Finite Group Databasebeta

← #13140030

Subgroups — #13140030

Order 1728.

715 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 121 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×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88430C212yesmaximal, complemented
3#11720573C211yesmaximal, characteristic, complemented
4#63657⋊C3C211yesmaximal, characteristic, complemented
5#11720317C212yesmaximal, complemented
6#10591159—31nomaximal
7#10591154—31nomaximal
8C23×(C3⋊D3)⋊C3C411yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C411yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—24no—
12C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#6180C2212yescomplemented
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88430—24no—
19C22×C4×C3⋊D3D311yescharacteristic
20C22×C3×C4×D3—31no—
21C2×#3712—32no—
22C2×#3711—32no—
23#63657D311yescharacteristic, complemented
24C3×#830—31no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63557—32no—
29(C2×D3)⋊(C3⋊C8)—32no—
30C23×H(3,𝔽3)C2×C411yescharacteristic
31C22×(C3⋊D3)⋊C3—22no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
35C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)C2×C411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C4×(C3⋊D3)⋊C3—22no—
43C4×(C3⋊D3)⋊C3—22no—
44C4×(C3⋊D3)⋊C3—22no—
45C4×(C3⋊D3)⋊C3D414yes—
46#6180—22no—
47(C3⋊(C3⋊C8))⋊C3—22no—
48#5207—91no—
49#4726—91no—
50#4726—91no—
51C23×C3⋊D3Dic311yescharacteristic
52C23×C3×D3—31no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C3⋊Dic3Dic311yescharacteristic
57C2×C4×C3⋊D3—24no—
58C2×C4×C3⋊D3Dic311yescharacteristic
59C2×C4×C3⋊D3Dic311yescharacteristic
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C22×C3×Dic3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—64no—
68C2×C3⋊(C3⋊C8)C2×D312yescomplemented
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—62no—
73#3712—64no—
74#3711—64no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)M1612yescomplemented
78C22×H(3,𝔽3)C22×C411yescharacteristic
79C2×(C3⋊D3)⋊C3—28no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3C22⋊C412yes—
85C2×(C3⋊D3)⋊C3M1612yescomplemented
86C4×H(3,𝔽3)—21no—
87C4×H(3,𝔽3)C2×D412yes—
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3C22⋊C412yes—
90C23×C3×C4—91no—
91C22×C4×D3—31no—
92C22×C4×D3—91no—
93C2×#208—92no—
94C2×ΓL(1,𝔽25)—92no—
95C2×ΓL(1,𝔽25)—92no—
96C3×C22⋊C8—91no—
97C3×C22⋊C8—91no—
98#830—31no—
99#830—31no—
100#830—31no—
101#830—61no—
102#830—91no—
103#830—92no—
104(C2×D3)⋊C8—92no—
105(C2×D3)⋊C8—92no—
106C23×C32—21no—
107C23×C32C4×D311yescharacteristic
108C23×C32C2×Dic311yescharacteristic
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3C2×Dic311yescharacteristic
113C22×C3⋊D3C2×Dic311yescharacteristic
114C22×C3×D3—31no—
115C22×C3×D3—31no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—42no—
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—22no—
129C2×C3⋊Dic3C2×Dic311yescharacteristic
130C2×C3⋊Dic3C2×Dic311yescharacteristic
131C4×C3⋊D3—22no—
132C4×C3⋊D3—22no—
133C4×C3⋊D3—22no—
134C4×C3⋊D3#6714yes—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—62no—
138C3×C4×D3—34no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3⋊(C3⋊C8)—22no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—62no—
146C3×C3⋊C8—122no—
147#407—271noSylow
148C2×H(3,𝔽3)—22no—
149C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
150C2×H(3,𝔽3)C2×M1611yescharacteristic
151C2×H(3,𝔽3)C2×M1611yescharacteristic
152(C3⋊D3)⋊C3—22no—
153(C3⋊D3)⋊C3—22no—
154C24×C3—91no—
155C23×D3—31no—
156C23×D3—91no—
157C22×C3×C4—31no—
158C22×C3×C4—31no—
159C22×C3×C4—61no—
160C22×C3×C4—91no—
161C22×C3×C4—91no—
162C22×C3×C4—91no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—184no—
166C22×C3×C4D3211yescharacteristic
167C22×Dic3—31no—
168C22×Dic3—91no—
169C2×C4×D3—31no—
170C2×C4×D3—31no—
171C2×C4×D3—31no—
172C2×C4×D3—31no—
173C2×C4×D3—64no—
174C2×C4×D3—91no—
175C2×C4×D3—91no—
176C2×C4×D3—91no—
177C2×C4×D3—91no—
178C2×C4×D3—184no—
179C2×C3×C8—92no—
180C2×C3×C8—92no—
181#208—184no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—32no—
184C2×C3⋊C8—32no—
185C2×C3⋊C8—62no—
186C2×C3⋊C8—92no—
187ΓL(1,𝔽25)—184no—
188ΓL(1,𝔽25)—184no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32ΓL(1,𝔽25)12yescomplemented
198C22×C32#20812yescomplemented
199C22×C32C2×C4×D311yescharacteristic
200C22×C32C22×Dic311yescharacteristic
201C2×C3⋊D3—28no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3#20812yescomplemented
207C2×C3⋊D3#21712yes—
208C2×C3×D3—32no—
209C2×C3×D3—32no—
210C2×C3×D3—68no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—22no—
218C32×C4—41no—
219C32×C4D3×D412yes—
220C32×C4C2×#6712yes—
221C3⋊Dic3—21no—
222C3⋊Dic3#21712yes—
223C3×Dic3—32no—
224C3×Dic3—61no—
225C23×C4—91no—
226C2×M16—272no—
227C22⋊C8—91no—
228C22⋊C8—91no—
229C22⋊C8—272no—
230H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—61no—
234C23×C3—91no—
235C23×C3—91no—
236C23×C3—182no—
237C23×C3—182no—
238C23×C3—182no—
239C23×C3D3×Dic311yescharacteristic
240C22×D3—31no—
241C22×D3—31no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—91no—
246C22×D3—91no—
247C22×D3—182no—
248C22×D3—182no—
249C22×D3—182no—
250C2×C3×C4—22no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—94no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—122no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4D3×Dic311yescharacteristic
268C2×C3×C4C2×D3211yescharacteristic
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—62no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—182no—
275C4×D3—34no—
276C4×D3—62no—
277C4×D3—62no—
278C4×D3—62no—
279C4×D3—94no—
280C4×D3—182no—
281C4×D3—182no—
282C4×D3—182no—
283C3×C8—182no—
284C3×C8—182no—
285C3⋊C8—62no—
286C3⋊C8—62no—
287C3⋊C8—62no—
288C3⋊C8—122no—
289C3⋊C8—182no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—42no—
296C2×C32D3×C22⋊C411yescharacteristic
297C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
298C2×C32D3×M1611yescharacteristic
299C2×C32C2×#20811yescharacteristic
300C2×C32C2×#20811yescharacteristic
301C2×C32C2×#21711yescharacteristic
302C3⋊D3—22no—
303C3⋊D3—22no—
304C3×D3—62no—
305C3×D3—62no—
306C24—91no—
307C22×C4—91no—
308C22×C4—91no—
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—184no—
313C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
314M16—544no—
315C2×C8—92no—
316C2×C8—92no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—32no—
322C22×C3—32no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—62no—
329C22×C3—92no—
330C22×C3—92no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—188no—
338C22×C3#371112yes—
339C22×C3C2×D3×Dic311yescharacteristic
340C2×D3—32no—
341C2×D3—32no—
342C2×D3—68no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—92no—
348C2×D3—92no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—188no—
354C3×C4—21no—
355C3×C4—32no—
356C3×C4—32no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—62no—
360C3×C4—92no—
361C3×C4—121no—
362C3×C4—181no—
363C3×C4D3×#6712yes—
364Dic3—32no—
365Dic3—61no—
366Dic3—92no—
367Dic3—181no—
368C32—21no—
369C32#472611yescharacteristic, complemented
370C32#520711yescharacteristic, complemented
371C23—91no—
372C23—91no—
373C23—182no—
374C23—182no—
375C23—182no—
376C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
377C2×C4—22no—
378C2×C4—91no—
379C2×C4—94no—
380C2×C4—91no—
381C2×C4—182no—
382C2×C4—182no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
386C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
387C8—182no—
388C8—182no—
389C2×C3—22no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—62no—
397C2×C3—62no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—122no—
402C2×C3—182no—
403C2×C3—182no—
404C2×C3D3×#20811yescharacteristic
405C2×C3C2×#371111yescharacteristic
406C2×C3D3×#21711yescharacteristic
407D3—62no—
408D3—62no—
409D3—182no—
410D3—182no—
411C22—21no—
412C22—21no—
413C22—92no—
414C22—92no—
415C22—188no—
416C22—181no—
417C22—181no—
418C22—181no—
419C22—181no—
420C22#8843012yescomplemented
421C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
422C4—21no—
423C4—92no—
424C4—181no—
425C4#8867612yes—
426C3—31no—
427C3—31no—
428C3—61no—
429C3#1059115411yescharacteristic
430C2—22no—
431C2—182no—
432C2—182no—
433C2#1172105311yescharacteristic
434C2C2×#8843011yescharacteristic
435C2#1172131011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC23×(C3⋊D3)⋊C3C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88430C2semidirect
3. #11720573C2semidirect
4. #63657⋊C3C2semidirect
5. #11720317C2semidirect
16. C2×#6180C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63657D3semidirect
68. C2×C3⋊(C3⋊C8)C2×D3semidirect
77. C22×H(3,𝔽3)M16semidirect
85. C2×(C3⋊D3)⋊C3M16semidirect
197. C22×C32ΓL(1,𝔽25)semidirect
198. C22×C32#208semidirect
206. C2×C3⋊D3#208semidirect
230. H(3,𝔽3)#407semidirect
369. C32#4726semidirect
370. C32#5207semidirect
420. C22#88430semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×#884302
#117205732
#63657⋊C32
#117203172
#105911593
#105911543