Finite Group Databasebeta

← #13148578

Subgroups — #13148578

Order 1728.

927 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 141 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
1#11721737C211yesmaximal, characteristic, complemented
2#64090⋊C3C212yesmaximal, complemented
3#11721322C212yesmaximal, complemented
4#11721318C211yesmaximal, characteristic, complemented
5C2×#88449C211yesmaximal, characteristic, complemented
6#11721142C211yesmaximal, characteristic, complemented
7C4×#6198C211yesmaximal, characteristic, complemented
8#11721109C212yesmaximal, complemented
9#11721100C212yesmaximal, complemented
10#11721098C211yesmaximal, characteristic, complemented
11#11721097C211yesmaximal, characteristic, complemented
12#10594218—32nomaximal
13C2×#6235C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#6200C2212yes—
20C2×#6198C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2211yescharacteristic
22#88542C2211yescharacteristic
23#88540C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88451C2212yes—
29#88450C2212yes—
30#88450C2212yes—
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88449C2212yes—
34#88449C2211yescharacteristic
35#88448C2212yes—
36#64090D312yescomplemented
37C3×#67⋊C4—32no—
38C3×#67⋊C4—32no—
39C3×#67⋊C4—61no—
40#63980—32no—
41#63978—32no—
42#63978—32no—
43C2×Dic3⋊Dic3—32no—
44#63906—32no—
45#63905—32no—
46C4×#626—32no—
47#63868—32no—
48#63867—32no—
49#63862—32no—
50#63855—32no—
51#63853—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×#272C2311yescharacteristic
54C22×(C3⋊D3)⋊C3C2312yes—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57#6235—22no—
58C2×#1084—22no—
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62C4×#272—22no—
63#638⋊C3—24no—
64C2×(C3⋊Dic3)⋊C3—42no—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C4×(C3⋊D3)⋊C3—42no—
69#6200—42no—
70#6198—22no—
71#5565—91no—
72#5540—92no—
73C2×#638C2×D312yes—
74C22×C3⋊Dic3C2×D312yes—
75C2×C4×C3⋊D3C2×D312yes—
76C2×C3×#67—32no—
77C2×C3×#67—32no—
78C2×C3×#67—61no—
79C22×C3×Dic3—32no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—61no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—61no—
85C2×#626—32no—
86C2×#626—32no—
87C2×#625—32no—
88C32×C22⋊C4—21no—
89C32×C22⋊C4C2×D312yes—
90#3752C2×D312yes—
91#3750C2×D312yes—
92C4×C3⋊Dic3C2×D312yes—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—61no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—61no—
102#3724—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110#3722—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C22×H(3,𝔽3)—21no—
116C22×H(3,𝔽3)—21no—
117C22×H(3,𝔽3)#4112yes—
118C22×H(3,𝔽3)C2411yescharacteristic
119C2×#272—21no—
120C2×#272—21no—
121C2×#272—21no—
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—21no—
128#1084—21no—
129#1084#4112yes—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133C3×C4×D4—92no—
134C3×C4×D4—91no—
135#939—92no—
136#939—92no—
137#939—92no—
138#930—91no—
139#930—91no—
140#930—92no—
141#929—92no—
142C4×#67—92no—
143#926—91no—
144C2×C4⋊Dic3—91no—
145C2×Dic3⋊C4—92no—
146#898—92no—
147#894—92no—
148#889—92no—
149#884—92no—
150#883—92no—
151#67⋊C4—32no—
152#67⋊C4—32no—
153#67⋊C4—61no—
154#67⋊C4—92no—
155#879—92no—
156#876—92no—
157C4×D12—91no—
158#872—92no—
159#870—91no—
160C23×C32—21no—
161C23×C32C22×D312yes—
162C22×C3⋊D3C22×D312yes—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D312yes—
170#638—24no—
171C2×C3⋊Dic3—42no—
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175C4×C3⋊D3—42no—
176C3×#67—64no—
177C3×#67—64no—
178C3×#67—64no—
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—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—122no—
190C2×C3×Dic3—122no—
191C3×C4×D3—64no—
192C3×C4×D3—122no—
193C3×C4×D3—122no—
194#626—64no—
195#626—122no—
196#625—122no—
197#553—271noSylow
198C2×H(3,𝔽3)—22no—
199C2×H(3,𝔽3)C2×#4111yescharacteristic
200C2×H(3,𝔽3)C2×#4111yescharacteristic
201C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
202#272—41no—
203(C3⋊D3)⋊C3—42no—
204C2×C3×D4—92no—
205C2×C3×D4—91no—
206C22×C3×C4—92no—
207C22×C3×C4—91no—
208C22×C3×C4—92no—
209C22×C3×C4—91no—
210C2×#67—32no—
211C2×#67—32no—
212C2×#67—61no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—92no—
216C2×#67—92no—
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—61no—
220C22×Dic3—92no—
221C2×D12—91no—
222C2×D12—92no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—61no—
226C2×C4×D3—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—32no—
230C3×C22⋊C4—61no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4D3211yescharacteristic
234C3×C42—92no—
235C3×C42—91no—
236#217—92no—
237#217—92no—
238#217—92no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—32no—
241(C2×D3)⋊C4—61no—
242(C2×D3)⋊C4—92no—
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—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254Dic3⋊C4—32no—
255Dic3⋊C4—32no—
256Dic3⋊C4—61no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265C4×Dic3—32no—
266C4×Dic3—32no—
267C4×Dic3—61no—
268C4×Dic3—92no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32#23514yes—
276C22×C32C23×D312yes—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—22no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—41no—
292C32×C4—41no—
293C3⋊Dic3—22no—
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3×Dic3—34no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—121no—
305#126—272no—
306#125—271no—
307#123—272no—
308#121—271no—
309#121—272no—
310#121—271no—
311C4×D4—91no—
312C4×D4—92no—
313C4×D4—271no—
314#117—271no—
315C2×C4⋊C4—271no—
316H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
317C23×C3—32no—
318C23×C3—61no—
319C23×C3—92no—
320C23×C3—91no—
321C23×C3C2×D3211yescharacteristic
322C22×D3—32no—
323C22×D3—32no—
324C22×D3—61no—
325C22×D3—92no—
326C3×D4—184no—
327C3×D4—182no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—91no—
333C2×C3×C4—92no—
334C2×C3×C4—92no—
335C2×C3×C4—92no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340C2×C3×C4—362no—
341C2×C3×C4—362no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344#67—64no—
345#67—64no—
346#67—64no—
347#67—184no—
348#67—184no—
349#67—362no—
350#67—362no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—64no—
361C2×Dic3—92no—
362C2×Dic3—92no—
363C2×Dic3—92no—
364C2×Dic3—122no—
365C2×Dic3—122no—
366C2×Dic3—362no—
367D12—182no—
368D12—362no—
369C4×D3—64no—
370C4×D3—122no—
371C4×D3—122no—
372C4×D3—362no—
373C2×C32—24no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—42no—
378C2×C32C2×#23512yes—
379C2×C32#100412yes—
380C2×C32D3×#4112yes—
381C3⋊D3—42no—
382C3×D3—122no—
383C3×D3—122no—
384C3×D3—122no—
385C2×D4—91no—
386C2×D4—92no—
387C2×D4—271no—
388C2×D4—272no—
389C22×C4—91no—
390C22×C4—92no—
391C22×C4—91no—
392C22×C4—92no—
393C4⋊C4—92no—
394C4⋊C4—91no—
395C4⋊C4—271no—
396C4⋊C4—271no—
397C4⋊C4—272no—
398C4⋊C4—271no—
399C4⋊C4—272no—
400C22⋊C4—92no—
401C22⋊C4—91no—
402C22⋊C4—272no—
403C22⋊C4—272no—
404C22⋊C4—272no—
405C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C42—92no—
407C42—91no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—34no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3—181no—
424C22×C3#379812yes—
425C22×C3C22×D3211yescharacteristic
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4—92no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—182no—
446C3×C4—182no—
447C3×C4—182no—
448C3×C4—181no—
449C3×C4—181no—
450Dic3—34no—
451Dic3—62no—
452Dic3—62no—
453Dic3—62no—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—121no—
459Dic3—182no—
460Dic3—182no—
461Dic3—182no—
462C32—21no—
463C32#554012yescomplemented
464C23—91no—
465C23—92no—
466C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
467D4—182no—
468D4—184no—
469D4—542no—
470D4—1082no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—91no—
476C2×C4—92no—
477C2×C4—182no—
478C2×C4—182no—
479C2×C4—362no—
480C2×C4—362no—
481C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
482C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
483C2×C3—22no—
484C2×C3—32no—
485C2×C3—32no—
486C2×C3—32no—
487C2×C3—64no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—122no—
492C2×C3—362no—
493C2×C3—361no—
494C2×C3C2×#379811yescharacteristic
495C2×C3#6431211yescharacteristic
496C2×C3#6431311yescharacteristic
497D3—122no—
498D3—122no—
499D3—122no—
500D3—362no—
501C22—21no—
502C22—21no—
503C22—181no—
504C22—182no—
505C22—182no—
506C22—181no—
507C22—181no—
508C22—182no—
509C22#8865012yes—
510C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
511C4—21no—
512C4—21no—
513C4—92no—
514C4—181no—
515C4—182no—
516C4—182no—
517C4—181no—
518C4—182no—
519C3—32no—
520C3—61no—
521C3#1059423111yescharacteristic
522C2—22no—
523C2—361no—
524C2—362no—
525C2C2×#8865011yescharacteristic
526C2#1172250511yescharacteristic
527C2#1172250611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#553
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. #11721737C2semidirect
2. #64090⋊C3C2semidirect
3. #11721322C2semidirect
4. #11721318C2semidirect
5. C2×#88449C2semidirect
6. #11721142C2semidirect
7. C4×#6198C2semidirect
8. #11721109C2semidirect
9. #11721100C2semidirect
10. #11721098C2semidirect
11. #11721097C2semidirect
36. #64090D3semidirect
316. H(3,𝔽3)#553semidirect
463. C32#5540semidirect

Maximal subgroups

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

subgroupindex
#117217372
#64090⋊C32
#117213222
#117213182
C2×#884492
#117211422
C4×#61982
#117211092
#117211002
#117210982
#117210972
#105942183