Finite Group Databasebeta

← #13150711

Subgroups — #13150711

Order 1728.

1193 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 159 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×#88672C212yesmaximal, complemented
2C2×C4×#1084C211yesmaximal, characteristic, complemented
3#64139⋊C3C212yesmaximal, complemented
4#11721318C212yesmaximal, complemented
5#11721145C212yesmaximal, complemented
6#11721144C211yesmaximal, characteristic, complemented
7#11721138C212yesmaximal, complemented
8#11721136C212yesmaximal, complemented
9#11721117C211yesmaximal, characteristic, complemented
10#10594910—32nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×#638⋊C3C2214yes—
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×#6198C2212yes—
17C2×#6198C2212yes—
18#88672—28no—
19C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
20C2×#6196C2212yes—
21C4×#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2214yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27#88450C2214yes—
28#88449C2212yes—
29C2×#3804—34no—
30#64139D312yescomplemented
31C3×#930—32no—
32C2×C3×C4×Dic3—32no—
33C2×C3×C4×Dic3—61no—
34#63978—34no—
35#63918—32no—
36#63911—32no—
37#63894—32no—
38Dic3×D12—32no—
39#63880—34no—
40#63852—34no—
41C23×H(3,𝔽3)C2311yescharacteristic
42C22×(C3⋊D3)⋊C3C2314yes—
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084—22no—
47C2×#1084D414yescomplemented
48C2×#1084C2311yescharacteristic
49C2×#1084C2311yescharacteristic
50C2×#1084C2311yescharacteristic
51C2×#1084C2311yescharacteristic
52#638⋊C3—28no—
53C2×(C3⋊Dic3)⋊C3C2314yes—
54C4⋊((C3⋊D3)⋊C3)—24no—
55#6198—24no—
56#6198—24no—
57((C3⋊D3)⋊C3)⋊C4—28no—
58#6196—24no—
59#5596—92no—
60#5477—91no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D312yes—
63C2×#638C2×D314yes—
64C2×C4⋊(C3⋊D3)C2×D312yes—
65C2×C3×#67—34no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3—61no—
70C2×C3×D12—32no—
71C2×#627—34no—
72C2×#626—34no—
73C2×#626—34no—
74C2×#624—34no—
75#3804—616no—
76C2×D3×Dic3—34no—
77#3752C2×D314yes—
78C4⋊(C3⋊Dic3)C2×D312yes—
79C3×(C2×D3)⋊C4—34no—
80C3×C4⋊Dic3—32no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—32no—
84C3×C4×Dic3—32no—
85C3×C4×Dic3—61no—
86C3×C4×Dic3—61no—
87C3×C4×Dic3—61no—
88C3×C4×Dic3—61no—
89Dic3⋊Dic3—34no—
90#3722—34no—
91(C2×D3)⋊Dic3—34no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)—21no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—24no—
98C2×(C3⋊D3)⋊C3—24no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)#4112yescomplemented
101#1084—21no—
102#1084C2×D412yes—
103#1084C2×D412yes—
104#1084#4112yescomplemented
105(C3⋊Dic3)⋊C3—24no—
106C2×#237—94no—
107C2×#235—92no—
108C3×#121—92no—
109C2×C3×C42—91no—
110#940—92no—
111#939—94no—
112D4×Dic3—92no—
113#930—32no—
114#930—92no—
115#930—92no—
116C2×C4×Dic3—32no—
117C2×C4×Dic3—61no—
118#893—92no—
119#888—92no—
120#885—94no—
121#67⋊C4—94no—
122#875—92no—
123#874—91no—
124C4×D12—92no—
125C4×D12—92no—
126#869—91no—
127C23×C32—21no—
128C23×C32C22×D312yes—
129C22×C3⋊D3C22×D314yes—
130C22×C3×D3—34no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D312yes—
136C2×C32×C4C22×D312yes—
137#638—28no—
138C2×C3⋊Dic3C22×D314yes—
139C4⋊(C3⋊D3)—24no—
140C3×#67—68no—
141C2×C3×Dic3—38no—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C3×D12—64no—
155#627—68no—
156#626—68no—
157#626—68no—
158#624—68no—
159D3×Dic3—68no—
160#532—271noSylow
161C2×H(3,𝔽3)—22no—
162C2×H(3,𝔽3)C22×D411yescharacteristic
163C2×H(3,𝔽3)C2×#4111yescharacteristic
164C2×H(3,𝔽3)C2×#4111yescharacteristic
165(C3⋊D3)⋊C3—44no—
166C2×C3×D4—92no—
167C2×C3×D4—94no—
168C22×C3×C4—32no—
169C22×C3×C4—61no—
170C22×C3×C4—91no—
171C22×C3×C4—91no—
172C22×C3×C4D3211yescharacteristic
173C2×#67—34no—
174C2×#67—94no—
175C2×#67—94no—
176C2×#67—94no—
177C22×Dic3—32no—
178C22×Dic3—32no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181C22×Dic3—94no—
182#237—1816no—
183#235—188no—
184C2×D12—32no—
185C2×D12—92no—
186C2×D12—92no—
187C2×D12—92no—
188C2×C4×D3—94no—
189C2×C4×D3—94no—
190C2×Dic6—92no—
191C2×Dic6—94no—
192C3×C4⋊C4—92no—
193C3×C22⋊C4—94no—
194C3×C42—91no—
195C3×C42—91no—
196C3×C42—91no—
197C3×C42—91no—
198#217—94no—
199(C2×D3)⋊C4—34no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—94no—
202(C2×D3)⋊C4—94no—
203C4⋊Dic3—32no—
204C4⋊Dic3—92no—
205C4⋊Dic3—92no—
206Dic3⋊C4—94no—
207C4×Dic3—32no—
208C4×Dic3—32no—
209C4×Dic3—32no—
210C4×Dic3—32no—
211C4×Dic3—61no—
212C4×Dic3—61no—
213C4×Dic3—61no—
214C4×Dic3—61no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32D3×D414yes—
222C22×C32C23×D312yes—
223C2×C3⋊D3—24no—
224C2×C3⋊D3—24no—
225C2×C3⋊D3—24no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4#23714yescomplemented
233C3⋊Dic3—24no—
234C3×Dic3—34no—
235C3×Dic3—34no—
236C3×Dic3—34no—
237C3×Dic3—62no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—64no—
242C3×Dic3—62no—
243C2×#41—272no—
244C4⋊Q8—271no—
245#127—271no—
246#124—272no—
247#121—92no—
248#121—272no—
249C4×D4—272no—
250C4×D4—272no—
251C2×C42—91no—
252H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
253C23×C3—32no—
254C23×C3—61no—
255C23×C3—94no—
256C23×C3C2×D3211yescharacteristic
257C22×D3—34no—
258C22×D3—94no—
259C3×D4—184no—
260C3×D4—188no—
261C2×C3×C4—22no—
262C2×C3×C4—32no—
263C2×C3×C4—32no—
264C2×C3×C4—64no—
265C2×C3×C4—64no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—94no—
272C2×C3×C4—94no—
273C2×C3×C4—91no—
274C2×C3×C4—182no—
275C2×C3×C4C2×D3211yescharacteristic
276C2×C3×C4C2×D3211yescharacteristic
277#67—68no—
278#67—188no—
279#67—188no—
280#67—188no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—38no—
285C2×Dic3—32no—
286C2×Dic3—34no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—64no—
292C2×Dic3—64no—
293C2×Dic3—64no—
294C2×Dic3—94no—
295C2×Dic3—188no—
296D12—64no—
297D12—184no—
298D12—184no—
299D12—184no—
300C4×D3—188no—
301C4×D3—188no—
302Dic6—188no—
303Dic6—184no—
304C2×C32—24no—
305C2×C32—24no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32C2×D3×D412yes—
310C2×C32C2×#23712yes—
311C2×C32D3×#4112yes—
312C3⋊D3—44no—
313C3×D3—124no—
314#41—548no—
315C2×Q8—272no—
316C2×D4—94no—
317C2×D4—92no—
318C2×D4—272no—
319C2×D4—272no—
320C22×C4—91no—
321C22×C4—91no—
322C22×C4—274no—
323C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C4⋊C4—92no—
325C4⋊C4—272no—
326C22⋊C4—94no—
327C22⋊C4—274no—
328C42—91no—
329C42—91no—
330C42—91no—
331C42—91no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—32no—
335C22×C3—34no—
336C22×C3—62no—
337C22×C3—62no—
338C22×C3—62no—
339C22×C3—62no—
340C22×C3—61no—
341C22×C3—62no—
342C22×C3—184no—
343C22×C3—184no—
344C22×C3—184no—
345C22×C3#380212yes—
346C22×C3C22×D3211yescharacteristic
347C2×D3—64no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—184no—
351C2×D3—184no—
352C2×D3—184no—
353C3×C4—21no—
354C3×C4—34no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—62no—
358C3×C4—92no—
359C3×C4—92no—
360C3×C4—92no—
361C3×C4—184no—
362C3×C4—181no—
363C3×C4#380512yes—
364Dic3—34no—
365Dic3—34no—
366Dic3—34no—
367Dic3—62no—
368Dic3—62no—
369Dic3—62no—
370Dic3—62no—
371Dic3—62no—
372Dic3—64no—
373Dic3—184no—
374C32—21no—
375C32#559612yescomplemented
376C23—94no—
377C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378Q8—544no—
379D4—188no—
380D4—184no—
381D4—544no—
382D4—544no—
383C2×C4—22no—
384C2×C4—91no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—94no—
388C2×C4—91no—
389C2×C4—94no—
390C2×C4—182no—
391C2×C4—548no—
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C2×C3—22no—
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—32no—
398C2×C3—64no—
399C2×C3—64no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—364no—
404C2×C3C2×#380211yescharacteristic
405C2×C3#6430611yescharacteristic
406C2×C3C2×#380511yescharacteristic
407D3—124no—
408D3—364no—
409C22—21no—
410C22—21no—
411C22—184no—
412C22—184no—
413C22—184no—
414C22#8865512yes—
415C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C4—21no—
417C4—92no—
418C4—92no—
419C4—92no—
420C4—184no—
421C4—181no—
422C4#8867212yescomplemented
423C3—32no—
424C3—61no—
425C3#1059492911yescharacteristic
426C2—22no—
427C2—364no—
428C2C2×#8865511yescharacteristic
429C2#1172249111yescharacteristic
430C2C2×#8867211yescharacteristic
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)#532
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. C2×#88672C2semidirect
2. C2×C4×#1084C2semidirect
3. #64139⋊C3C2semidirect
4. #11721318C2semidirect
5. #11721145C2semidirect
6. #11721144C2semidirect
7. #11721138C2semidirect
8. #11721136C2semidirect
9. #11721117C2semidirect
30. #64139D3semidirect
47. C2×#1084D4semidirect
100. C4×H(3,𝔽3)#41semidirect
104. #1084#41semidirect
232. C32×C4#237semidirect
252. H(3,𝔽3)#532semidirect
375. C32#5596semidirect
422. C4#88672semidirect

Maximal subgroups

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

subgroupindex
C2×#886722
C2×C4×#10842
#64139⋊C32
#117213182
#117211452
#117211442
#117211382
#117211362
#117211172
#105949103