Finite Group Databasebeta

← #13150705

Subgroups — #13150705

Order 1728.

1391 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 151 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, complemented
2C2×#88672C211yesmaximal, characteristic, complemented
3#64138⋊C3C212yesmaximal, complemented
4#11721323C212yesmaximal, complemented
5#11721298C211yesmaximal, characteristic, complemented
6#11721142C212yesmaximal, complemented
7#11721124C214yesmaximal, complemented
8#11721096C212yesmaximal, complemented
9#10594896—32nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C2×C4×#272—24no—
14C2×C4×#272C2212yes—
15C2×C4×#272C2212yes—
16C2×#638⋊C3C2212yes—
17C2×#6200C2214yes—
18C2×#6198C2211yescharacteristic, complemented
19#88672—24no—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
21C2×#6196C2211yescharacteristic, complemented
22#3757⋊C3C2212yes—
23#3752⋊C3C2214yescomplemented
24#3750⋊C3C2214yescomplemented
25#88451C2214yes—
26#88449C2212yescomplemented
27#88447C2212yescomplemented
28C22×C3×C4×D3—32no—
29C22×C3×C4×D3—61no—
30C2×#3804—32no—
31#64138D312yescomplemented
32C3×#929—32no—
33#63977—32no—
34#63976—32no—
35#63955—32no—
36#63906—34no—
37#63904—34no—
38#63879—34no—
39#63851—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272—22no—
42C22×#272—24no—
43C22×#272C2312yes—
44C22×(C3⋊D3)⋊C3C2312yes—
45C2×C4×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×#1084D414yescomplemented
48C2×#1084C2311yescharacteristic
49C2×#1084C2311yescharacteristic
50C4×#272—24no—
51C4×#272—24no—
52#638⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2314yes—
55#6200—28no—
56#6198—22no—
57((C3⋊D3)⋊C3)⋊C4—24no—
58#6196—22no—
59#5791—91no—
60#5656—92no—
61C23×C3×D3—32no—
62C23×C3×D3—61no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D312yes—
65C2×#638C2×D312yes—
66C2×C3×#67—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C2×C3×C4×D3—34no—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—68no—
72C2×C3×C4×D3—62no—
73C2×C3×C4×D3—62no—
74C2×C3×C4×D3—68no—
75C2×#627—32no—
76C2×#626—32no—
77C2×#626—34no—
78C2×#625—34no—
79C2×#624—32no—
80#3804—68no—
81C2×D3×Dic3—32no—
82#3757C2×D312yes—
83#3752C2×D314yescomplemented
84#3750C2×D314yescomplemented
85C3×#217—32no—
86C3×(C2×D3)⋊C4—34no—
87C3×Dic3⋊C4—34no—
88Dic3⋊Dic3—34no—
89(C2×D3)⋊Dic3—34no—
90(C2×D3)⋊Dic3—34no—
91Dic32—34no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)C2×D412yes—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×#272—28no—
96C2×#272—22no—
97C2×#272—22no—
98C2×#272#4114yescomplemented
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—24no—
101C4×H(3,𝔽3)—22no—
102#1084C2×D412yes—
103#1084C2×D412yes—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—24no—
106C23×C3×C4—91no—
107C2×#237—92no—
108C2×#235—91no—
109C22×C4×D3—32no—
110C22×C4×D3—61no—
111C3×#123—92no—
112#953—92no—
113#937—92no—
114#930—92no—
115#929—32no—
116#929—92no—
117C4×#67—94no—
118#926—91no—
119#924—92no—
120#896—94no—
121#893—94no—
122#884—94no—
123#882—92no—
124#67⋊C4—94no—
125#877—92no—
126C23×C32—21no—
127C23×C32C22×D312yes—
128C22×C3⋊D3C22×D312yes—
129C22×C3×D3—34no—
130C22×C3×D3—32no—
131C22×C3×D3—62no—
132C22×C3×D3—68no—
133C22×C3×D3—68no—
134C22×C3×D3—64no—
135C22×C3×D3—64no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—22no—
139C2×C32×C4C22×D314yes—
140#638—24no—
141C2×C3⋊Dic3C22×D314yes—
142C2×C3⋊Dic3C22×D312yes—
143C3×#67—64no—
144C2×C3×Dic3—38no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C3×C4×D3—68no—
153C3×C4×D3—68no—
154C3×C4×D3—68no—
155C3×C4×D3—68no—
156#627—64no—
157#626—64no—
158#626—68no—
159#625—68no—
160#624—64no—
161D3×Dic3—64no—
162#525—271noSylow
163C2×H(3,𝔽3)—22no—
164C2×H(3,𝔽3)C22×D411yescharacteristic
165C2×H(3,𝔽3)C2×#4112yes—
166#272—24no—
167(C3⋊D3)⋊C3—42no—
168C24×C3—91no—
169C23×D3—32no—
170C23×D3—61no—
171C2×C3×D4—92no—
172C22×C3×C4—32no—
173C22×C3×C4—61no—
174C22×C3×C4—91no—
175C22×C3×C4—92no—
176C22×C3×C4—92no—
177C22×C3×C4—184no—
178C22×C3×C4D3211yescharacteristic
179C2×#67—32no—
180C2×#67—92no—
181C2×#67—94no—
182C2×#67—92no—
183C2×#67—94no—
184C22×Dic3—32no—
185C22×Dic3—61no—
186C22×Dic3—92no—
187#237—188no—
188#235—184no—
189C2×D12—91no—
190C2×D12—94no—
191C2×C4×D3—34no—
192C2×C4×D3—34no—
193C2×C4×D3—62no—
194C2×C4×D3—62no—
195C2×C4×D3—68no—
196C2×C4×D3—68no—
197C2×C4×D3—92no—
198C2×C4×D3—92no—
199C2×Dic6—91no—
200C2×Dic6—92no—
201C3×C4⋊C4—94no—
202C3×C22⋊C4—94no—
203C3×C22⋊C4—92no—
204#217—32no—
205#217—92no—
206#217—94no—
207(C2×D3)⋊C4—34no—
208(C2×D3)⋊C4—94no—
209(C2×D3)⋊C4—94no—
210(C2×D3)⋊C4—94no—
211C4⋊Dic3—92no—
212Dic3⋊C4—34no—
213Dic3⋊C4—94no—
214Dic3⋊C4—94no—
215C4×Dic3—92no—
216C4×Dic3—94no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32D3×D414yes—
222C22×C32C23×D312yes—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—24no—
225C2×C3×D3—38no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—616no—
229C2×C3×D3—616no—
230C2×C3×D3—64no—
231C2×C3×D3—64no—
232C2×C3×D3—64no—
233C2×C3×D3—62no—
234C2×C3×D3—64no—
235C32×C4—24no—
236C32×C4—24no—
237C3⋊Dic3—22no—
238C3⋊Dic3—24no—
239C3×Dic3—34no—
240C3×Dic3—34no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C3×Dic3—64no—
245C2×#41—271no—
246C23×C4—91no—
247#123—92no—
248C22⋊Q8—272no—
249#121—272no—
250#120—272no—
251C4×D4—274no—
252#117—271no—
253H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
254C23×C3—32no—
255C23×C3—61no—
256C23×C3—92no—
257C23×C3—92no—
258C23×C3—184no—
259C23×C3—182no—
260C23×C3C2×D3211yescharacteristic
261C22×D3—34no—
262C22×D3—32no—
263C22×D3—62no—
264C22×D3—68no—
265C22×D3—64no—
266C22×D3—64no—
267C22×D3—68no—
268C22×D3—92no—
269C3×D4—184no—
270C2×C3×C4—22no—
271C2×C3×C4—34no—
272C2×C3×C4—64no—
273C2×C3×C4—64no—
274C2×C3×C4—62no—
275C2×C3×C4—91no—
276C2×C3×C4—91no—
277C2×C3×C4—94no—
278C2×C3×C4—94no—
279C2×C3×C4—92no—
280C2×C3×C4—184no—
281C2×C3×C4—184no—
282C2×C3×C4C2×D3212yes—
283#67—64no—
284#67—184no—
285#67—184no—
286#67—188no—
287#67—188no—
288C2×Dic3—32no—
289C2×Dic3—32no—
290C2×Dic3—38no—
291C2×Dic3—34no—
292C2×Dic3—32no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—64no—
296C2×Dic3—94no—
297C2×Dic3—92no—
298C2×Dic3—184no—
299D12—182no—
300D12—188no—
301C4×D3—68no—
302C4×D3—68no—
303C4×D3—68no—
304C4×D3—68no—
305C4×D3—184no—
306C4×D3—184no—
307Dic6—184no—
308Dic6—182no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32C2×D3×D412yes—
314C2×C32D3×#4114yes—
315C3⋊D3—42no—
316C3×D3—68no—
317C3×D3—68no—
318C3×D3—122no—
319C24—91no—
320#41—544no—
321C2×Q8—271no—
322C2×D4—92no—
323C2×D4—274no—
324C2×D4—271no—
325C22×C4—91no—
326C22×C4—92no—
327C22×C4—92no—
328C22×C4—184no—
329C22×C4—272no—
330C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C4⋊C4—94no—
332C4⋊C4—272no—
333C22⋊C4—92no—
334C22⋊C4—94no—
335C22⋊C4—274no—
336C42—272no—
337C22×C3—22no—
338C22×C3—32no—
339C22×C3—34no—
340C22×C3—64no—
341C22×C3—64no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—94no—
345C22×C3—188no—
346C22×C3—182no—
347C22×C3—184no—
348C22×C3—182no—
349C22×C3—182no—
350C22×C3#381112yes—
351C22×C3C22×D3211yescharacteristic
352C2×D3—38no—
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—64no—
357C2×D3—616no—
358C2×D3—616no—
359C2×D3—64no—
360C2×D3—64no—
361C2×D3—62no—
362C2×D3—182no—
363C2×D3—184no—
364C3×C4—22no—
365C3×C4—64no—
366C3×C4—64no—
367C3×C4—92no—
368C3×C4—92no—
369C3×C4—182no—
370C3×C4—184no—
371Dic3—34no—
372Dic3—34no—
373Dic3—62no—
374Dic3—62no—
375Dic3—64no—
376Dic3—62no—
377Dic3—182no—
378Dic3—184no—
379C32—21no—
380C32#565612yescomplemented
381C23—92no—
382C23—92no—
383C23—184no—
384C23—182no—
385C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
386Q8—542no—
387D4—184no—
388D4—542no—
389D4—548no—
390C2×C4—22no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—94no—
394C2×C4—94no—
395C2×C4—92no—
396C2×C4—184no—
397C2×C4—184no—
398C2×C4—544no—
399C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
400C2×C3—22no—
401C2×C3—32no—
402C2×C3—34no—
403C2×C3—64no—
404C2×C3—64no—
405C2×C3—61no—
406C2×C3—62no—
407C2×C3—184no—
408C2×C3—362no—
409C2×C3#6430612yes—
410C2×C3C2×#381111yescharacteristic
411D3—68no—
412D3—68no—
413D3—122no—
414D3—362no—
415C22—22no—
416C22—94no—
417C22—188no—
418C22—182no—
419C22—182no—
420C22—182no—
421C22—184no—
422C22#8867712yes—
423C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C4—22no—
425C4—92no—
426C4—92no—
427C4—182no—
428C4—184no—
429C3—32no—
430C3—61no—
431C3#1059492011yescharacteristic
432C2—22no—
433C2—184no—
434C2—362no—
435C2#1172249112yes—
436C2C2×#8867711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#525
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
1. C22×C4×#272C2semidirect
2. C2×#88672C2semidirect
3. #64138⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721298C2semidirect
6. #11721142C2semidirect
7. #11721124C2semidirect
8. #11721096C2semidirect
18. C2×#6198C22semidirect
20. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
21. C2×#6196C22semidirect
23. #3752⋊C3C22semidirect
24. #3750⋊C3C22semidirect
26. #88449C22semidirect
27. #88447C22semidirect
31. #64138D3semidirect
47. C2×#1084D4semidirect
83. #3752C2×D3semidirect
84. #3750C2×D3semidirect
98. C2×#272#41semidirect
253. H(3,𝔽3)#525semidirect
380. C32#5656semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886722
#64138⋊C32
#117213232
#117212982
#117211422
#117211242
#117210962
#105948963