Finite Group Databasebeta

← #13147810

Subgroups — #13147810

Order 1728.

927 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C42×#272C211yesmaximal, characteristic, complemented
2#11721387C212yesmaximal, complemented
3#11721145C211yesmaximal, characteristic, complemented
4#11721142C211yesmaximal, characteristic
5#11721138C211yesmaximal, characteristic, complemented
6#11721124C212yesmaximal, complemented
7#11721121C212yesmaximal, complemented
8#11721104C212yesmaximal, complemented
9#11721098C211yesmaximal, characteristic, complemented
10#11721096C211yesmaximal, characteristic, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10593975—32nomaximal
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×#6200C2212yes—
17C2×#6198C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2211yescharacteristic
20C42×H(3,𝔽3)C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#88451C2212yes—
31#88450C2212yes—
32#88449C2211yescharacteristic
33#88449C2211yescharacteristic
34#88448C2212yes—
35#88447C2212yes—
36#64085D312yescomplemented
37C3×#876—32no—
38C3×C42×D3—32no—
39C3×C42×D3—61no—
40#63906—32no—
41#63905—32no—
42#63904—32no—
43#63880—32no—
44#63879—32no—
45#63877—32no—
46#63867—32no—
47#63852—32no—
48#63851—32no—
49#63850—32no—
50#63837—32no—
51#63836—32no—
52C22×#272C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C4×#272—22no—
61C4×#272—22no—
62C4×#272—22no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66#6200—24no—
67#6198—22no—
68((C3⋊D3)⋊C3)⋊C4—24no—
69#6196—22no—
70#5706—92no—
71#5479—91no—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×#627—32no—
79C2×#626—32no—
80C2×#626—32no—
81C2×#625—32no—
82C2×#624—32no—
83C2×D3×Dic3—32no—
84C32×C42—21no—
85C32×C42C2×D312yes—
86#3752C2×D312yes—
87#3752C2×D312yes—
88#3752C2×D312yes—
89#3750C2×D312yes—
90#3750C2×D312yes—
91#3750C2×D312yes—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3—61no—
104#3724—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112Dic32—32no—
113Dic32—32no—
114C22×H(3,𝔽3)C2411yescharacteristic
115C2×#272—21no—
116C2×#272—21no—
117C2×#272#4112yes—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—22no—
121C4×H(3,𝔽3)—21no—
122C4×H(3,𝔽3)—21no—
123C4×H(3,𝔽3)—21no—
124#1084—21no—
125#1084#4112yes—
126#1084#4112yes—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3—22no—
130C3×#126—92no—
131C2×C3×C42—91no—
132#930—91no—
133#929—92no—
134C4×#67—92no—
135#924—91no—
136#896—92no—
137#894—92no—
138#893—92no—
139#892—92no—
140#889—92no—
141#887—92no—
142#885—92no—
143#884—92no—
144#883—92no—
145#67⋊C4—92no—
146#67⋊C4—92no—
147#877—92no—
148#876—32no—
149#876—92no—
150#875—91no—
151C4×D12—91no—
152#872—92no—
153C42×D3—32no—
154C42×D3—61no—
155#870—91no—
156C4×Dic6—91no—
157C22×C3⋊D3C22×D312yes—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C2×C3⋊Dic3C22×D312yes—
169C2×C3⋊Dic3C22×D312yes—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C3×C4×D3—64no—
180C3×C4×D3—64no—
181C3×C4×D3—64no—
182C3×C4×D3—64no—
183C3×C4×D3—64no—
184C3×C4×D3—64no—
185#627—64no—
186#626—64no—
187#626—64no—
188#625—64no—
189#624—64no—
190D3×Dic3—64no—
191#529—271noSylow
192C2×H(3,𝔽3)C2×#4111yescharacteristic
193C2×H(3,𝔽3)C2×#4111yescharacteristic
194C2×H(3,𝔽3)C2×#4111yescharacteristic
195#272—22no—
196(C3⋊D3)⋊C3—42no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C2×#67—92no—
201C2×#67—92no—
202C2×#67—92no—
203C22×Dic3—92no—
204C2×D12—91no—
205C2×D12—92no—
206C2×C4×D3—32no—
207C2×C4×D3—32no—
208C2×C4×D3—32no—
209C2×C4×D3—61no—
210C2×C4×D3—61no—
211C2×C4×D3—61no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×Dic6—91no—
215C2×Dic6—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C42—32no—
223C3×C42—61no—
224C3×C42—91no—
225C3×C42—91no—
226C3×C42—91no—
227C3×C42D3211yescharacteristic
228#217—92no—
229#217—92no—
230(C2×D3)⋊C4—32no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—92no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253C4×Dic3—32no—
254C4×Dic3—32no—
255C4×Dic3—32no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3—61no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C22×C32—21no—
263C22×C32C23×D312yes—
264C2×C3⋊D3—22no—
265C2×C3⋊D3—22no—
266C2×C3⋊D3—22no—
267C2×C3×D3—34no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—22no—
281C32×C4—22no—
282C3⋊Dic3—22no—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3×Dic3—34no—
286C3×Dic3—34no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294#126—92no—
295#125—271no—
296#124—271no—
297#123—272no—
298C22⋊Q8—271no—
299#121—271no—
300C4×Q8—271no—
301C4×D4—271no—
302C4×D4—272no—
303#117—272no—
304C2×C42—91no—
305H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
306C23×C3—92no—
307C23×C3—91no—
308C22×D3—32no—
309C22×D3—32no—
310C22×D3—61no—
311C22×D3—92no—
312C2×C3×C4—32no—
313C2×C3×C4—32no—
314C2×C3×C4—32no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—92no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4—91no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330#67—184no—
331#67—184no—
332#67—184no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—32no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—184no—
346D12—182no—
347D12—184no—
348C4×D3—64no—
349C4×D3—64no—
350C4×D3—64no—
351C4×D3—64no—
352C4×D3—64no—
353C4×D3—64no—
354C4×D3—184no—
355C4×D3—184no—
356Dic6—184no—
357Dic6—182no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32D3×#4112yes—
362C2×C32D3×#4112yes—
363C2×C32D3×#4112yes—
364C3⋊D3—42no—
365C3×D3—64no—
366C3×D3—64no—
367C3×D3—122no—
368C2×Q8—271no—
369C2×D4—272no—
370C2×D4—271no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—272no—
375C4⋊C4—92no—
376C4⋊C4—92no—
377C4⋊C4—92no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C22⋊C4—92no—
382C22⋊C4—92no—
383C22⋊C4—92no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C42—91no—
387C42—91no—
388C42—91no—
389C42—272no—
390C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C22×C3—32no—
392C22×C3—61no—
393C22×C3—92no—
394C22×C3—182no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3C22×D3211yescharacteristic
400C2×D3—34no—
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—182no—
410C2×D3—182no—
411C2×D3—182no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—62no—
420C3×C4—62no—
421C3×C4—92no—
422C3×C4—92no—
423C3×C4—182no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4—181no—
427Dic3—34no—
428Dic3—34no—
429Dic3—62no—
430Dic3—62no—
431Dic3—62no—
432Dic3—62no—
433Dic3—62no—
434Dic3—62no—
435Dic3—62no—
436Dic3—182no—
437Dic3—182no—
438Dic3—182no—
439C32—21no—
440C32#570612yescomplemented
441C23—91no—
442C23—92no—
443Q8—542no—
444D4—542no—
445D4—544no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—182no—
453C2×C4—182no—
454C2×C4—182no—
455C2×C4—544no—
456C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
457C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
458C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
459C2×C3—32no—
460C2×C3—32no—
461C2×C3—32no—
462C2×C3—61no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—182no—
466C2×C3—362no—
467C2×C3#6430611yescharacteristic
468C2×C3#6430611yescharacteristic
469C2×C3#6430611yescharacteristic
470D3—64no—
471D3—64no—
472D3—122no—
473D3—362no—
474C22—92no—
475C22—181no—
476C22—181no—
477C22—182no—
478C22—182no—
479C22—182no—
480C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—92no—
485C4—92no—
486C4—181no—
487C4—182no—
488C4—182no—
489C4—182no—
490C3—32no—
491C3—61no—
492C3#1059397911yescharacteristic
493C2—182no—
494C2—362no—
495C2#1172249111yescharacteristic
496C2#1172249111yescharacteristic
497C2#1172249111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721142C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C42×#272C2semidirect
2. #11721387C2semidirect
3. #11721145C2semidirect
5. #11721138C2semidirect
6. #11721124C2semidirect
7. #11721121C2semidirect
8. #11721104C2semidirect
9. #11721098C2semidirect
10. #11721096C2semidirect
11. #11721078C2semidirect
36. #64085D3semidirect
305. H(3,𝔽3)#529semidirect
440. C32#5706semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
#117213872
#117211452
#117211422
#117211382
#117211242
#117211212
#117211042
#117210982
#117210962
#117210782
#105939753