Finite Group Databasebeta

← #13141178

Subgroups — #13141178

Order 1728.

1237 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1237 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#6235C211yesmaximal, characteristic
2C22×#6199C211yesmaximal, characteristic, complemented
3C2×#3757⋊C3C212yesmaximal, complemented
4C2×#88451C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591721—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C23×#272C2211yescharacteristic
9C2×#6235—22no—
10C2×#6235—22no—
11C2×#6235C414yescomplemented
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6199—22no—
16C2×#6199—24no—
17#3757⋊C3—24no—
18#88451—22no—
19C22×C3×#67—32no—
20C22×C3×#67—61no—
21C22×D3×Dic3—32no—
22C2×#3757D312yescomplemented
23C2×C3×#217—32no—
24C2×(C2×D3)⋊Dic3—32no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)—21no—
28C23×H(3,𝔽3)—21no—
29C23×H(3,𝔽3)—22no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)C2×C412yescomplemented
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272—22no—
34C22×#272—22no—
35C22×#272C2×C412yes—
36C22×#272D412yes—
37C22×#272D412yes—
38#6235—28no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084C2×C412yescomplemented
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3D414yescomplemented
48#6199—28no—
49#5220—91no—
50#4956—92no—
51C24×C32—21no—
52C24×C32C2×D312yes—
53C23×C3×D3—32no—
54C23×C3×D3—61no—
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D312yes—
57C2×C3×#67—38no—
58C2×C3×#67—64no—
59C2×C3×#67—64no—
60C2×C3×#67—64no—
61C2×C3×#67—64no—
62C2×C3×#67—64no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—61no—
67C2×D3×Dic3—64no—
68C2×D3×Dic3—64no—
69C2×D3×Dic3—64no—
70#3757—24no—
71C3×#217—64no—
72(C2×D3)⋊Dic3—64no—
73C22×H(3,𝔽3)—22no—
74C22×H(3,𝔽3)—24no—
75C22×H(3,𝔽3)—41no—
76C22×H(3,𝔽3)—42no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)C22×C411yescharacteristic
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C22×H(3,𝔽3)C2×D412yes—
82C22×H(3,𝔽3)#4112yes—
83C2×#272—24no—
84C2×#272—22no—
85C2×#272—24no—
86C2×#272—22no—
87C2×#272C22⋊C414yescomplemented
88#1084—22no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—42no—
91C22×C3×D4—91no—
92C22×#67—32no—
93C22×#67—61no—
94C23×Dic3—91no—
95C22×C4×D3—92no—
96C2×C3×C22⋊C4—92no—
97C2×#217—32no—
98C2×#217—92no—
99C2×#217—91no—
100C2×(C2×D3)⋊C4—92no—
101C3⋊#95—92no—
102C3⋊#95—92no—
103C3⋊#95—92no—
104C23×C32—22no—
105C23×C32—22no—
106C23×C32—22no—
107C23×C32—22no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—21no—
114C23×C32—42no—
115C23×C32—42no—
116C23×C32C4×D314yescomplemented
117C23×C32C22×D312yes—
118C22×C3×D3—34no—
119C22×C3×D3—34no—
120C22×C3×D3—34no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—122no—
129C22×C3×D3—122no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3#6714yescomplemented
136C3×#67—616no—
137C3×#67—616no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150D3×Dic3—616no—
151#386—271noSylow
152C2×H(3,𝔽3)—42no—
153C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
154C2×H(3,𝔽3)C4×D412yes—
155C2×H(3,𝔽3)#12011yescharacteristic
156C2×H(3,𝔽3)#12112yes—
157C2×H(3,𝔽3)#12311yescharacteristic
158#272—24no—
159C24×C3—32no—
160C24×C3—61no—
161C24×C3—91no—
162C24×C3D3211yescharacteristic
163C23×D3—32no—
164C23×D3—61no—
165C2×C3×D4—94no—
166C2×C3×D4—182no—
167C2×C3×D4—182no—
168C22×C3×C4—92no—
169C22×C3×C4—91no—
170C22×C3×C4—92no—
171C2×#67—38no—
172C2×#67—64no—
173C2×#67—64no—
174C2×#67—64no—
175C2×#67—64no—
176C2×#67—64no—
177C22×Dic3—32no—
178C22×Dic3—32no—
179C22×Dic3—32no—
180C22×Dic3—61no—
181C22×Dic3—92no—
182C22×Dic3—92no—
183C22×Dic3—184no—
184C22×Dic3—182no—
185C2×C4×D3—184no—
186C2×C4×D3—184no—
187C2×C4×D3—184no—
188C3×C22⋊C4—184no—
189#217—64no—
190#217—184no—
191#217—182no—
192(C2×D3)⋊C4—184no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—22no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—44no—
203C22×C32—44no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—42no—
210C22×C32C2×C4×D312yes—
211C22×C32#23512yes—
212C22×C32D3×D412yes—
213C22×C32D3×D412yes—
214C22×C32D3×D412yes—
215C22×C32#23712yes—
216C22×C32C2×#6712yes—
217C2×C3×D3—38no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C2×C3×D3—64no—
224C2×C3×D3—64no—
225C2×C3×D3—124no—
226C2×C3×D3—124no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C3⋊Dic3—24no—
230C3⋊Dic3—42no—
231C3×Dic3—64no—
232C3×Dic3—64no—
233C3×Dic3—64no—
234C3×Dic3—122no—
235C22×D4—91no—
236C23×C4—271no—
237C2×C22⋊C4—92no—
238C2×C22⋊C4—271no—
239#95—272no—
240H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—34no—
246C23×C3—32no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—62no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3—62no—
255C23×C3—62no—
256C23×C3—61no—
257C23×C3—92no—
258C23×C3—92no—
259C23×C3—92no—
260C23×C3—122no—
261C23×C3—122no—
262C23×C3—182no—
263C23×C3—182no—
264C23×C3#62412yes—
265C23×C3C2×D3211yescharacteristic
266C22×D3—34no—
267C22×D3—34no—
268C22×D3—34no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—62no—
275C22×D3—62no—
276C22×D3—122no—
277C22×D3—122no—
278C3×D4—188no—
279C2×C3×C4—94no—
280C2×C3×C4—92no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—182no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288#67—616no—
289#67—616no—
290C2×Dic3—34no—
291C2×Dic3—34no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—94no—
303C2×Dic3—182no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306C2×Dic3—182no—
307C2×Dic3—182no—
308C2×Dic3—188no—
309C4×D3—1816no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—44no—
316C2×C32—44no—
317C2×C32D3×C22⋊C412yes—
318C2×C32#67⋊C412yes—
319C2×C32#88312yes—
320C2×C32#88412yes—
321C2×C32C4×#6712yes—
322C2×C32#93712yes—
323C2×C32#93812yes—
324C3×D3—68no—
325C3×D3—124no—
326C24—91no—
327C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C2×D4—94no—
329C2×D4—182no—
330C2×D4—182no—
331C22×C4—91no—
332C22×C4—92no—
333C22×C4—92no—
334C22×C4—544no—
335C22×C4—542no—
336C22⋊C4—184no—
337C22⋊C4—542no—
338C22×C3—24no—
339C22×C3—22no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—42no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—64no—
351C22×C3—64no—
352C22×C3—64no—
353C22×C3—64no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—62no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—94no—
360C22×C3—124no—
361C22×C3—122no—
362C22×C3—122no—
363C22×C3—122no—
364C22×C3—122no—
365C22×C3—124no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3—184no—
369C22×C3—184no—
370C22×C3—182no—
371C22×C3—182no—
372C22×C3#380412yes—
373C22×C3C2×#62411yescharacteristic
374C22×C3D3×#6712yes—
375C22×C3#381111yescharacteristic
376C22×C3#381111yescharacteristic
377C2×D3—38no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3—64no—
384C2×D3—64no—
385C2×D3—124no—
386C2×D3—124no—
387C2×D3—122no—
388C2×D3—122no—
389C3×C4—184no—
390C3×C4—182no—
391C3×C4—362no—
392Dic3—64no—
393Dic3—64no—
394Dic3—64no—
395Dic3—122no—
396Dic3—184no—
397Dic3—362no—
398C32—21no—
399C32#495612yescomplemented
400C23—22no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—92no—
405C23—92no—
406C23—92no—
407C23—182no—
408C23—182no—
409C23#619912yescomplemented
410C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411D4—188no—
412C2×C4—92no—
413C2×C4—94no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—182no—
421C2×C4—548no—
422C2×C3—32no—
423C2×C3—32no—
424C2×C3—32no—
425C2×C3—32no—
426C2×C3—32no—
427C2×C3—32no—
428C2×C3—32no—
429C2×C3—42no—
430C2×C3—62no—
431C2×C3—62no—
432C2×C3—61no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—124no—
436C2×C3—124no—
437C2×C3—184no—
438C2×C3#6395212yes—
439C2×C3#6395812yes—
440C2×C3#6397411yescharacteristic
441C2×C3#6397511yescharacteristic
442C2×C3#6398311yescharacteristic
443D3—68no—
444D3—124no—
445C22—24no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—94no—
451C22—184no—
452C22—184no—
453C22—182no—
454C22—182no—
455C22#8867112yes—
456C22C2×#619911yescharacteristic
457C22#8867612yes—
458C22#8867711yescharacteristic
459C22#8867711yescharacteristic
460C4—182no—
461C4—184no—
462C4—362no—
463C3—32no—
464C3—61no—
465C3#1059172211yescharacteristic
466C2—42no—
467C2—184no—
468C2#1172129312yes—
469C2#1172130312yes—
470C2#1172131911yescharacteristic
471C2#1172132011yescharacteristic
472C2#1172132411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC24×H(3,𝔽3)C22
prime radicalC22×#6235C2
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6199C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C2×#88451C2semidirect
5. #11720334C2semidirect
11. C2×#6235C4semidirect
22. C2×#3757D3semidirect
31. C23×H(3,𝔽3)C2×C4semidirect
41. C2×#1084C2×C4semidirect
47. C2×(C3⋊Dic3)⋊C3D4semidirect
87. C2×#272C22⋊C4semidirect
116. C23×C32C4×D3semidirect
135. C2×C3⋊Dic3#67semidirect
240. H(3,𝔽3)#386semidirect
399. C32#4956semidirect
409. C23#6199semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#61992
C2×#3757⋊C32
C2×#884512
#117203342
#105917213