Finite Group Databasebeta

← #13140529

Subgroups — #13140529

Order 1728.

605 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#3830⋊C3C211yesmaximal, characteristic
2#11721296C212yesmaximal, complemented
3#11720580C211yesmaximal, characteristic, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#11720333C212yesmaximal, complemented
6#10591529—31nomaximal
7#10591528—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3C411yescharacteristic, complemented
10C2×#638⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C411yescharacteristic, complemented
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13#3830⋊C3—22no—
14#3830⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
16C2×#634⋊C3C411yescharacteristic, complemented
17#88552C2212yes—
18#3757⋊C3C2212yes—
19#88448—22no—
20#88447—22no—
21C2×#3830D311yescharacteristic
22C2×C3×#237—31no—
23#63957—32no—
24#63955—32no—
25#63661D311yescharacteristic, complemented
26C3×#834—31no—
27C3×#834—31no—
28C3×#834—31no—
29C3×#834—61no—
30#63581—32no—
31#63580—32no—
32C23×H(3,𝔽3)C2×C411yescharacteristic
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
35D4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)C2×C411yescharacteristic
37C2×#1084—22no—
38#638⋊C3—22no—
39#638⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D412yes—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3—22no—
51#5240—91no—
52#4740—91no—
53#4740—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D311yescharacteristic
57C2×#638Dic311yescharacteristic, complemented
58C2×#638C2×D311yescharacteristic
59C22×C3⋊Dic3Dic311yescharacteristic, complemented
60C22×C3⋊Dic3C2×D311yescharacteristic
61#3830—22no—
62#3830—22no—
63C2×C4×C3⋊D3Dic311yescharacteristic, complemented
64C2×#634Dic311yescharacteristic, complemented
65C2×C3×#67—31no—
66C2×C3×#67—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C3×#237—62no—
70C3×#237—62no—
71C2×C3×C4×D3—31no—
72C2×C3×Dic6—31no—
73#3757C2×D312yes—
74C3×#217—32no—
75C3×#217—32no—
76C3×#217—32no—
77C3×#217—62no—
78#3724—62no—
79Dic3⋊Dic3—62no—
80Dic32—62no—
81Dic32—62no—
82C22×H(3,𝔽3)—21no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)C22×C411yescharacteristic
86C22×H(3,𝔽3)C2×D412yes—
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—21no—
90#1084—42no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—42no—
93(C3⋊Dic3)⋊C3C22⋊C412yes—
94(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
95(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
96C2×C3×#41—91no—
97C2×#237—31no—
98C2×#237—91no—
99#926—92no—
100#877—92no—
101#877—92no—
102C3×#99—91no—
103C3×#99—91no—
104#834—31no—
105#834—31no—
106#834—31no—
107#834—61no—
108#834—91no—
109#834—92no—
110#806—92no—
111#806—92no—
112C23×C32—21no—
113C23×C32—21no—
114C23×C32C4×D311yescharacteristic
115C23×C32C2×Dic311yescharacteristic
116C23×C32C22×D311yescharacteristic
117C23×C32C22×D311yescharacteristic
118C22×C3⋊D3C2×Dic311yescharacteristic
119C22×C3×D3—31no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4—42no—
123C2×C32×C4—21no—
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4C2×Dic311yescharacteristic
126#638—22no—
127#638—22no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—22no—
131C2×C3⋊Dic3C2×Dic311yescharacteristic
132C2×C3⋊Dic3C2×Dic311yescharacteristic
133C2×C3⋊Dic3C2×Dic311yescharacteristic
134C2×C3⋊Dic3#6712yes—
135C2×C3⋊Dic3#6712yes—
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138#634—22no—
139C3×#67—62no—
140C3×#67—62no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—122no—
152C3×C4×D3—61no—
153C3×C4×D3—61no—
154C3×Dic6—62no—
155#410—271noSylow
156C2×H(3,𝔽3)—22no—
157C2×H(3,𝔽3)—21no—
158C2×H(3,𝔽3)—41no—
159C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
160(C3⋊D3)⋊C3—41no—
161C3×#41—182no—
162C3×#41—182no—
163C2×C3×Q8—91no—
164C2×C3×D4—31no—
165C2×C3×D4—31no—
166C2×C3×D4—61no—
167C2×C3×D4—91no—
168C2×C3×D4—91no—
169C2×C3×D4D3211yescharacteristic
170C22×C3×C4—91no—
171C22×C3×C4—91no—
172C22×C3×C4—91no—
173C2×#67—31no—
174C2×#67—31no—
175C2×#67—91no—
176C2×#67—91no—
177C22×Dic3—31no—
178C22×Dic3—31no—
179C22×Dic3—91no—
180C22×Dic3—91no—
181#237—62no—
182#237—62no—
183#237—182no—
184#237—182no—
185C2×C4×D3—31no—
186C2×C4×D3—91no—
187C2×Dic6—31no—
188C2×Dic6—91no—
189C3×C22⋊C4—92no—
190C3×C22⋊C4—92no—
191#217—32no—
192#217—32no—
193#217—32no—
194#217—62no—
195#217—92no—
196C4⋊Dic3—182no—
197Dic3⋊C4—182no—
198Dic3⋊C4—182no—
199C4×Dic3—182no—
200C4×Dic3—182no—
201C4×Dic3—182no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—81no—
212C22×C32—81no—
213C22×C32C2×C4×D311yescharacteristic
214C22×C32D3×D412yes—
215C22×C32C22×Dic311yescharacteristic
216C22×C32C2×#6712yes—
217C2×C3⋊D3—21no—
218C2×C3⋊D3—41no—
219C2×C3×D3—61no—
220C2×C3×D3—121no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C3⋊Dic3—21no—
225C3⋊Dic3—42no—
226C3⋊Dic3#21712yes—
227C3⋊Dic3#21711yescharacteristic
228C3⋊Dic3#21711yescharacteristic
229C3×Dic3—32no—
230C3×Dic3—31no—
231C3×Dic3—31no—
232C3×Dic3—61no—
233C3×Dic3—122no—
234C3×Dic3—122no—
235C3×Dic3—122no—
236C3×Dic3—242no—
237C2×#41—91no—
238#117—272no—
239#99—91no—
240#99—91no—
241#99—272no—
242H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—61no—
248C23×C3—61no—
249C23×C3—91no—
250C23×C3D3×Dic311yescharacteristic
251C23×C3C2×D3211yescharacteristic
252C22×D3—31no—
253C22×D3—91no—
254C3×Q8—182no—
255C3×D4—22no—
256C3×D4—62no—
257C3×D4—62no—
258C3×D4—122no—
259C3×D4—182no—
260C3×D4—182no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—92no—
265C2×C3×C4—92no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×C3×C4D3×Dic311yescharacteristic
276#67—62no—
277#67—62no—
278#67—182no—
279#67—182no—
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—31no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—91no—
291C2×Dic3—91no—
292C2×Dic3—92no—
293C2×Dic3—92no—
294C2×Dic3—91no—
295C2×Dic3—122no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—182no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—181no—
302C4×D3—181no—
303Dic6—62no—
304Dic6—182no—
305C2×C32—22no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—42no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—81no—
315C2×C32D3×C22⋊C411yescharacteristic
316C2×C32C2×#21711yescharacteristic
317C3⋊D3—41no—
318C3×D3—121no—
319#41—182no—
320#41—182no—
321C2×Q8—91no—
322C2×D4—91no—
323C2×D4—91no—
324C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C22×C4—91no—
326C22×C4—91no—
327C22×C4—91no—
328C4⋊C4—542no—
329C22⋊C4—92no—
330C22⋊C4—92no—
331C42—542no—
332C22×C3—21no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—32no—
336C22×C3—32no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—62no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—181no—
349C22×C3—241no—
350C22×C3—241no—
351C22×C3—361no—
352C22×C3C2×D3×Dic311yescharacteristic
353C22×C3D3×#6712yes—
354C2×D3—61no—
355C2×D3—121no—
356C2×D3—181no—
357C2×D3—361no—
358C3×C4—21no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—91no—
362C3×C4—91no—
363C3×C4—92no—
364C3×C4—121no—
365C3×C4—181no—
366C3×C4—362no—
367C3×C4—362no—
368Dic3—32no—
369Dic3—31no—
370Dic3—31no—
371Dic3—61no—
372Dic3—92no—
373Dic3—91no—
374Dic3—91no—
375Dic3—122no—
376Dic3—122no—
377Dic3—122no—
378Dic3—181no—
379Dic3—242no—
380Dic3—362no—
381C32—21no—
382C32#474011yescharacteristic, complemented
383C32#524011yescharacteristic, complemented
384C23—91no—
385C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
386C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387Q8—182no—
388D4—22no—
389D4—182no—
390D4—182no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—92no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—41no—
408C2×C3—62no—
409C2×C3—62no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—122no—
414C2×C3—121no—
415C2×C3—121no—
416C2×C3—121no—
417C2×C3—241no—
418C2×C3—361no—
419C2×C3D3×#21711yescharacteristic
420D3—121no—
421D3—361no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—181no—
426C22—361no—
427C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
428C22#8867612yes—
429C4—21no—
430C4—91no—
431C4—91no—
432C4—92no—
433C4—181no—
434C4—362no—
435C4—362no—
436C3—31no—
437C3—31no—
438C3—61no—
439C3#1059152811yescharacteristic
440C2—22no—
441C2—21no—
442C2—41no—
443C2—361no—
444C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC2×#3830⋊C3C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#410
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721296C2semidirect
3. #11720580C2semidirect
4. #63661⋊C3C2semidirect
5. #11720333C2semidirect
9. C2×#638⋊C3C4semidirect
11. C22×(C3⋊Dic3)⋊C3C4semidirect
15. C2×C4×(C3⋊D3)⋊C3C4semidirect
16. C2×#634⋊C3C4semidirect
25. #63661D3semidirect
57. C2×#638Dic3semidirect
59. C22×C3⋊Dic3Dic3semidirect
63. C2×C4×C3⋊D3Dic3semidirect
64. C2×#634Dic3semidirect
242. H(3,𝔽3)#410semidirect
382. C32#4740semidirect
383. C32#5240semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
#117212962
#117205802
#63661⋊C32
#117203332
#105915293
#105915283