Finite Group Databasebeta

← C2×#11724302 #13181064

Subgroups — C2×#11724302 #13181064

Order 1728.

968 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 968 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×#89040C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic
3C2×C4×C3⋊#157C211yesmaximal, characteristic
4C2×#88934C214yesmaximal
5#11724302C218yesmaximal, direct product factor
6C2×#64222—31nomaximal
7C22×C3⋊#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634—21no—
10C2×C32×Dic6—21no—
11C2×#6296—21no—
12C2×#6294C2212yes—
13#89040—22no—
14#89040C2214yes—
15C2×C3×#627—21no—
16C2×C3×#624C2212yes—
17C3×#3797—22no—
18C3×#3797C2214yes—
19C4×C3⋊#157C2214yes—
20#88934C22116yes—
21C2×C3×D3×Q8—61no—
22C2×#3797—31no—
23C2×#3797D311yescharacteristic
24C2×D3×Dic6—61no—
25C2×C4×#157—31no—
26C2×#3773—34no—
27#64222—38no—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157Q818yescomplemented
30C2×C3⋊#157C2314yes—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3D412yes—
35C3×C4×C3⋊D3C2312yes—
36C3×#634—24no—
37C2×C32×Dic3—22no—
38C32×Dic6—24no—
39#6296—24no—
40#6294—22no—
41#6294C2314yes—
42C3×#627—24no—
43C3×#624—22no—
44C3×#624C2314yes—
45C2×#944—91nomaximal
46C22×#157—32no—
47C2×C32×Q8—61no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C2×#634—21no—
50C3×D3×Q8—64no—
51C3×D3×Q8—64no—
52C2×C3×C4×D3—62no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—41no—
58C2×C3×Dic6—62no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—61no—
61C2×C3×Dic6—121no—
62C2×#627—21no—
63C2×#627—62no—
64C2×#627—61no—
65C2×#624—32no—
66C2×#624C2×D312yes—
67C2×D3×Dic3—62no—
68#3797—22no—
69#3797—34no—
70#3797—62no—
71#3797C2×D314yes—
72D3×Dic6—64no—
73D3×Dic6—64no—
74C4×#157—34no—
75#3773—316no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C2×C3×C3⋊D3C2×Q814yes—
79C2×C3×C3⋊D3C2411yescharacteristic
80C3⋊#157C2×Q818yescomplemented
81C33×C4C2×D412yes—
82C3×C3⋊Dic3—24no—
83C3×C3⋊Dic3C2×D412yes—
84C32×Dic3—24no—
85C22×C3×Q8—91no—
86C2×D3×Q8—61no—
87C2×D3×Q8—181no—
88C22×Dic6—91no—
89#944—98no—
90C2×Dic3⋊C4—94no—
91C2×C4×Dic3—91no—
92C22×C3⋊D3C22×D311yescharacteristic
93C22×C3×D3—61no—
94C22×C3×D3—61no—
95C2×#157—38no—
96C2×#157—34no—
97C32×Q8—64no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4#6711yescharacteristic
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3#6711yescharacteristic
106C4×C3⋊D3#6712yes—
107C4×C3⋊D3C22×D312yes—
108#634—24no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—122no—
117C3×C4×D3—64no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—64no—
123C3×Dic6—24no—
124C3×Dic6—24no—
125C3×Dic6—44no—
126C3×Dic6—68no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129C3×Dic6—124no—
130#627—24no—
131#627—68no—
132#627—64no—
133#624—22no—
134#624—34no—
135#624—62no—
136#624C22×D314yes—
137D3×Dic3—64no—
138D3×Dic3—64no—
139C2×C4⋊Q8—271noSylow
140C2×C33C4⋊Q812yescomplemented
141C2×C33C22×D411yescharacteristic
142C3×C3⋊D3C4⋊Q812yescomplemented
143C3×C3⋊D3C22×Q812yes—
144C2×C3×Q8—61no—
145C2×C3×Q8—61no—
146C2×C3×Q8—94no—
147C2×C3×Q8—121no—
148C2×C3×Q8—181no—
149C2×C3×Q8—181no—
150C2×C3×Q8—182no—
151C22×C3×C4—92no—
152C22×C3×C4—91no—
153C22×Dic3—92no—
154C22×Dic3—92no—
155D3×Q8—64no—
156D3×Q8—64no—
157D3×Q8—184no—
158D3×Q8—184no—
159C2×C4×D3—62no—
160C2×C4×D3—61no—
161C2×C4×D3—61no—
162C2×C4×D3—182no—
163C2×Dic6—21no—
164C2×Dic6—62no—
165C2×Dic6—61no—
166C2×Dic6—61no—
167C2×Dic6—94no—
168C2×Dic6—121no—
169C2×Dic6—182no—
170C2×Dic6—181no—
171C2×Dic6—182no—
172Dic3⋊C4—916no—
173C4×Dic3—94no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32C2×#6711yescharacteristic
179C2×C3⋊D3D3×Q814yes—
180C2×C3⋊D3C2×#6711yescharacteristic
181C2×C3⋊D3C23×D311yescharacteristic
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—64no—
185C2×C3×D3—64no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188#157—38no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—42no—
193C32×C4—64no—
194C32×C4C2×#6712yes—
195C3⋊Dic3—24no—
196C3⋊Dic3C2×#6712yes—
197C3×Dic3—24no—
198C3×Dic3—24no—
199C3×Dic3—44no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—64no—
203C3×Dic3—64no—
204C3×Dic3—124no—
205C22×Q8—91no—
206C22×Q8—271no—
207C4⋊Q8—278no—
208C2×C4⋊C4—274no—
209C2×C42—271no—
210C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—61no—
213C22×D3—61no—
214C3×Q8—64no—
215C3×Q8—64no—
216C3×Q8—124no—
217C3×Q8—184no—
218C3×Q8—184no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—94no—
226C2×C3×C4—91no—
227C2×C3×C4—92no—
228C2×C3×C4—92no—
229C2×C3×C4—122no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4GO+(4,𝔽2)11yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—94no—
239C2×Dic3—98no—
240C2×Dic3—94no—
241C2×Dic3—122no—
242C2×Dic3—182no—
243C4×D3—64no—
244C4×D3—62no—
245C4×D3—62no—
246C4×D3—62no—
247C4×D3—62no—
248C4×D3—64no—
249C4×D3—184no—
250C4×D3—184no—
251Dic6—24no—
252Dic6—68no—
253Dic6—64no—
254Dic6—64no—
255Dic6—124no—
256Dic6—184no—
257Dic6—188no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—42no—
263C2×C32—42no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32#94412yescomplemented
267C2×C32C22×#6711yescharacteristic
268C3⋊D3#94412yescomplemented
269C3⋊D3C2×D3×Q812yes—
270C3×D3—62no—
271C3×D3—62no—
272C3×D3—62no—
273C3×D3—62no—
274C2×Q8—61no—
275C2×Q8—94no—
276C2×Q8—181no—
277C2×Q8—182no—
278C2×Q8—181no—
279C2×Q8—274no—
280C2×Q8—541no—
281C2×Q8—542no—
282C22×C4—92no—
283C22×C4—91no—
284C22×C4—272no—
285C22×C4—272no—
286C4⋊C4—2716no—
287C42—274no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—91no—
293C22×C3—91no—
294C22×C3—94no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—61no—
297C2×D3—61no—
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—64no—
301C2×D3—64no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4—64no—
307C3×C4—64no—
308C3×C4—92no—
309C3×C4—124no—
310C3×C4—184no—
311C3×C4C2×GO+(4,𝔽2)12yes—
312Dic3—24no—
313Dic3—62no—
314Dic3—62no—
315Dic3—64no—
316Dic3—64no—
317Dic3—98no—
318Dic3—124no—
319C32—21no—
320C32—21no—
321C32—41no—
322C32—41no—
323C32C2×#94411yescharacteristic, complemented
324C23—91no—
325Q8—64no—
326Q8—184no—
327Q8—184no—
328Q8—544no—
329C2×C4—62no—
330C2×C4—94no—
331C2×C4—91no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—274no—
337C2×C4—274no—
338C2×C4—278no—
339C2×C4—542no—
340C2×C4D32⋊D311yescharacteristic
341C2×C3—22no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—42no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—92no—
350C2×C3—92no—
351C2×C3#6422212yescomplemented
352C2×C3C22×GO+(4,𝔽2)11yescharacteristic
353D3—62no—
354D3—62no—
355D3—62no—
356D3—62no—
357C22—94no—
358C22—91no—
359C22—91no—
360C22C2×D32⋊D311yescharacteristic
361C4—64no—
362C4—92no—
363C4—184no—
364C4—278no—
365C4C2×D32⋊D312yes—
366C3—21no—
367C3—21no—
368C3—41no—
369C3—41no—
370C3C2×#6422211yescharacteristic, complemented
371C2—92no—
372C2—92no—
373C2#1172430212yesdirect product factor
374C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×C4⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
5. #11724302C2direct (N × H)
29. C2×C3⋊#157Q8semidirect
80. C3⋊#157C2×Q8semidirect
140. C2×C33C4⋊Q8semidirect
142. C3×C3⋊D3C4⋊Q8semidirect
210. C33C2×C4⋊Q8semidirect
266. C2×C32#944semidirect
268. C3⋊D3#944semidirect
323. C32C2×#944semidirect
351. C2×C3#64222semidirect
370. C3C2×#64222semidirect
373. C2#11724302direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37972
C2×C4×C3⋊#1572
C2×#889342
#117243022
C2×#642223
C2×#9449