Finite Group Databasebeta

← #13141170

Subgroups — #13141170

Order 1728.

1139 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1139 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
2C2×#3757⋊C3C212yesmaximal, complemented
3C2×#88451C212yesmaximal, complemented
4C2×#88449C211yesmaximal, characteristic, complemented
5C2×#88447C211yesmaximal, characteristic, complemented
6#10591707—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—
15#3757⋊C3—24no—
16#88451—24no—
17#88449—22no—
18#88447—22no—
19C22×C3×#67—32no—
20C22×C3×#67—61no—
21C2×#3757D312yescomplemented
22C2×C3×#217—32no—
23C2×Dic3⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—32no—
25C2×(C2×D3)⋊Dic3—32no—
26C2×Dic32—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×C412yes—
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272—21no—
34C22×#272—21no—
35C22×#272—22no—
36C22×#272—22no—
37C22×#272C2×C412yes—
38#6235—28no—
39C2×#1084C2×C412yes—
40C2×#1084D412yes—
41C2×#1084D412yes—
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#5221—91no—
49#4957—92no—
50C24×C32—21no—
51C24×C32C2×D312yes—
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D312yes—
56C2×C3×#67—38no—
57C2×C3×#67—64no—
58C2×C3×#67—64no—
59C2×C3×#67—64no—
60C2×C3×#67—64no—
61C2×C3×#67—64no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—61no—
66#3757—24no—
67C3×#217—64no—
68Dic3⋊Dic3—64no—
69(C2×D3)⋊Dic3—64no—
70(C2×D3)⋊Dic3—64no—
71Dic32—64no—
72C22×H(3,𝔽3)—22no—
73C22×H(3,𝔽3)—24no—
74C22×H(3,𝔽3)—42no—
75C22×H(3,𝔽3)—41no—
76C22×H(3,𝔽3)—41no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)#4112yes—
82C2×#272—22no—
83C2×#272—24no—
84C2×#272—42no—
85C2×#272—41no—
86C2×#272—41no—
87#1084C22⋊C414yescomplemented
88(C3⋊Dic3)⋊C3—24no—
89(C3⋊Dic3)⋊C3—42no—
90C22×C3×D4—91no—
91C22×#67—32no—
92C22×#67—61no—
93C2×C3×C22⋊C4—92no—
94C2×#217—32no—
95C2×#217—92no—
96C2×#217—92no—
97C2×(C2×D3)⋊C4—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×C4⋊Dic3—91no—
100C2×Dic3⋊C4—92no—
101C2×C4×Dic3—91no—
102C2×C4×Dic3—92no—
103C23×C32—22no—
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—21no—
113C23×C32—42no—
114C23×C32—42no—
115C23×C32C4×D314yes—
116C23×C32C22×D312yes—
117C22×C3×D3—34no—
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—62no—
127C22×C3×D3—122no—
128C22×C3×D3—122no—
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3#6714yescomplemented
135C3×#67—616no—
136C3×#67—616no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
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—
149#390—271noSylow
150C2×H(3,𝔽3)—42no—
151C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
152C2×H(3,𝔽3)C4×D412yes—
153C2×H(3,𝔽3)#12112yes—
154C2×H(3,𝔽3)#12411yescharacteristic
155C2×H(3,𝔽3)#12711yescharacteristic
156#272—42no—
157C24×C3—32no—
158C24×C3—61no—
159C24×C3—91no—
160C24×C3D3211yescharacteristic
161C23×D3—32no—
162C23×D3—61no—
163C2×C3×D4—94no—
164C2×C3×D4—182no—
165C2×C3×D4—182no—
166C22×C3×C4—91no—
167C22×C3×C4—92no—
168C22×C3×C4—92no—
169C2×#67—38no—
170C2×#67—64no—
171C2×#67—64no—
172C2×#67—64no—
173C2×#67—64no—
174C2×#67—64no—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—32no—
178C22×Dic3—61no—
179C22×Dic3—92no—
180C22×Dic3—92no—
181C3×C22⋊C4—184no—
182#217—64no—
183#217—184no—
184#217—184no—
185(C2×D3)⋊C4—184no—
186(C2×D3)⋊C4—184no—
187C4⋊Dic3—182no—
188Dic3⋊C4—184no—
189C4×Dic3—184no—
190C4×Dic3—182no—
191C22×C32—24no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—44no—
201C22×C32—42no—
202C22×C32—44no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32C2×C4×D312yes—
209C22×C32#23512yes—
210C22×C32D3×D412yes—
211C22×C32D3×D412yes—
212C22×C32D3×D412yes—
213C22×C32#23712yes—
214C22×C32C2×#6712yes—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—124no—
220C2×C3×D3—122no—
221C2×C3×D3—124no—
222C2×C3×D3—122no—
223C2×C3×D3—122no—
224C2×C3×D3—122no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C3⋊Dic3—24no—
228C3⋊Dic3—42no—
229C3×Dic3—38no—
230C3×Dic3—64no—
231C3×Dic3—64no—
232C3×Dic3—122no—
233C22×D4—91no—
234C2×C4⋊C4—271no—
235C2×C22⋊C4—92no—
236C2×C22⋊C4—272no—
237C2×C42—271no—
238H(3,𝔽3)#39011yescharacteristic, complemented, Sylow
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—34no—
244C23×C3—32no—
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—62no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3—61no—
255C23×C3—92no—
256C23×C3—122no—
257C23×C3—122no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3—181no—
261C23×C3—181no—
262C23×C3#62412yes—
263C23×C3C2×D3211yescharacteristic
264C22×D3—34no—
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—
276C3×D4—188no—
277C2×C3×C4—92no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—94no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—182no—
284C2×C3×C4—182no—
285C2×C3×C4—182no—
286#67—616no—
287#67—616no—
288C2×Dic3—34no—
289C2×Dic3—34no—
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—94no—
301C2×Dic3—182no—
302C2×Dic3—182no—
303C2×Dic3—182no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306C2×C32—22no—
307C2×C32—21no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—44no—
312C2×C32—44no—
313C2×C32D3×C22⋊C412yes—
314C2×C32#67⋊C412yes—
315C2×C32#88412yes—
316C2×C32#88512yes—
317C2×C32C4×#6712yes—
318C2×C32#93912yes—
319C2×C32#94012yes—
320C3×D3—124no—
321C3×D3—124no—
322C24—91no—
323C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C2×D4—94no—
325C2×D4—182no—
326C2×D4—182no—
327C22×C4—91no—
328C22×C4—92no—
329C22×C4—92no—
330C4⋊C4—542no—
331C22⋊C4—184no—
332C22⋊C4—544no—
333C42—542no—
334C22×C3—24no—
335C22×C3—22no—
336C22×C3—32no—
337C22×C3—32no—
338C22×C3—32no—
339C22×C3—32no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—41no—
344C22×C3—42no—
345C22×C3—41no—
346C22×C3—64no—
347C22×C3—64no—
348C22×C3—64no—
349C22×C3—64no—
350C22×C3—62no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—124no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—124no—
362C22×C3—122no—
363C22×C3—184no—
364C22×C3—182no—
365C22×C3—361no—
366C22×C3—362no—
367C22×C3—361no—
368C22×C3#380412yes—
369C22×C3C2×#62411yescharacteristic
370C22×C3D3×#6712yes—
371C22×C3#381111yescharacteristic
372C22×C3#381111yescharacteristic
373C2×D3—64no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—124no—
378C2×D3—122no—
379C2×D3—122no—
380C2×D3—122no—
381C2×D3—122no—
382C2×D3—122no—
383C2×D3—124no—
384C2×D3—122no—
385C3×C4—94no—
386C3×C4—184no—
387C3×C4—362no—
388Dic3—38no—
389Dic3—64no—
390Dic3—64no—
391Dic3—122no—
392Dic3—184no—
393Dic3—362no—
394C32—21no—
395C32#495712yescomplemented
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—92no—
401C23—182no—
402C23—182no—
403C23—181no—
404C23—181no—
405C23#619912yes—
406C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
407D4—188no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—94no—
412C2×C4—182no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—182no—
417C2×C3—32no—
418C2×C3—32no—
419C2×C3—32no—
420C2×C3—32no—
421C2×C3—32no—
422C2×C3—32no—
423C2×C3—32no—
424C2×C3—42no—
425C2×C3—62no—
426C2×C3—61no—
427C2×C3—62no—
428C2×C3—61no—
429C2×C3—61no—
430C2×C3—124no—
431C2×C3—124no—
432C2×C3—362no—
433C2×C3#6395212yes—
434C2×C3#6395311yescharacteristic
435C2×C3#6397112yes—
436C2×C3#6397411yescharacteristic
437C2×C3#6397911yescharacteristic
438D3—124no—
439D3—124no—
440C22—24no—
441C22—22no—
442C22—41no—
443C22—42no—
444C22—41no—
445C22—184no—
446C22—182no—
447C22—361no—
448C22—362no—
449C22—361no—
450C22#8867112yes—
451C22C2×#619911yescharacteristic
452C22#8867612yes—
453C22#8867711yescharacteristic
454C22#8867711yescharacteristic
455C4—94no—
456C4—184no—
457C4—362no—
458C3—32no—
459C3—61no—
460C3#1059170411yescharacteristic
461C2—42no—
462C2—362no—
463C2#1172129312yes—
464C2#1172129411yescharacteristic
465C2#1172131212yes—
466C2#1172131711yescharacteristic
467C2#1172132011yescharacteristic
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)#390
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. C2×#3757⋊C3C2semidirect
3. C2×#88451C2semidirect
4. C2×#88449C2semidirect
5. C2×#88447C2semidirect
11. C2×#6235C4semidirect
21. C2×#3757D3semidirect
47. C2×(C3⋊Dic3)⋊C3D4semidirect
87. #1084C22⋊C4semidirect
134. C2×C3⋊Dic3#67semidirect
238. H(3,𝔽3)#390semidirect
395. C32#4957semidirect

Maximal subgroups

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

subgroupindex
C22×#62352
C2×#3757⋊C32
C2×#884512
C2×#884492
C2×#884472
#105917073