Finite Group Databasebeta

← #13148477

Subgroups — #13148477

Order 1728.

892 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 149 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
2#64087⋊C3C212yesmaximal, complemented
3#11721309C211yesmaximal, characteristic, complemented
4C2×#88449C211yesmaximal, characteristic, complemented
5#11721293C212yesmaximal
6#11721152C211yesmaximal, characteristic, complemented
7#11721148C212yesmaximal, complemented
8#11721099C211yesmaximal, characteristic, complemented
9#11721096C211yesmaximal, characteristic, complemented
10#11721086C212yesmaximal, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10594146—32nomaximal
13C2×#6235C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#634⋊C3C2212yes—
18C2×#6199C2212yes—
19C2×#6196C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88542C2211yescharacteristic
22#88540C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24#3757⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#88451C2212yes—
29#88449C2212yes—
30#88449C2211yescharacteristic
31#88449C2212yes—
32#88449C2211yescharacteristic
33#88449C2211yescharacteristic
34#88448C2212yes—
35#88447C2212yes—
36#64087D312yescomplemented
37C3×#67⋊C4—32no—
38C3×#67⋊C4—61no—
39C3×#878—32no—
40#67⋊Dic3—32no—
41Dic3×#67—32no—
42C2×Dic3⋊Dic3—32no—
43#63964—32no—
44#63896—32no—
45D3×C4⋊Dic3—32no—
46#63887—32no—
47#63866—32no—
48#63851—32no—
49#63846—32no—
50#63845—32no—
51#63837—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×#272C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56#6235Q814yescomplemented
57C2×#1084—22no—
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C4×#272—22no—
62C2×(C3⋊Dic3)⋊C3—42no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67#634⋊C3—42no—
68#6199—42no—
69#6196—22no—
70#5636—92no—
71#5543—91no—
72C22×C3⋊Dic3C2×D312yes—
73C2×#634C2×D312yes—
74C2×C3×#67—32no—
75C2×C3×#67—61no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—61no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—32no—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C22⋊C4—21no—
86C32×C22⋊C4C2×D312yes—
87#3757C2×D312yes—
88C4⋊(C3⋊Dic3)C2×D312yes—
89#3750C2×D312yes—
90#3750C2×D312yes—
91C3×#217—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—61no—
101#3724—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—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)C2×Q812yes—
117C22×H(3,𝔽3)C2411yescharacteristic
118C2×#272—21no—
119C2×#272—21no—
120C2×#272C2×Q812yes—
121C4×H(3,𝔽3)—21no—
122C4×H(3,𝔽3)—21no—
123#1084—21no—
124#1084C2×Q812yes—
125#1084#4112yes—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3—22no—
130C3×C22⋊Q8—92no—
131C3×C4×D4—91no—
132#946—92no—
133#944—92no—
134D4×Dic3—92no—
135C4×#67—92no—
136C2×C4⋊Dic3—91no—
137C2×C4⋊Dic3—91no—
138#924—91no—
139#924—91no—
140C2×Dic3⋊C4—92no—
141#896—92no—
142#896—92no—
143D3×C4⋊C4—92no—
144#890—92no—
145#889—92no—
146#889—92no—
147#887—92no—
148#67⋊C4—32no—
149#67⋊C4—61no—
150#67⋊C4—92no—
151#878—32no—
152#878—92no—
153#878—92no—
154#878—92no—
155#869—91no—
156C4×Dic6—91no—
157C23×C32—21no—
158C23×C32C22×D312yes—
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4C22×D312yes—
164C2×C32×C4C22×D312yes—
165C2×C3⋊Dic3—42no—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C2×C3⋊Dic3C22×D312yes—
169C2×C3⋊Dic3C22×D312yes—
170#634—42no—
171C3×#67—38no—
172C3×#67—64no—
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—32no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—122no—
185C2×C3×Dic3—122no—
186C3×C4×D3—64no—
187C3×C4×D3—122no—
188C3×Dic6—122no—
189#627—64no—
190D3×Dic3—122no—
191D3×Dic3—122no—
192#554—271noSylow
193C2×H(3,𝔽3)—22no—
194C2×H(3,𝔽3)C22×Q811yescharacteristic
195C2×H(3,𝔽3)C2×#4111yescharacteristic
196C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
197#272—22no—
198C2×C3×Q8—92no—
199C2×C3×D4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—92no—
202C22×C3×C4—91no—
203C2×#67—32no—
204C2×#67—61no—
205C22×Dic3—32no—
206C22×Dic3—32no—
207C22×Dic3—61no—
208C22×Dic3—92no—
209C22×Dic3—92no—
210C2×C4×D3—32no—
211C2×C4×D3—61no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×Dic6—32no—
215C2×Dic6—92no—
216C2×Dic6—91no—
217C2×Dic6—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—91no—
222C3×C22⋊C4—32no—
223C3×C22⋊C4—61no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—91no—
226C3×C22⋊C4D3211yescharacteristic
227C3×C42—91no—
228#217—32no—
229#217—92no—
230#217—92no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235C4⋊Dic3—32no—
236C4⋊Dic3—92no—
237C4⋊Dic3—91no—
238C4⋊Dic3—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—61no—
247Dic3⋊C4—92no—
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—
258C4×Dic3—32no—
259C4×Dic3—61no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32D3×Q814yes—
270C22×C32C23×D312yes—
271C2×C3×D3—34no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—41no—
280C32×C4—41no—
281C3⋊Dic3—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—
294C3×Dic3—121no—
295C4⋊Q8—271no—
296#125—272no—
297C22⋊Q8—92no—
298C22⋊Q8—271no—
299C22⋊Q8—272no—
300C22⋊Q8—271no—
301C4×Q8—271no—
302C4×D4—91no—
303C4×D4—272no—
304C2×C4⋊C4—271no—
305C2×C4⋊C4—271no—
306H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
307C23×C3—32no—
308C23×C3—61no—
309C23×C3—91no—
310C23×C3C2×D3211yescharacteristic
311C22×D3—32no—
312C22×D3—61no—
313C3×Q8—362no—
314C3×D4—94no—
315C2×C3×C4—32no—
316C2×C3×C4—32no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—91no—
326C2×C3×C4—182no—
327C2×C3×C4—182no—
328C2×C3×C4—362no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331#67—38no—
332#67—64no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—32no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—64no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—122no—
349C2×Dic3—122no—
350C2×Dic3—362no—
351C2×Dic3—362no—
352C4×D3—64no—
353C4×D3—122no—
354C4×D3—362no—
355C4×D3—362no—
356Dic6—122no—
357Dic6—184no—
358Dic6—182no—
359Dic6—362no—
360C2×C32—24no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—42no—
365C2×C32#100412yes—
366C2×C32C2×D3×Q812yes—
367C2×C32D3×#4112yes—
368C3×D3—64no—
369C3×D3—122no—
370C2×Q8—92no—
371C2×Q8—271no—
372C2×D4—91no—
373C22×C4—91no—
374C22×C4—92no—
375C22×C4—91no—
376C22×C4—272no—
377C4⋊C4—92no—
378C4⋊C4—91no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4—271no—
382C4⋊C4—272no—
383C4⋊C4—271no—
384C4⋊C4—271no—
385C4⋊C4—272no—
386C4⋊C4—272no—
387C22⋊C4—92no—
388C22⋊C4—91no—
389C22⋊C4—272no—
390C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C42—91no—
392C42—272no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—32no—
396C22×C3—34no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—61no—
400C22×C3—62no—
401C22×C3—92no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—181no—
405C22×C3—181no—
406C22×C3#379712yes—
407C22×C3C22×D3211yescharacteristic
408C2×D3—34no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—121no—
413C2×D3—121no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—92no—
419C3×C4—92no—
420C3×C4—121no—
421C3×C4—121no—
422C3×C4—182no—
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—121no—
437Dic3—182no—
438Dic3—182no—
439Dic3—182no—
440Dic3—182no—
441C32—21no—
442C32#563612yescomplemented
443C23—91no—
444C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
445Q8—362no—
446Q8—542no—
447D4—94no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—91no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—182no—
456C2×C4—182no—
457C2×C4—362no—
458C2×C4—1082no—
459C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
461C2×C3—22no—
462C2×C3—32no—
463C2×C3—32no—
464C2×C3—32no—
465C2×C3—64no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—122no—
470C2×C3—182no—
471C2×C3C2×#379711yescharacteristic
472C2×C3#6431211yescharacteristic
473C2×C3#6431311yescharacteristic
474D3—64no—
475D3—122no—
476C22—21no—
477C22—21no—
478C22—92no—
479C22—181no—
480C22—181no—
481C22#8865112yes—
482C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
483C4—21no—
484C4—21no—
485C4—92no—
486C4—92no—
487C4—181no—
488C4—182no—
489C4—182no—
490C4—182no—
491C4—182no—
492C3—32no—
493C3—61no—
494C3#1059416811yescharacteristic
495C2—22no—
496C2—182no—
497C2C2×#8865111yescharacteristic
498C2#1172250511yescharacteristic
499C2#1172250611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#6235C22
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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
2. #64087⋊C3C2semidirect
3. #11721309C2semidirect
4. C2×#88449C2semidirect
6. #11721152C2semidirect
7. #11721148C2semidirect
8. #11721099C2semidirect
9. #11721096C2semidirect
10. #11721086C2semidirect
11. #11721078C2semidirect
36. #64087D3semidirect
56. #6235Q8semidirect
306. H(3,𝔽3)#554semidirect
442. C32#5636semidirect

Maximal subgroups

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

subgroupindex
#117217372
#64087⋊C32
#117213092
C2×#884492
#117212932
#117211522
#117211482
#117210992
#117210962
#117210862
#117210782
#105941463