Finite Group Databasebeta

← #13159574

Subgroups — #13159574

Order 1728.

988 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 170 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×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723452C211yesmaximal, characteristic
3C2×#88788C211yesmaximal, characteristic
4C3×#63641C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592538—31nomaximal
9#10590701—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C8×C3⋊(C3⋊D3)—22no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806C411yescharacteristic
19#88806C411yescharacteristic
20#88804C411yescharacteristic
21#88804C411yescharacteristic
22#88788—22no—
23C2×#3743—31no—
24C2×C8×C3⋊D3—31no—
25C2×C8×C3⋊D3—34no—
26C2×C8×C3⋊D3—64no—
27#64081—31no—
28#63874—34no—
29C2×#3712—34no—
30C32×C4⋊C8—31no—
31#63641—31no—
32#63641D311yescharacteristic, complemented
33C4⋊(C3⋊(C3⋊C8))—31no—
34C3×Dic3⋊C8—34no—
35C3×Dic3⋊C8D314yescomplemented
36Dic3⋊(C3⋊C8)—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C4×C3⋊(C3⋊D3)D412yes—
42C4×C3⋊(C3⋊D3)Q812yes—
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C32×Dic3C2×C411yescharacteristic
46C2×C32×Dic3C2×C411yescharacteristic
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#4839—94no—
52#4839—91no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—34no—
55C2×C4×C3⋊D3—64no—
56C2×C32×C8—24no—
57C2×C32×C8—31no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60C32×C42—31no—
61#3752—31no—
62#3752—31no—
63#3750—31no—
64#3750—31no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70#3743—62no—
71C8×C3⋊D3—62no—
72C8×C3⋊D3—68no—
73C8×C3⋊D3—616no—
74C3×C4×Dic3—34no—
75C3×C4×Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—34no—
79#3724—34no—
80#3722—34no—
81#3722—34no—
82#3712—68no—
83C22×C33C22×C411yescharacteristic
84C2×C3⋊(C3⋊D3)—21no—
85C2×C3⋊(C3⋊D3)—21no—
86C2×C3⋊(C3⋊D3)C4⋊C412yes—
87C33×C4C2×D411yescharacteristic
88C33×C4C2×Q811yescharacteristic
89C3⋊(C3⋊Dic3)C4⋊C412yes—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—21no—
92C32×Dic3—21no—
93C32×Dic3—21no—
94C2×ΓL(1,𝔽25)—94no—
95C2×ΓL(1,𝔽25)—91no—
96C2×D3×C8—94no—
97C2×D3×C8—91no—
98C2×D3×C8—184no—
99#872—94no—
100#872—91no—
101C3×C4⋊C8—34no—
102C3×C4⋊C8—91no—
103Dic3⋊C8—34no—
104Dic3⋊C8—94no—
105Dic3⋊C8—91no—
106Dic3⋊C8C3⋊D311yescharacteristic, complemented
107C4⋊(C3⋊C8)—94no—
108C4⋊(C3⋊C8)—91no—
109C22×C3⋊D3—31no—
110C22×C3⋊D3—34no—
111C22×C3⋊D3—64no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3C4×D311yescharacteristic
121C2×C3⋊Dic3C4×D311yescharacteristic
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—38no—
125C4×C3⋊D3—38no—
126C4×C3⋊D3—68no—
127C4×C3⋊D3—68no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3C4×D314yes—
131C2×C3×Dic3C4×D314yes—
132C32×C8—28no—
133C32×C8—24no—
134C32×C8—21no—
135C32×C8—61no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—68no—
138C3⋊(C3⋊C8)—64no—
139C3⋊(C3⋊C8)—61no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—64no—
142#424—271noSylow
143C2×C33C2×C4⋊C411yescharacteristic
144C2×C33#13111yescharacteristic
145C2×C33#13111yescharacteristic
146C3⋊(C3⋊D3)—22no—
147C2×C4×D3—94no—
148C2×C4×D3—91no—
149C2×C4×D3—184no—
150C2×C3×C8—24no—
151C2×C3×C8—34no—
152C2×C3×C8—91no—
153C2×C3×C8D3214yes—
154C2×C3×C8C2×C3⋊D311yescharacteristic
155C3×C42—34no—
156C3×C42—91no—
157(C2×D3)⋊C4—94no—
158(C2×D3)⋊C4—94no—
159(C2×D3)⋊C4—91no—
160(C2×D3)⋊C4—91no—
161Dic3⋊C4—94no—
162Dic3⋊C4—94no—
163Dic3⋊C4—91no—
164Dic3⋊C4—91no—
165C4×Dic3—34no—
166C4×Dic3C2×C3⋊D311yescharacteristic
167C2×C3⋊C8—34no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171C2×C3⋊C8C2×C3⋊D311yescharacteristic
172ΓL(1,𝔽25)—182no—
173ΓL(1,𝔽25)—188no—
174D3×C8—188no—
175D3×C8—182no—
176D3×C8—1816no—
177C22×C32—24no—
178C22×C32C2×C4×D314yes—
179C22×C32C2×C4×D311yescharacteristic
180C2×C3⋊D3—32no—
181C2×C3⋊D3—38no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—68no—
184C2×C3⋊D3—68no—
185C2×C3⋊D3—64no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—64no—
188C2×C3⋊D3—61no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—61no—
192C32×C4—61no—
193C32×C4D3×D414yes—
194C32×C4D3×D411yescharacteristic
195C32×C4D3×Q814yes—
196C32×C4D3×Q811yescharacteristic
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—32no—
200C3⋊Dic3—38no—
201C3⋊Dic3—68no—
202C3×Dic3—24no—
203C3×Dic3—24no—
204C3×Dic3—64no—
205C3×Dic3—64no—
206C2×M16—271no—
207C22×C8—271no—
208#117—271no—
209C4⋊C8—31no—
210C4⋊C8—91no—
211C4⋊C8—271no—
212C4⋊C8—271no—
213C33#42411yescharacteristic, complemented, Sylow
214C22×D3—94no—
215C22×D3—91no—
216C22×D3—184no—
217C2×C3×C4—24no—
218C2×C3×C4—34no—
219C2×C3×C4—34no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4C2×D3214yes—
223C2×C3×C4C22×C3⋊D311yescharacteristic
224C2×Dic3—34no—
225C2×Dic3—34no—
226C2×Dic3—94no—
227C2×Dic3—91no—
228C2×Dic3—184no—
229C2×Dic3C4×C3⋊D311yescharacteristic
230C2×Dic3C4×C3⋊D311yescharacteristic
231C4×D3—98no—
232C4×D3—98no—
233C4×D3—92no—
234C4×D3—92no—
235C4×D3—188no—
236C4×D3—188no—
237C3×C8—28no—
238C3×C8—24no—
239C3×C8—21no—
240C3×C8—64no—
241C3×C8—181no—
242C3⋊C8—21no—
243C3⋊C8—64no—
244C3⋊C8—188no—
245C3⋊C8—184no—
246C3⋊C8—181no—
247C2×C32—24no—
248C2×C32—24no—
249C2×C32—24no—
250C2×C32D3×C4⋊C414yes—
251C2×C32D3×C4⋊C411yescharacteristic
252C2×C32#90114yes—
253C2×C32#90111yescharacteristic
254C2×C32#90714yes—
255C2×C32#90711yescharacteristic
256C3⋊D3—616no—
257C3⋊D3—68no—
258C3⋊D3—62no—
259C22×C4—271no—
260M16—542no—
261C2×C8—31no—
262C2×C8—91no—
263C2×C8—271no—
264C2×C8—542no—
265C2×C8D3×C3⋊D311yescharacteristic
266C4⋊C4—271no—
267C4⋊C4—271no—
268C22⋊C4—271no—
269C22⋊C4—271no—
270C42—31no—
271C42—91no—
272C22×C3—24no—
273C22×C3C4×D3214yes—
274C22×C3C2×C4×C3⋊D311yescharacteristic
275C2×D3—92no—
276C2×D3—98no—
277C2×D3—188no—
278C2×D3—188no—
279C2×D3—188no—
280C2×D3—184no—
281C2×D3—184no—
282C2×D3—181no—
283C2×D3—181no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—64no—
287C3×C4—64no—
288C3×C4—181no—
289C3×C4—181no—
290C3×C4#379714yes—
291C3×C4#381114yes—
292C3×C4D4×C3⋊D311yescharacteristic
293C3×C4Q8×C3⋊D311yescharacteristic
294Dic3—21no—
295Dic3—21no—
296Dic3—64no—
297Dic3—64no—
298Dic3—92no—
299Dic3—98no—
300Dic3—188no—
301C32—24no—
302C32#483914yescomplemented
303C32#483911yescharacteristic, complemented
304C23—271no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—271no—
310C2×C4—272no—
311C2×C4—272no—
312C2×C4C2×D3×C3⋊D311yescharacteristic
313C8—21no—
314C8—61no—
315C8—181no—
316C8—541no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3#6380314yes—
321C2×C3#6380514yes—
322C2×C3#6388314yes—
323C2×C3C4⋊C4×C3⋊D311yescharacteristic
324C2×C3#6411011yescharacteristic
325C2×C3#6411611yescharacteristic
326D3—1816no—
327D3—182no—
328D3—188no—
329C22—272no—
330C22—541no—
331C22—541no—
332C22C4×D3×C3⋊D311yescharacteristic
333C4—61no—
334C4—61no—
335C4—181no—
336C4—181no—
337C4—272no—
338C4#8901911yescharacteristic
339C4#8903911yescharacteristic
340C3—24no—
341C3#1059070114yescomplemented
342C3#1059253811yescharacteristic, complemented
343C2—542no—
344C2#1172333111yescharacteristic
345C2#1172333411yescharacteristic
346C2#1172347011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63641C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. #11722051C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
32. #63641D3semidirect
35. C3×Dic3⋊C8D3semidirect
106. Dic3⋊C8C3⋊D3semidirect
213. C33#424semidirect
302. C32#4839semidirect
303. C32#4839semidirect
341. C3#10590701semidirect
342. C3#10592538semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117234522
C2×#887882
C3×#636412
C32×Dic3⋊C82
#117220512
Dic3⋊(C3⋊(C3⋊C8))2
#105925383
#105907013