Finite Group Databasebeta

← #13147836

Subgroups — #13147836

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×#88650C212yesmaximal, complemented
2C42×#272C211yesmaximal, characteristic, complemented
3C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
4#11721159C212yesmaximal, complemented
5#11721145C212yesmaximal, complemented
6#11721144C211yesmaximal, characteristic, complemented
7#11721142C212yesmaximal, complemented
8C4×#6198C212yesmaximal, complemented
9#11721117C211yesmaximal, characteristic, complemented
10#10593968—32nomaximal
11C2×C4×#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×#6200C2214yes—
17C2×#6198C2212yes—
18C2×#6198C2212yes—
19C2×#6196C2212yes—
20#88650—28no—
21C42×H(3,𝔽3)C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2214yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27#88450C2214yes—
28#88449C2212yes—
29C2×#3798—34no—
30C4×C4⋊(C3⋊D3)D312yescomplemented
31C3×C4×D12—32no—
32C3×C42×D3—32no—
33C3×C42×D3—61no—
34#63918—32no—
35#63911—32no—
36#63909—32no—
37#63908—32no—
38#63906—34no—
39C4×#626—34no—
40#63852—34no—
41C22×#272C2311yescharacteristic
42C22×(C3⋊D3)⋊C3C2314yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084C2311yescharacteristic
47C2×#1084C2311yescharacteristic
48C2×#1084C2311yescharacteristic
49C4×#272—22no—
50C4×#272—22no—
51C4×#272D414yescomplemented
52C2×(C3⋊Dic3)⋊C3C2314yes—
53C4⋊((C3⋊D3)⋊C3)—24no—
54C4×(C3⋊D3)⋊C3—28no—
55#6200—28no—
56#6198—24no—
57#6198—24no—
58#6196—24no—
59#5548—92no—
60#5477—91no—
61C2×C4⋊(C3⋊D3)C2×D312yes—
62C2×C4×C3⋊D3C2×D314yes—
63C2×C3×D12—32no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—34no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×#627—34no—
72C2×#626—34no—
73C2×#626—34no—
74C2×#626—34no—
75C2×#625—34no—
76#3798—616no—
77C32×C42—21no—
78C32×C42C2×D312yes—
79#3752C2×D314yes—
80C4⋊(C3⋊Dic3)C2×D312yes—
81C3×(C2×D3)⋊C4—34no—
82C3×C4⋊Dic3—32no—
83C3×C4×Dic3—32no—
84C3×C4×Dic3—32no—
85C3×C4×Dic3—32no—
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)C2411yescharacteristic
93C2×#272—21no—
94C2×#272—21no—
95C2×#272C2×D412yes—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—24no—
98C2×(C3⋊D3)⋊C3—24no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)C2×D412yes—
101C4×H(3,𝔽3)#4112yescomplemented
102#1084—21no—
103#1084C2×D412yes—
104#1084#4112yescomplemented
105(C3⋊Dic3)⋊C3—24no—
106C2×#235—92no—
107C2×#235—94no—
108C3×C4×D4—92no—
109C2×C3×C42—91no—
110#940—92no—
111#938—92no—
112#930—92no—
113#930—92no—
114C4×#67—94no—
115#895—92no—
116#888—92no—
117#885—94no—
118#884—94no—
119#875—92no—
120#874—91no—
121C4×D12—32no—
122C4×D12—92no—
123C4×D12—92no—
124C42×D3—32no—
125C42×D3—61no—
126#869—91no—
127C22×C3⋊D3C22×D314yes—
128C22×C3×D3—32no—
129C22×C3×D3—34no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D312yes—
135C2×C32×C4C22×D312yes—
136C2×C32×C4C22×D312yes—
137C2×C3⋊Dic3C22×D314yes—
138C4⋊(C3⋊D3)—24no—
139C4×C3⋊D3—28no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—64no—
148C3×C4×D3—38no—
149C3×C4×D3—64no—
150C3×C4×D3—64no—
151C3×C4×D3—64no—
152C3×C4×D3—64no—
153C3×C4×D3—64no—
154C3×C4×D3—68no—
155#627—68no—
156#626—68no—
157#626—68no—
158#626—68no—
159#625—68no—
160#532—271noSylow
161C2×H(3,𝔽3)C22×D411yescharacteristic
162C2×H(3,𝔽3)C2×#4111yescharacteristic
163C2×H(3,𝔽3)C2×#4111yescharacteristic
164#272—22no—
165(C3⋊D3)⋊C3—44no—
166C2×C3×D4—92no—
167C22×C3×C4—94no—
168C22×C3×C4—91no—
169C22×C3×C4—91no—
170C22×C3×C4—91no—
171C2×#67—94no—
172C2×#67—94no—
173C2×#67—94no—
174C2×#67—94no—
175#235—1816no—
176#235—188no—
177C2×D12—32no—
178C2×D12—92no—
179C2×D12—92no—
180C2×D12—92no—
181C2×D12—94no—
182C2×C4×D3—32no—
183C2×C4×D3—32no—
184C2×C4×D3—32no—
185C2×C4×D3—34no—
186C2×C4×D3—61no—
187C2×C4×D3—61no—
188C2×C4×D3—61no—
189C2×C4×D3—94no—
190C2×Dic6—92no—
191C2×Dic6—94no—
192C3×C4⋊C4—92no—
193C3×C22⋊C4—94no—
194C3×C42—32no—
195C3×C42—61no—
196C3×C42—91no—
197C3×C42—91no—
198C3×C42—91no—
199C3×C42D3211yescharacteristic
200#217—94no—
201(C2×D3)⋊C4—34no—
202(C2×D3)⋊C4—94no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—94no—
205C4⋊Dic3—32no—
206C4⋊Dic3—92no—
207C4⋊Dic3—92no—
208Dic3⋊C4—94no—
209C4×Dic3—32no—
210C4×Dic3—32no—
211C4×Dic3—32no—
212C4×Dic3—61no—
213C4×Dic3—61no—
214C4×Dic3—61no—
215C22×C32—21no—
216C22×C32C23×D312yes—
217C2×C3⋊D3—24no—
218C2×C3⋊D3—24no—
219C2×C3⋊D3—24no—
220C2×C3×D3—34no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—64no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4#23514yescomplemented
234C32×C4D3×D414yes—
235C3⋊Dic3—24no—
236C3×Dic3—34no—
237C3×Dic3—34no—
238C3×Dic3—62no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—64no—
243C2×#41—272no—
244C4⋊Q8—271no—
245#127—271no—
246#124—272no—
247#121—272no—
248#121—272no—
249C4×D4—92no—
250C4×D4—272no—
251C2×C42—91no—
252H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
253C23×C3—94no—
254C23×C3—91no—
255C22×D3—32no—
256C22×D3—34no—
257C22×D3—61no—
258C22×D3—94no—
259C3×D4—184no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—32no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—94no—
269C2×C3×C4—94no—
270C2×C3×C4—91no—
271C2×C3×C4—182no—
272C2×C3×C4—188no—
273C2×C3×C4—182no—
274C2×C3×C4C2×D3211yescharacteristic
275C2×C3×C4C2×D3211yescharacteristic
276C2×C3×C4C2×D3211yescharacteristic
277#67—188no—
278#67—188no—
279#67—188no—
280#67—188no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—34no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—94no—
289D12—64no—
290D12—184no—
291D12—184no—
292D12—184no—
293D12—188no—
294C4×D3—38no—
295C4×D3—64no—
296C4×D3—64no—
297C4×D3—64no—
298C4×D3—64no—
299C4×D3—64no—
300C4×D3—68no—
301C4×D3—188no—
302Dic6—188no—
303Dic6—184no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32C2×#23512yes—
308C2×C32C2×D3×D412yes—
309C2×C32D3×#4112yes—
310C3⋊D3—44no—
311C3×D3—64no—
312C3×D3—64no—
313C3×D3—124no—
314#41—548no—
315C2×Q8—272no—
316C2×D4—92no—
317C2×D4—272no—
318C2×D4—274no—
319C2×D4—272no—
320C22×C4—91no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—94no—
324C4⋊C4—92no—
325C4⋊C4—272no—
326C22⋊C4—94no—
327C22⋊C4—274no—
328C42—91no—
329C42—91no—
330C42—91no—
331C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
332C22×C3—32no—
333C22×C3—61no—
334C22×C3—92no—
335C22×C3—184no—
336C22×C3—184no—
337C22×C3—184no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3C22×D3211yescharacteristic
341C2×D3—34no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—62no—
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—34no—
356C3×C4—62no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—92no—
361C3×C4—92no—
362C3×C4—184no—
363C3×C4—181no—
364C3×C4#379812yes—
365C3×C4#380212yes—
366Dic3—34no—
367Dic3—34no—
368Dic3—62no—
369Dic3—62no—
370Dic3—62no—
371Dic3—62no—
372Dic3—64no—
373Dic3—184no—
374C32—21no—
375C32#554812yescomplemented
376C23—91no—
377C23—94no—
378Q8—544no—
379D4—184no—
380D4—544no—
381D4—544no—
382D4—548no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—94no—
386C2×C4—91no—
387C2×C4—94no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4—188no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C2×C3—32no—
395C2×C3—32no—
396C2×C3—32no—
397C2×C3—61no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—182no—
401C2×C3—364no—
402C2×C3C2×#379811yescharacteristic
403C2×C3C2×#380211yescharacteristic
404C2×C3#6430611yescharacteristic
405D3—64no—
406D3—64no—
407D3—124no—
408D3—364no—
409C22—92no—
410C22—181no—
411C22—181no—
412C22—184no—
413C22—184no—
414C22—184no—
415C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C4—21no—
417C4—92no—
418C4—92no—
419C4—181no—
420C4—184no—
421C4#8865012yescomplemented
422C4#8865512yes—
423C3—32no—
424C3—61no—
425C3#1059397211yescharacteristic
426C2—182no—
427C2—364no—
428C2C2×#8865011yescharacteristic
429C2C2×#8865511yescharacteristic
430C2#1172249111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88650C2semidirect
2. C42×#272C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #11721159C2semidirect
5. #11721145C2semidirect
6. #11721144C2semidirect
7. #11721142C2semidirect
8. C4×#6198C2semidirect
9. #11721117C2semidirect
30. C4×C4⋊(C3⋊D3)D3semidirect
51. C4×#272D4semidirect
101. C4×H(3,𝔽3)#41semidirect
104. #1084#41semidirect
233. C32×C4#235semidirect
252. H(3,𝔽3)#532semidirect
375. C32#5548semidirect
421. C4#88650semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C42×#2722
C4×C4⋊((C3⋊D3)⋊C3)2
#117211592
#117211452
#117211442
#117211422
C4×#61982
#117211172
#105939683