Finite Group Databasebeta

← #13148721

Subgroups — #13148721

Order 1728.

1329 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1329 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88677C211yesmaximal, characteristic
2C22×#6199C211yesmaximal, characteristic, complemented
3C22⋊C4×#272C211yesmaximal, characteristic, complemented
4#64095⋊C3C212yesmaximal, complemented
5#11721306C212yesmaximal, complemented
6#11721161C212yesmaximal, complemented
7#11721152C212yesmaximal, complemented
8#11721102C214yesmaximal, complemented
9D3×#886—32nomaximal
10C23×#272C2211yescharacteristic
11C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
12C2×C4×#272C2212yes—
13C2×#638⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2212yescomplemented
15#88677—24no—
16C2×#6200C2212yes—
17C2×#6199—24no—
18C2×#6199C2212yes—
19C2×#6199C2211yescharacteristic
20C2×#6199C2212yescomplemented
21C22⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
22#88552C2211yescharacteristic, complemented
23#88542C2212yescomplemented
24#3752⋊C3C2214yescomplemented
25C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
26#88450C2214yescomplemented
27#88449C2214yescomplemented
28C2×D3×#67—32no—
29C22×D3×Dic3—32no—
30#64095D312yescomplemented
31C3×#886—32no—
32C3×D3×C22⋊C4—32no—
33C3×D3×C22⋊C4—61no—
34#63966—32no—
35#63962—32no—
36D3×(C2×D3)⋊C4—34no—
37#63891—34no—
38#63890—34no—
39D3×C4⋊Dic3—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272—24no—
42C22×#272D414yes—
43C22×#272C2311yescharacteristic
44C22×#272C2311yescharacteristic
45C22×(C3⋊D3)⋊C3C2312yes—
46C2×((C3⋊D3)⋊C3)⋊C2—22no—
47C2×((C3⋊D3)⋊C3)⋊C2—24no—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×#1084C2312yes—
50C4×#272—24no—
51#638⋊C3—24no—
52C2×(C3⋊Dic3)⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3C2314yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55#6200—24no—
56#6199—24no—
57#6199—24no—
58#6199—22no—
59D3×#123—92no—
60C2×#886—91no—
61C23×C3×D3—32no—
62C23×C3×D3—61no—
63C22×D32—32no—
64C2×#638C2×D312yes—
65C22×C3⋊Dic3C2×D312yescomplemented
66C2×C3×#67—32no—
67C22×C3×Dic3—32no—
68C2×C3×C4×D3—34no—
69C2×C3×C4×D3—62no—
70D3×#67—68no—
71C2×#626—32no—
72C2×#625—32no—
73C2×D3×Dic3—34no—
74C2×D3×Dic3—34no—
75C2×D3×Dic3—32no—
76C2×D3×Dic3—68no—
77C32×C22⋊C4—21no—
78C32×C22⋊C4C2×D312yescomplemented
79#3752C2×D314yescomplemented
80C4⋊(C3⋊Dic3)C2×D314yescomplemented
81C3×#217—32no—
82C3×#217—61no—
83C3×(C2×D3)⋊C4—34no—
84C3×(C2×D3)⋊C4—34no—
85C3×(C2×D3)⋊C4—62no—
86C3×C4⋊Dic3—34no—
87Dic3⋊Dic3—34no—
88Dic3⋊Dic3—34no—
89#3722—34no—
90(C2×D3)⋊Dic3—34no—
91C22×H(3,𝔽3)—22no—
92C22×H(3,𝔽3)C2×D412yes—
93C22×H(3,𝔽3)C2411yescharacteristic
94C2×#272—22no—
95C2×#272—28no—
96C2×#272C2×D412yes—
97C2×#272C2×D412yes—
98C2×#272#4114yes—
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—24no—
101((C3⋊D3)⋊C3)⋊C2—44no—
102C4×H(3,𝔽3)—22no—
103#1084—22no—
104(C3⋊Dic3)⋊C3—24no—
105(C3⋊Dic3)⋊C3—22no—
106C22×#67—91no—
107C23×Dic3—91no—
108C2×D3×D4—92no—
109C22×C4×D3—92no—
110C3×#123—92no—
111C2×C3×C22⋊C4—91no—
112#935—92no—
113#929—92no—
114C2×(C2×D3)⋊C4—92no—
115C2×C4⋊Dic3—92no—
116#894—94no—
117D3×C4⋊C4—94no—
118#886—32no—
119#886—92no—
120#886—94no—
121#886—92no—
122#883—94no—
123D3×C22⋊C4—32no—
124D3×C22⋊C4—61no—
125D3×C22⋊C4—94no—
126C23×C32—21no—
127C23×C32C22×D312yes—
128C22×C3⋊D3C22×D312yes—
129C22×C3×D3—32no—
130C22×C3×D3—38no—
131C22×C3×D3—32no—
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—64no—
135C22×C3×D3—61no—
136C22×C3×D3—68no—
137C22×C3×D3—124no—
138C2×D32—68no—
139C2×D32—64no—
140C2×C32×C4—22no—
141C2×C32×C4C22×D314yes—
142#638—24no—
143C2×C3⋊Dic3—24no—
144C2×C3⋊Dic3C22×D314yes—
145C2×C3⋊Dic3C22×D312yes—
146C3×#67—64no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—64no—
152C3×C4×D3—68no—
153C3×C4×D3—124no—
154#626—64no—
155#625—64no—
156D3×Dic3—68no—
157D3×Dic3—68no—
158D3×Dic3—64no—
159C2×#123—271noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)C22×D411yescharacteristic
162C2×H(3,𝔽3)C2×#4112yes—
163#272—22no—
164#272#12314yescomplemented
165(C3⋊D3)⋊C3—42no—
166C24×C3—91no—
167C23×D3—32no—
168C23×D3—61no—
169C23×D3—91no—
170C23×D3—92no—
171C2×C3×D4—92no—
172C22×C3×C4—92no—
173C22×C3×C4—92no—
174C2×#67—32no—
175C2×#67—92no—
176C2×#67—92no—
177C2×#67—92no—
178C2×#67—184no—
179C22×Dic3—32no—
180C22×Dic3—92no—
181C22×Dic3—92no—
182C22×Dic3—91no—
183C22×Dic3—92no—
184C22×Dic3—184no—
185D3×D4—188no—
186C2×D12—92no—
187C2×C4×D3—34no—
188C2×C4×D3—62no—
189C2×C4×D3—94no—
190C2×C4×D3—94no—
191C2×C4×D3—92no—
192C2×C4×D3—188no—
193C3×C4⋊C4—94no—
194C3×C22⋊C4—32no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—94no—
197C3×C22⋊C4—91no—
198C3×C22⋊C4—92no—
199C3×C22⋊C4D3211yescharacteristic
200#217—32no—
201#217—61no—
202#217—94no—
203(C2×D3)⋊C4—34no—
204(C2×D3)⋊C4—34no—
205(C2×D3)⋊C4—62no—
206(C2×D3)⋊C4—94no—
207(C2×D3)⋊C4—94no—
208(C2×D3)⋊C4—94no—
209C4⋊Dic3—34no—
210C4⋊Dic3—94no—
211C4⋊Dic3—94no—
212Dic3⋊C4—94no—
213Dic3⋊C4—94no—
214C22×C32—24no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—42no—
218C22×C32D3×D414yes—
219C22×C32C23×D312yes—
220C2×C3⋊D3—22no—
221C2×C3⋊D3—24no—
222C2×C3×D3—34no—
223C2×C3×D3—38no—
224C2×C3×D3—34no—
225C2×C3×D3—62no—
226C2×C3×D3—64no—
227C2×C3×D3—62no—
228C2×C3×D3—616no—
229C2×C3×D3—64no—
230C2×C3×D3—62no—
231C2×C3×D3—64no—
232C2×C3×D3—128no—
233C2×C3×D3—122no—
234D32—128no—
235C32×C4—24no—
236C32×C4—42no—
237C3⋊Dic3—24no—
238C3⋊Dic3—22no—
239C3×Dic3—64no—
240C3×Dic3—62no—
241C3×Dic3—64no—
242C3×Dic3—122no—
243C22×D4—271no—
244C23×C4—271no—
245#123—92no—
246#123—274no—
247#123—272no—
248C2×C4⋊C4—272no—
249C2×C22⋊C4—91no—
250C2×C22⋊C4—272no—
251H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
252C23×C3—32no—
253C23×C3—61no—
254C23×C3—92no—
255C23×C3—91no—
256C23×C3—91no—
257C23×C3—94no—
258C23×C3—184no—
259C23×C3C2×D3211yescharacteristic
260C22×D3—32no—
261C22×D3—32no—
262C22×D3—38no—
263C22×D3—32no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—64no—
267C22×D3—68no—
268C22×D3—92no—
269C22×D3—124no—
270C22×D3—184no—
271C22×D3—188no—
272C22×D3—182no—
273C22×D3—184no—
274C3×D4—184no—
275C2×C3×C4—34no—
276C2×C3×C4—62no—
277C2×C3×C4—92no—
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—184no—
287C2×Dic3—34no—
288C2×Dic3—32no—
289C2×Dic3—34no—
290C2×Dic3—62no—
291C2×Dic3—64no—
292C2×Dic3—92no—
293C2×Dic3—94no—
294C2×Dic3—184no—
295C2×Dic3—184no—
296C2×Dic3—184no—
297C2×Dic3—182no—
298D12—184no—
299C4×D3—68no—
300C4×D3—124no—
301C4×D3—188no—
302C4×D3—188no—
303C4×D3—184no—
304C2×C32—24no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—42no—
308C2×C32C2×D3×D412yes—
309C2×C32D3×#4114yes—
310C3⋊D3—42no—
311C3×D3—38no—
312C3×D3—64no—
313C3×D3—64no—
314C3×D3—122no—
315C3×D3—122no—
316C24—91no—
317C24—271no—
318C2×D4—92no—
319C2×D4—272no—
320C2×D4—544no—
321C22×C4—92no—
322C22×C4—92no—
323C22×C4—272no—
324C22×C4—271no—
325C22×C4—272no—
326C22×C4—544no—
327C4⋊C4—94no—
328C4⋊C4—274no—
329C22⋊C4—91no—
330C22⋊C4—92no—
331C22⋊C4—94no—
332C22⋊C4—274no—
333C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
334C22×C3—22no—
335C22×C3—32no—
336C22×C3—34no—
337C22×C3—64no—
338C22×C3—61no—
339C22×C3—62no—
340C22×C3—92no—
341C22×C3—94no—
342C22×C3—92no—
343C22×C3—122no—
344C22×C3—188no—
345C22×C3—182no—
346C22×C3—184no—
347C22×C3—182no—
348C22×C3#380212yes—
349C22×C3C22×D3211yescharacteristic
350C2×D3—38no—
351C2×D3—34no—
352C2×D3—34no—
353C2×D3—64no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—616no—
357C2×D3—64no—
358C2×D3—62no—
359C2×D3—64no—
360C2×D3—128no—
361C2×D3—122no—
362C2×D3—182no—
363C2×D3—184no—
364C2×D3—368no—
365C2×D3—364no—
366C3×C4—22no—
367C3×C4—64no—
368C3×C4—122no—
369C3×C4—184no—
370C3×C4—182no—
371C3×C4—182no—
372Dic3—64no—
373Dic3—64no—
374Dic3—62no—
375Dic3—122no—
376Dic3—184no—
377Dic3—182no—
378C32—21no—
379C32D3×#12312yescomplemented
380C23—91no—
381C23—91no—
382C23—94no—
383C23—92no—
384C23—184no—
385C23—542no—
386C23—544no—
387C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
388D4—184no—
389D4—544no—
390C2×C4—92no—
391C2×C4—94no—
392C2×C4—92no—
393C2×C4—184no—
394C2×C4—184no—
395C2×C4—544no—
396C2×C4—544no—
397C2×C4—542no—
398C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
399C2×C3—22no—
400C2×C3—34no—
401C2×C3—32no—
402C2×C3—64no—
403C2×C3—62no—
404C2×C3—61no—
405C2×C3—94no—
406C2×C3—122no—
407C2×C3—182no—
408C2×C3—362no—
409C2×C3C2×#380211yescharacteristic
410C2×C3#6431212yes—
411D3—38no—
412D3—64no—
413D3—64no—
414D3—122no—
415D3—122no—
416D3—362no—
417C22—22no—
418C22—94no—
419C22—92no—
420C22—92no—
421C22—188no—
422C22—182no—
423C22—184no—
424C22—182no—
425C22—1084no—
426C22#8865512yes—
427C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C4—22no—
429C4—182no—
430C4—184no—
431C4—182no—
432C3—32no—
433C3—61no—
434C3#1059429511yescharacteristic
435C2—22no—
436C2—94no—
437C2—182no—
438C2—362no—
439C2C2×#8865511yescharacteristic
440C2#1172250512yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#88677C2
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
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
2. C22×#6199C2semidirect
3. C22⋊C4×#272C2semidirect
4. #64095⋊C3C2semidirect
5. #11721306C2semidirect
6. #11721161C2semidirect
7. #11721152C2semidirect
8. #11721102C2semidirect
14. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×#6199C22semidirect
21. C22⋊C4×H(3,𝔽3)C22semidirect
22. #88552C22semidirect
23. #88542C22semidirect
24. #3752⋊C3C22semidirect
25. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
26. #88450C22semidirect
27. #88449C22semidirect
30. #64095D3semidirect
65. C22×C3⋊Dic3C2×D3semidirect
78. C32×C22⋊C4C2×D3semidirect
79. #3752C2×D3semidirect
80. C4⋊(C3⋊Dic3)C2×D3semidirect
164. #272#123semidirect
251. H(3,𝔽3)C2×#123semidirect
333. C22⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
379. C32D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#886772
C22×#61992
C22⋊C4×#2722
#64095⋊C32
#117213062
#117211612
#117211522
#117211022
D3×#8863