Finite Group Databasebeta

← #13140255

Subgroups — #13140255

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×C4⋊#272C211yesmaximal, characteristic
2C22×#6199C211yesmaximal, characteristic, complemented
3C2×#88451C212yesmaximal, complemented
4C2×#88451C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10591376—32nomaximal
7C23×#272C2211yescharacteristic
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272—22no—
11C2×C4⋊#272—22no—
12C2×C4⋊#272C414yescomplemented
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C2×#6199—22no—
16C2×#6199—24no—
17#88451—22no—
18#88451—24no—
19C22×C3×D12—32no—
20C22×C3×D12—61no—
21C22×D3×Dic3—32no—
22C2×(C2×D3)⋊Dic3—32no—
23C2×(C2×D3)⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—32no—
25#63658D312yescomplemented
26C3×C3⋊#95—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×#272—22no—
29C22×#272—22no—
30C22×#272—21no—
31C22×#272—21no—
32C22×#272—22no—
33C22×#272—22no—
34C22×#272C2×C412yes—
35C22×#272C2×C412yescomplemented
36C22×#272D412yes—
37C22×#272D412yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2×C412yescomplemented
41C4⋊#272—28no—
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#4665—92no—
51C23×C3×D3—32no—
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C23×C3×D3—61no—
55C22×C32×C4—21no—
56C22×C32×C4C2×D312yes—
57C22×C3⋊Dic3C2×D312yes—
58C22×C3⋊Dic3C2×D312yes—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C2×C3×D12—38no—
62C2×C3×D12—64no—
63C2×C3×D12—64no—
64C2×C3×D12—64no—
65C2×C3×D12—64no—
66C2×C3×D12—64no—
67C2×D3×Dic3—64no—
68C2×D3×Dic3—64no—
69C2×D3×Dic3—64no—
70(C2×D3)⋊Dic3—64no—
71(C2×D3)⋊Dic3—64no—
72(C2×D3)⋊Dic3—64no—
73C22×H(3,𝔽3)C22×C411yescharacteristic
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)C2×D412yes—
77C22×H(3,𝔽3)#4112yes—
78C2×#272—24no—
79C2×#272—22no—
80C2×#272—24no—
81C2×#272—22no—
82C2×#272—22no—
83C2×#272—24no—
84C2×#272—41no—
85C2×#272—42no—
86C2×#272—41no—
87C2×#272C22⋊C414yescomplemented
88C4×H(3,𝔽3)—22no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—42no—
91C22×C3×D4—91no—
92C23×Dic3—91no—
93C22×D12—32no—
94C22×D12—61no—
95C22×C4×D3—92no—
96C2×#217—92no—
97C2×#217—91no—
98C2×(C2×D3)⋊C4—92no—
99C2×(C2×D3)⋊C4—92no—
100C2×(C2×D3)⋊C4—92no—
101C3×#95—92no—
102C3⋊#95—32no—
103C3⋊#95—92no—
104C23×C32—21no—
105C23×C32C22×D312yes—
106C22×C3×D3—34no—
107C22×C3×D3—34no—
108C22×C3×D3—34no—
109C22×C3×D3—34no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C22×C3×D3—62no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—122no—
127C22×C3×D3—122no—
128C22×C3×D3—122no—
129C22×C3×D3—122no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4C4×D314yescomplemented
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3D1214yescomplemented
142C2×C3×Dic3—34no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C3×D12—616no—
149C3×D12—616no—
150D3×Dic3—616no—
151#386—271noSylow
152C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
153C2×H(3,𝔽3)C4×D412yes—
154C2×H(3,𝔽3)#12011yescharacteristic
155C2×H(3,𝔽3)#12112yes—
156C2×H(3,𝔽3)#12311yescharacteristic
157#272—24no—
158#272—42no—
159C24×C3—91no—
160C24×C3—91no—
161C23×D3—32no—
162C23×D3—32no—
163C23×D3—61no—
164C23×D3—61no—
165C2×C3×D4—94no—
166C2×C3×D4—182no—
167C2×C3×D4—182no—
168C22×C3×C4—32no—
169C22×C3×C4—61no—
170C22×C3×C4—92no—
171C22×C3×C4—92no—
172C22×C3×C4D3211yescharacteristic
173C22×Dic3—32no—
174C22×Dic3—32no—
175C22×Dic3—92no—
176C22×Dic3—92no—
177C22×Dic3—184no—
178C22×Dic3—182no—
179C2×D12—38no—
180C2×D12—64no—
181C2×D12—64no—
182C2×D12—64no—
183C2×D12—64no—
184C2×D12—64no—
185C2×C4×D3—184no—
186C2×C4×D3—184no—
187C2×C4×D3—184no—
188#217—184no—
189#217—182no—
190(C2×D3)⋊C4—184no—
191(C2×D3)⋊C4—184no—
192(C2×D3)⋊C4—184no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2×C4×D312yes—
199C22×C32C2×D1212yes—
200C22×C32#23512yes—
201C22×C32D3×D412yes—
202C22×C32D3×D412yes—
203C22×C32D3×D412yes—
204C22×C32#23912yes—
205C2×C3×D3—38no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—124no—
218C2×C3×D3—122no—
219C2×C3×D3—124no—
220C2×C3×D3—122no—
221C2×C3×D3—124no—
222C2×C3×D3—122no—
223C2×C3×D3—124no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C32×C4—24no—
230C32×C4—24no—
231C3⋊Dic3—24no—
232C3⋊Dic3—42no—
233C3×Dic3—64no—
234C3×Dic3—122no—
235C22×D4—91no—
236C23×C4—271no—
237C2×C22⋊C4—272no—
238C2×C22⋊C4—271no—
239#95—92no—
240H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
241C23×C3—32no—
242C23×C3—61no—
243C23×C3—92no—
244C23×C3—92no—
245C23×C3—92no—
246C23×C3—92no—
247C23×C3—182no—
248C23×C3—182no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3—181no—
252C23×C3—181no—
253C23×C3C2×D3211yescharacteristic
254C22×D3—34no—
255C22×D3—34no—
256C22×D3—34no—
257C22×D3—34no—
258C22×D3—62no—
259C22×D3—62no—
260C22×D3—62no—
261C22×D3—62no—
262C22×D3—62no—
263C22×D3—62no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—122no—
275C22×D3—122no—
276C22×D3—122no—
277C22×D3—122no—
278C3×D4—188no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—34no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—94no—
288C2×C3×C4—182no—
289C2×C3×C4—182no—
290C2×C3×C4—182no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×C3×C4#62412yes—
294C2×Dic3—34no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—94no—
301C2×Dic3—182no—
302C2×Dic3—182no—
303C2×Dic3—182no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306C2×Dic3—188no—
307D12—616no—
308D12—616no—
309C4×D3—1816no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32C4×D1212yes—
316C2×C32D3×C22⋊C412yes—
317C2×C32#88212yes—
318C2×C32#88412yes—
319C2×C32#89312yes—
320C2×C32#89412yes—
321C2×C32#89512yes—
322C3×D3—68no—
323C3×D3—124no—
324C3×D3—124no—
325C3×D3—124no—
326C24—91no—
327C24—91no—
328C2×D4—94no—
329C2×D4—182no—
330C2×D4—182no—
331C22×C4—92no—
332C22×C4—92no—
333C22×C4—544no—
334C22×C4—542no—
335C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C22⋊C4—544no—
337C22⋊C4—542no—
338C22×C3—32no—
339C22×C3—32no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—62no—
346C22×C3—62no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—94no—
351C22×C3—184no—
352C22×C3—182no—
353C22×C3—184no—
354C22×C3—182no—
355C22×C3—184no—
356C22×C3—182no—
357C22×C3—361no—
358C22×C3—362no—
359C22×C3—361no—
360C22×C3#379912yes—
361C22×C3D3×D1212yes—
362C22×C3C2×#62411yescharacteristic
363C22×C3#381111yescharacteristic
364C22×C3#381111yescharacteristic
365C2×D3—38no—
366C2×D3—64no—
367C2×D3—64no—
368C2×D3—64no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—64no—
372C2×D3—64no—
373C2×D3—64no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—124no—
378C2×D3—122no—
379C2×D3—124no—
380C2×D3—122no—
381C2×D3—124no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—124no—
388C2×D3—122no—
389C3×C4—22no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—184no—
393C3×C4—362no—
394Dic3—64no—
395Dic3—122no—
396Dic3—184no—
397Dic3—362no—
398C32—21no—
399C32#466512yescomplemented
400C23—92no—
401C23—92no—
402C23—92no—
403C23—92no—
404C23—182no—
405C23—182no—
406C23—182no—
407C23—182no—
408C23—181no—
409C23—181no—
410C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411D4—188no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—94no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—548no—
421C2×C4#619912yescomplemented
422C2×C3—32no—
423C2×C3—32no—
424C2×C3—32no—
425C2×C3—32no—
426C2×C3—32no—
427C2×C3—32no—
428C2×C3—32no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—61no—
432C2×C3—62no—
433C2×C3—61no—
434C2×C3—184no—
435C2×C3—362no—
436C2×C3#6389412yes—
437C2×C3#6389711yescharacteristic
438C2×C3#6391012yes—
439C2×C3#6392311yescharacteristic
440C2×C3#6397411yescharacteristic
441D3—68no—
442D3—124no—
443D3—124no—
444D3—124no—
445C22—94no—
446C22—184no—
447C22—182no—
448C22—184no—
449C22—182no—
450C22—182no—
451C22—184no—
452C22—361no—
453C22—362no—
454C22—361no—
455C22#8865212yes—
456C22#8865412yes—
457C22C2×#619911yescharacteristic
458C22#8867711yescharacteristic
459C22#8867711yescharacteristic
460C4—22no—
461C4—184no—
462C4—362no—
463C3—32no—
464C3—61no—
465C3#1059138011yescharacteristic
466C2—184no—
467C2—362no—
468C2#1172114912yes—
469C2#1172115111yescharacteristic
470C2#1172115712yes—
471C2#1172116311yescharacteristic
472C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC22×C4⋊#272C2
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6199C2semidirect
3. C2×#88451C2semidirect
4. C2×#88451C2semidirect
5. #63658⋊C3C2semidirect
12. C2×C4⋊#272C4semidirect
25. #63658D3semidirect
35. C22×#272C2×C4semidirect
40. C2×C4×H(3,𝔽3)C2×C4semidirect
47. C2×(C3⋊Dic3)⋊C3D4semidirect
87. C2×#272C22⋊C4semidirect
135. C2×C32×C4C4×D3semidirect
141. C2×C3⋊Dic3D12semidirect
240. H(3,𝔽3)#386semidirect
399. C32#4665semidirect
421. C2×C4#6199semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×#61992
C2×#884512
#63658⋊C32
#105913763