Finite Group Databasebeta

← #13167411

Subgroups — #13167411

Order 1728.

776 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 79 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×C4⋊#1113C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3#11723673C211yesmaximal, characteristic, complemented
4#11722793C214yesmaximal, complemented
5#10593363—31nomaximal
6C22×#1113—21no—
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6295C2212yes—
10C4⋊#1113—22no—
11C4⋊#1113C414yescomplemented
12C2×C3×#622C2212yescomplemented
13C3×#3788—22no—
14C3×C4×C3⋊Dic3—21no—
15#88815C2212yescomplemented
16#88814—21no—
17C2×#3802—31no—
18C2×D3×D12—61no—
19C2×#3788D311yescharacteristic, complemented
20#63874—31no—
21#63855—61no—
22#63739—34no—
23C22×C3×C3⋊D3—22no—
24C22×C3×C3⋊D3D411yescharacteristic
25C2×#1113—24no—
26C2×#1113—44no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4⋊(C3⋊D3)—24no—
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3D412yes—
33#6295—22no—
34#6295C2×C414yes—
35C3×#622—22no—
36#5131—91nomaximal
37C22×D32—62no—
38C22×D32—61no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×D12—61no—
42C2×C3×D12—61no—
43C2×C3×D12—121no—
44C2×C3×C4×D3—61no—
45C2×C3×C4×D3—61no—
46C2×#626—62no—
47C2×#625—32no—
48#3802—34no—
49#3802—62no—
50D3×D12—64no—
51D3×D12—64no—
52C2×#622C2×D312yescomplemented
53#3788—22no—
54C4×C3⋊Dic3—21no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57C3×C4×Dic3—121no—
58#3724—32no—
59Dic3⋊Dic3—62no—
60#3722—61no—
61(C2×D3)⋊Dic3—62no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3—22no—
64C2×C3×C3⋊D3—24no—
65C2×C3×C3⋊D3—44no—
66C2×C3×C3⋊D3C22⋊C412yes—
67#1113—48no—
68C33×C4C22⋊C412yes—
69C3×C3⋊Dic3—42no—
70C3×C3⋊Dic3C22×C411yescharacteristic
71C3×C3⋊Dic3C2×D411yescharacteristic
72C2×D3×D4—181no—
73C22×D12—91no—
74C2×C3×M16—91no—
75#926—91no—
76#872—181no—
77#821—94no—
78C22×C3⋊D3—22no—
79C22×C3⋊D3D1211yescharacteristic
80C22×C3×D3—61no—
81C22×C3×D3—61no—
82C22×C3×D3—62no—
83C22×C3×D3—62no—
84C22×C3×D3—122no—
85C2×D32—68no—
86C2×D32—64no—
87C2×D32—128no—
88C2×D32—124no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4D1211yescharacteristic
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3C22×D311yescharacteristic
96C4⋊(C3⋊D3)—24no—
97C4×C3⋊D3C4×D312yes—
98C4×C3⋊D3#6712yes—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C3×D12—64no—
105C3×D12—64no—
106C3×D12—124no—
107C3×C4×D3—62no—
108C3×C4×D3—62no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111#626—64no—
112#626—64no—
113#625—34no—
114#625—62no—
115#622—22no—
116#418—271noSylow
117C2×C33C2×C22⋊C411yescharacteristic
118C2×C33AGL(1,ℤ/8)12yes—
119C3×C3⋊D3—22no—
120C3×C3⋊D3—44no—
121C23×D3—181no—
122C23×D3—182no—
123C2×C3×D4—181no—
124C22×C3×C4—91no—
125C2×#67—182no—
126D3×D4—184no—
127D3×D4—184no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—94no—
131C2×D12—92no—
132C2×D12—121no—
133C2×D12—182no—
134C2×C4×D3—61no—
135C2×C4×D3—61no—
136C3×M16—182no—
137C2×C3×C8—92no—
138C3×C42—181no—
139#217—181no—
140(C2×D3)⋊C4—182no—
141C4⋊Dic3—92no—
142Dic3⋊C4—182no—
143C4×Dic3—61no—
144C4×Dic3—61no—
145C4×Dic3—121no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32C2×D1211yescharacteristic
151C2×C3⋊D3—22no—
152C2×C3⋊D3—24no—
153C2×C3⋊D3—44no—
154C2×C3⋊D3(C2×D3)⋊C412yes—
155C2×C3×D3—62no—
156C2×C3×D3—62no—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—122no—
161C2×C3×D3—124no—
162C2×C3×D3—124no—
163C2×C3×D3—124no—
164C2×C3×D3—244no—
165D32—1216no—
166D32—128no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4(C2×D3)⋊C412yes—
172C3⋊Dic3—42no—
173C3⋊Dic3C2×C4×D311yescharacteristic
174C3⋊Dic3C2×#6711yescharacteristic
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—122no—
180C3×Dic3—124no—
181C3×Dic3—122no—
182C22×D4—271no—
183C2×M16—91no—
184#117—271no—
185#102—274no—
186C33#41811yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C23×C3—182no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—62no—
192C22×D3—62no—
193C22×D3—122no—
194C22×D3—184no—
195C22×D3—188no—
196C22×D3—368no—
197C22×D3—364no—
198C3×D4—184no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—91no—
204C2×C3×C4—92no—
205C2×C3×C4—92no—
206C2×C3×C4—182no—
207C2×C3×C4GO+(4,𝔽2)11yescharacteristic
208#67—184no—
209#67—184no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—122no—
215D12—64no—
216D12—64no—
217D12—94no—
218D12—124no—
219D12—182no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C3×C8—182no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32—42no—
233C2×C32#90812yes—
234C2×C32C2×(C2×D3)⋊C411yescharacteristic
235C3⋊D3—22no—
236C3⋊D3—44no—
237C3×D3—64no—
238C3×D3—122no—
239C3×D3—124no—
240C3×D3—124no—
241C3×D3—244no—
242C24—541no—
243C2×D4—181no—
244C2×D4—274no—
245C2×D4—272no—
246C2×D4—542no—
247C22×C4—91no—
248M16—182no—
249C2×C8—92no—
250C4⋊C4—272no—
251C22⋊C4—541no—
252C42—181no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—92no—
258C22×C3—184no—
259C22×C3—182no—
260C22×C3—364no—
261C22×C3C2×GO+(4,𝔽2)11yescharacteristic
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—64no—
267C2×D3—122no—
268C2×D3—124no—
269C2×D3—124no—
270C2×D3—124no—
271C2×D3—244no—
272C2×D3—3616no—
273C2×D3—368no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—91no—
279C3×C4—91no—
280C3×C4—362no—
281C3×C4D32⋊C412yes—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285Dic3—61no—
286Dic3—122no—
287Dic3—124no—
288Dic3—122no—
289C32—21no—
290C32—21no—
291C32—41no—
292C32—41no—
293C32#513111yescharacteristic, complemented
294C23—91no—
295C23—182no—
296C23—544no—
297C23—1084no—
298D4—184no—
299D4—274no—
300D4—542no—
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—182no—
305C2×C4#157⋊D311yescharacteristic
306C8—182no—
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—42no—
312C2×C3—42no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—182no—
316C2×C3—364no—
317C2×C3#6422412yes—
318C2×C3C2×D32⋊C411yescharacteristic
319D3—64no—
320D3—122no—
321D3—124no—
322D3—124no—
323D3—244no—
324C22—92no—
325C22—182no—
326C22—184no—
327C22—364no—
328C22—1088no—
329C22C2×#157⋊D311yescharacteristic
330C4—91no—
331C4—91no—
332C4—362no—
333C4#8893912yes—
334C3—21no—
335C3—21no—
336C3—41no—
337C3—41no—
338C3#1059336311yescharacteristic, complemented
339C2—182no—
340C2—364no—
341C2#1172434012yes—
342C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#418
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3788C2semidirect
3. #11723673C2semidirect
4. #11722793C2semidirect
11. C4⋊#1113C4semidirect
12. C2×C3×#622C22semidirect
15. #88815C22semidirect
19. C2×#3788D3semidirect
52. C2×#622C2×D3semidirect
186. C33#418semidirect
293. C32#5131semidirect
338. C3#10593363semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×C3×#37882
#117236732
#117227932
#105933633
#51319