C32×C4—62no—
142#638—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—62no—
146C2×C3⋊Dic3—62no—
147C2×C3⋊Dic3—122no—
148C2×C3⋊Dic3—122no—
149C2×C3⋊Dic3#6711yescharacteristic
150C3×#67—22no—
151C3×#67—22no—
152C3×#67—22no—
153C3×#67—22no—
154C3×#67—42no—
155C3×#67—42no—
156C3×#67—62no—
157C3×#67—62no—
158C3×#67—62no—
159C3×#67—62no—
160C3×#67—122no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—122no—
173C2×C3×Dic3#6712yes—
174C2×C3×Dic3#6712yes—
175C3×D12—62no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×Dic6—62no—
179#627—21no—
180#626—22no—
181#625—21no—
182#624—21no—
183D3×Dic3—22no—
184#458—271noSylow
185C2×C33—21no—
186C2×C33—22no—
187C2×C33#12011yescharacteristic
188C3×C3⋊D3—41no—
189C32×D3—42no—
190C3×#41—182no—
191C3×#41—181no—
192C2×C3×D4—32no—
193C2×C3×D4—32no—
194C2×C3×D4—62no—
195C2×C3×D4—91no—
196C2×C3×D4—181no—
197C2×C3×D4—182no—
198C22×C3×C4—32no—
199C22×C3×C4—61no—
200C2×#67—32no—
201C2×#67—61no—
202C2×#67—62no—
203C2×#67D3212yes—
204C22×Dic3D3211yescharacteristic, complemented
205#237—62no—
206#237—62no—
207D3×D4—62no—
208D3×D4—62no—
209#235—62no—
210C3×C4⋊C4—182no—
211C3×C22⋊C4—92no—
212C3×C22⋊C4—182no—
213#217—32no—
214#217—32no—
215#217—62no—
216#217—62no—
217#217—92no—
218#217—91no—
219#217—181no—
220#217—182no—
221#217—181no—
222#217—361no—
223(C2×D3)⋊C4—62no—
224(C2×D3)⋊C4—182no—
225(C2×D3)⋊C4—182no—
226Dic3⋊C4—62no—
227Dic3⋊C4—62no—
228Dic3⋊C4—181no—
229Dic3⋊C4—182no—
230Dic3⋊C4—182no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—41no—
237C22×C32—42no—
238C22×C32—41no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—62no—
242C22×C32—82no—
243C22×C32—82no—
244C22×C32—122no—
245C22×C32D3×D412yes—
246C22×C32D3×D412yes—
247C22×C32C2×#6712yes—
248C22×C32C2×#6712yes—
249C22×C32C2×#6711yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—41no—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—42no—
255C2×C3×D3—42no—
256C2×C3×D3—42no—
257C2×C3×D3—62no—
258C2×C3×D3—82no—
259C2×C3×D3—121no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—242no—
263D32—81no—
264C32×C4—62no—
265C32×C4—121no—
266C3⋊Dic3—21no—
267C3⋊Dic3—42no—
268C3⋊Dic3—121no—
269C3⋊Dic3—122no—
270C3⋊Dic3—122no—
271C3⋊Dic3—242no—
272C3⋊Dic3—242no—
273C3×Dic3—22no—
274C3×Dic3—22no—
275C3×Dic3—42no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—62no—
279C3×Dic3—121no—
280C3×Dic3—122no—
281C3×Dic3—122no—
282C3×Dic3—242no—
283H(5,𝔽2)—91no—
284#123—92no—
285#123—271no—
286#99—271no—
287#99—272no—
288C33#45811yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—22no—
291C23×C3—32no—
292C23×C3—32no—
293C23×C3—41no—
294C23×C3—62no—
295C23×C3—91no—
296C23×C3—181no—
297C23×C3C2×D3212yes—
298C23×C3C2×D3211yescharacteristic
299C22×D3—32no—
300C22×D3—61no—
301C22×D3—62no—
302C22×D3#62512yes—
303C3×Q8—181no—
304C3×D4—62no—
305C3×D4—62no—
306C3×D4—62no—
307C3×D4—62no—
308C3×D4—122no—
309C3×D4—122no—
310C3×D4—182no—
311C3×D4—181no—
312C3×D4—182no—
313C2×C3×C4—32no—
314C2×C3×C4—32no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—91no—
321C2×C3×C4—122no—
322C2×C3×C4—181no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325#67—22no—
326#67—22no—
327#67—62no—
328#67—62no—
329#67—62no—
330#67—62no—
331#67—122no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—32no—
335C2×Dic3—61no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—122no—
340C2×Dic3—182no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—362no—
344C2×Dic3—362no—
345C2×Dic3#62512yes—
346D12—62no—
347C4×D3—62no—
348C4×D3—62no—
349Dic6—62no—
350C2×C32—21no—
351C2×C32—22no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—41no—
359C2×C32—42no—
360C2×C32—42no—
361C2×C32—42no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—82no—
365C2×C32—122no—
366C2×C32#93712yes—
367C2×C32#95311yescharacteristic
368C3⋊D3—41no—
369C3×D3—42no—
370C3×D3—42no—
371C3×D3—82no—
372C3×D3—122no—
373C3×D3—241no—
374#41—182no—
375#41—181no—
376C2×D4—32no—
377C2×D4—91no—
378C2×D4—181no—
379C2×D4—182no—
380C22×C4—31no—
381C4⋊C4—182no—
382C4⋊C4—541no—
383C22⋊C4—92no—
384C22⋊C4—182no—
385C22⋊C4—271no—
386C22⋊C4—541no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—122no—
402C22×C3—122no—
403C22×C3—122no—
404C22×C3—181no—
405C22×C3—242no—
406C22×C3—361no—
407C22×C3—721no—
408C22×C3C2×#62512yes—
409C22×C3D3×#6712yes—
410C22×C3D3×#6712yes—
411C22×C3D3×#6712yes—
412C22×C3#381111yescharacteristic
413C2×D3—22no—
414C2×D3—42no—
415C2×D3—62no—
416C2×D3—121no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—242no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—122no—
423C3×C4—122no—
424C3×C4—122no—
425C3×C4—181no—
426C3×C4—362no—
427Dic3—22no—
428Dic3—41no—
429Dic3—62no—
430Dic3—121no—
431Dic3—122no—
432Dic3—122no—
433Dic3—242no—
434Dic3—361no—
435Dic3—362no—
436Dic3—362no—
437Dic3—722no—
438Dic3—722no—
439C32—22no—
440C32—21no—
441C32—41no—
442C32#505312yescomplemented
443C32#524311yescharacteristic, complemented
444C23—32no—
445C23—91no—
446C23—181no—
447C23D3311yescharacteristic
448Q8—181no—
449D4—62no—
450D4—62no—
451D4—182no—
452D4—181no—
453D4—182no—
454C2×C4—32no—
455C2×C4—62no—
456C2×C4—61no—
457C2×C4—91no—
458C2×C4—181no—
459C2×C4—182no—
460C2×C4—182no—
461C2×C4—541no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—42no—
471C2×C3—42no—
472C2×C3—42no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—41no—
476C2×C3—82no—
477C2×C3—122no—
478C2×C3—122no—
479C2×C3—242no—
480C2×C3—361no—
481C2×C3#6397612yes—
482C2×C3#6398311yescharacteristic
483D3—42no—
484D3—122no—
485D3—241no—
486C22—41no—
487C22—62no—
488C22—122no—
489C22—181no—
490C22—361no—
491C22—721no—
492C22D3×#62512yes—
493C22#8895211yescharacteristic
494C4—62no—
495C4—121no—
496C4—181no—
497C4—362no—
498C4—1081no—
499C3—21no—
500C3—22no—
501C3—41no—
502C3#1059139811yescharacteristic, complemented
503C3#1059153312yescomplemented
504C2—22no—
505C2—21no—
506C2—122no—
507C2—361no—
508C2#1172294611yescharacteristic
subgroupG/H
centerC2#11722946
commutator, Fitting, cosocleC23×C33C23
Frattini, hypercenterC23D33
socleC2×C33#120
prime radicalC3×#64330C2
hypocenterC33#458
LCSG ▷ C23×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723605C2semidirect
3. C3×#63961C2semidirect
4. #11722083C2semidirect
5. #11722062C2semidirect
10. C22×C32×Dic3C22semidirect
15. #88864C22semidirect
16. C3×#3757C22semidirect
28. #63961D3semidirect
65. C22×C3×Dic3C2×D3semidirect
80. #3757C2×D3semidirect
204. C22×Dic3D32semidirect
288. C33#458semidirect
442. C32#5053semidirect
443. C32#5243semidirect
502. C3#10591398semidirect
503. C3#10591533semidirect

Maximal subgroups

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

subgroupindex
C3×#643302
#117236052
C3×#639612
#117220832
#117220622
#105915333
#105913983