Finite Group Databasebeta

← #13150736

Subgroups — #13150736

Order 1728.

927 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×C4×#1084C211yesmaximal, characteristic, complemented
2#64138⋊C3C212yesmaximal, complemented
3#11721318C211yesmaximal, characteristic
4#11721315C212yesmaximal, complemented
5#11721309C211yesmaximal, characteristic, complemented
6#11721145C211yesmaximal, characteristic, complemented
7#11721138C211yesmaximal, characteristic, complemented
8#11721121C212yesmaximal, complemented
9#11721112C212yesmaximal, complemented
10#11721098C211yesmaximal, characteristic, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10594948—32nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C2×#6198C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2211yescharacteristic
20C4×#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24#3757⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#88450C2212yes—
30#88450C2212yes—
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#64138D312yescomplemented
37C3×#929—32no—
38C2×C3×C4×Dic3—32no—
39C2×C3×C4×Dic3—61no—
40#63978—32no—
41Dic3×#67—32no—
42#63964—32no—
43#63952—32no—
44#63880—32no—
45#63877—32no—
46#63868—32no—
47#63867—32no—
48#63859—32no—
49#63852—32no—
50#63837—32no—
51#63836—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084—22no—
58C2×#1084—22no—
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62C2×#1084C2311yescharacteristic
63#638⋊C3—24no—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67#6198—22no—
68((C3⋊D3)⋊C3)⋊C4—24no—
69#6196—22no—
70#5649—92no—
71#5479—91no—
72C22×C32×C4—21no—
73C22×C32×C4C2×D312yes—
74C2×#638C2×D312yes—
75C2×C3×#67—32no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—61no—
79C22×C3×Dic3—61no—
80C2×#627—32no—
81C2×#626—32no—
82C2×#624—32no—
83C2×D3×Dic3—32no—
84#3757C2×D312yes—
85#3752C2×D312yes—
86#3752C2×D312yes—
87#3750C2×D312yes—
88#3750C2×D312yes—
89C3×#217—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102#3724—32no—
103#3724—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108#3722—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112Dic32—32no—
113Dic32—32no—
114C22×H(3,𝔽3)—21no—
115C22×H(3,𝔽3)—21no—
116C22×H(3,𝔽3)#4112yes—
117C22×H(3,𝔽3)C2411yescharacteristic
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—
123#1084—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×#123—92no—
131C2×C3×C42—91no—
132#939—92no—
133D4×Dic3—92no—
134#930—91no—
135#929—32no—
136#929—92no—
137C4×#67—92no—
138#924—91no—
139C2×C4×Dic3—32no—
140C2×C4×Dic3—61no—
141#898—92no—
142#892—92no—
143#889—92no—
144#887—92no—
145#885—92no—
146#67⋊C4—92no—
147#67⋊C4—92no—
148#879—92no—
149#878—92no—
150#877—92no—
151#876—92no—
152#875—91no—
153C4×D12—91no—
154#872—92no—
155#870—91no—
156C4×Dic6—91no—
157C23×C32—21no—
158C23×C32C22×D312yes—
159C22×C3⋊D3C22×D312yes—
160C22×C3×D3—32no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D312yes—
167#638—24no—
168C2×C3⋊Dic3C22×D312yes—
169C2×C3⋊Dic3C22×D312yes—
170C2×C3⋊Dic3C22×D312yes—
171C3×#67—64no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—64no—
187#627—64no—
188#626—64no—
189#624—64no—
190D3×Dic3—64no—
191#529—271noSylow
192C2×H(3,𝔽3)—22no—
193C2×H(3,𝔽3)C2×#4111yescharacteristic
194C2×H(3,𝔽3)C2×#4111yescharacteristic
195C2×H(3,𝔽3)C2×#4111yescharacteristic
196(C3⋊D3)⋊C3—42no—
197C2×C3×D4—92no—
198C22×C3×C4—32no—
199C22×C3×C4—61no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4D3211yescharacteristic
203C2×#67—32no—
204C2×#67—92no—
205C2×#67—92no—
206C22×Dic3—32no—
207C22×Dic3—32no—
208C22×Dic3—61no—
209C22×Dic3—61no—
210C22×Dic3—92no—
211C2×D12—91no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×Dic6—91no—
215C2×Dic6—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C22⋊C4—92no—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—92no—
221C3×C42—91no—
222C3×C42—91no—
223C3×C42—91no—
224C3×C42—91no—
225#217—32no—
226#217—92no—
227#217—92no—
228#217—92no—
229(C2×D3)⋊C4—32no—
230(C2×D3)⋊C4—32no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237C4⋊Dic3—91no—
238C4⋊Dic3—91no—
239C4⋊Dic3—92no—
240C4⋊Dic3—92no—
241Dic3⋊C4—32no—
242Dic3⋊C4—32no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251C4×Dic3—32no—
252C4×Dic3—32no—
253C4×Dic3—32no—
254C4×Dic3—32no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3—61no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—22no—
268C22×C32#23714yes—
269C22×C32C23×D312yes—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—22no—
280C3⋊Dic3—22no—
281C3⋊Dic3—22no—
282C3⋊Dic3—22no—
283C3×Dic3—34no—
284C3×Dic3—34no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294#126—272no—
295#125—271no—
296#124—271no—
297#123—92no—
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—32no—
307C23×C3—61no—
308C23×C3—92no—
309C23×C3C2×D3211yescharacteristic
310C22×D3—32no—
311C22×D3—92no—
312C3×D4—184no—
313C2×C3×C4—22no—
314C2×C3×C4—32no—
315C2×C3×C4—32no—
316C2×C3×C4—64no—
317C2×C3×C4—64no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—91no—
321C2×C3×C4—91no—
322C2×C3×C4—92no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—182no—
328C2×C3×C4—182no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331#67—64no—
332#67—184no—
333#67—184no—
334C2×Dic3—32no—
335C2×Dic3—32no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×Dic3—64no—
348C2×Dic3—64no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—184no—
353D12—182no—
354C4×D3—184no—
355C4×D3—184no—
356Dic6—184no—
357Dic6—182no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32C2×#23712yes—
364C2×C32D3×#4112yes—
365C2×C32D3×#4112yes—
366C3⋊D3—42no—
367C3×D3—122no—
368C2×Q8—271no—
369C2×D4—92no—
370C2×D4—271no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—272no—
374C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C4⋊C4—92no—
376C4⋊C4—92no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—272no—
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—91no—
390C42—272no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—32no—
394C22×C3—34no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—61no—
400C22×C3—62no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3—182no—
404C22×C3#380512yes—
405C22×C3C22×D3211yescharacteristic
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—62no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—92no—
419C3×C4—92no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4—182no—
423C3×C4—181no—
424C3×C4—181no—
425Dic3—34no—
426Dic3—34no—
427Dic3—62no—
428Dic3—62no—
429Dic3—62no—
430Dic3—62no—
431Dic3—62no—
432Dic3—62no—
433Dic3—62no—
434Dic3—62no—
435Dic3—62no—
436Dic3—182no—
437Dic3—182no—
438Dic3—182no—
439C32—21no—
440C32#564912yescomplemented
441C23—92no—
442C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443Q8—542no—
444D4—184no—
445D4—542no—
446C2×C4—22no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—92no—
453C2×C4—92no—
454C2×C4—182no—
455C2×C4—182no—
456C2×C4—544no—
457C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
458C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
459C2×C3—22no—
460C2×C3—32no—
461C2×C3—32no—
462C2×C3—32no—
463C2×C3—64no—
464C2×C3—64no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—362no—
469C2×C3#6430611yescharacteristic
470C2×C3#6430611yescharacteristic
471C2×C3C2×#380511yescharacteristic
472D3—122no—
473D3—362no—
474C22—21no—
475C22—21no—
476C22—182no—
477C22—182no—
478C22—182no—
479C22#8867212yes—
480C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C4—21no—
482C4—21no—
483C4—92no—
484C4—92no—
485C4—182no—
486C4—181no—
487C4—181no—
488C4—182no—
489C4—182no—
490C3—32no—
491C3—61no—
492C3#1059496311yescharacteristic
493C2—22no—
494C2—362no—
495C2#1172249111yescharacteristic
496C2#1172249111yescharacteristic
497C2C2×#8867211yescharacteristic
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#11721318C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#529
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×C4×#1084C2semidirect
2. #64138⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721309C2semidirect
6. #11721145C2semidirect
7. #11721138C2semidirect
8. #11721121C2semidirect
9. #11721112C2semidirect
10. #11721098C2semidirect
11. #11721078C2semidirect
36. #64138D3semidirect
305. H(3,𝔽3)#529semidirect
440. C32#5649semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#64138⋊C32
#117213182
#117213152
#117213092
#117211452
#117211382
#117211212
#117211122
#117210982
#117210782
#105949483