Finite Group Databasebeta

← C2×#11724326 #13181119

Subgroups — C2×#11724326 #13181119

Order 1728.

944 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 127 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×#89046C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic, complemented
3C2×#88993C211yesmaximal, characteristic, complemented
4C2×#88938C212yesmaximal, complemented
5#11724326C218yesmaximal, direct product factor
6C2×#88936C212yesmaximal, complemented
7C2×#64239—31nomaximal
8C2×C3×#638—21no—
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3—21no—
11C2×#6296C2211yescharacteristic
12C2×#6295C2211yescharacteristic
13#89046—22no—
14#89046C2214yes—
15C2×#6294—21no—
16C2×C3×#627—22no—
17C2×C3×#627—21no—
18C2×C3×#627C2212yes—
19C2×C3×#627C2214yescomplemented
20#88993C2214yescomplemented
21C2×C3⋊#622C2212yes—
22#88938C2218yescomplemented
23#88936C2218yescomplemented
24C22×C3×Dic6—61no—
25C22×#627D311yescharacteristic, complemented
26C2×#3805—31no—
27C2×#3804—61no—
28C2×#3792—31no—
29C2×#3776—32no—
30#64239—38no—
31C2×#3775—32no—
32C23×C33D411yescharacteristic
33C22×C3×C3⋊D3—21no—
34C3×#638—24no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D412yes—
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C2×C3×C3⋊Dic3C2312yes—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—24no—
42#6296—21no—
43#6296C2312yes—
44#6295—21no—
45#6295C2312yes—
46#6294—24no—
47C3×#627—22no—
48C3×#627—24no—
49C3×#627C2314yes—
50C3⋊#622C2314yes—
51C2×#942—91nomaximal
52C22×C32×C4—61no—
53C2×#638—21no—
54C22×C3⋊Dic3C2×D311yescharacteristic
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C2×C3×#67—121no—
58C22×C3×Dic3—21no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—41no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C2×C3×Dic6—62no—
64C2×C3×Dic6—64no—
65C2×C3×Dic6—64no—
66C2×C3×Dic6—62no—
67C2×#627—22no—
68C2×#627—21no—
69C2×#627—31no—
70C2×#627—61no—
71C2×#627C2×D312yes—
72C2×#627C2×D314yescomplemented
73C2×#626—61no—
74C2×#625—31no—
75C2×#624—61no—
76#3805—34no—
77#3805—62no—
78#3804—64no—
79#3804—64no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82#3792—34no—
83C2×#622—32no—
84#3776—38no—
85#3775—38no—
86C22×C33—22no—
87C22×C33C2×D412yes—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—42no—
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3C2×D412yes—
93C3×C3⋊Dic3C2×D411yescharacteristic
94C3×C3⋊Dic3C2411yescharacteristic
95C32×Dic3—24no—
96C22×C3×Q8—91no—
97C2×#237—181no—
98C2×#235—91no—
99C22×Dic6—61no—
100C2×#216—92no—
101#942—98no—
102C2×#215—92no—
103C2×#208—91no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32—41no—
107C23×C32—41no—
108C23×C32#6711yescharacteristic
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C22×C3×D3—61no—
112C22×C3×D3—121no—
113C2×C32×C4—62no—
114C2×C32×C4—64no—
115#638—24no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6712yes—
119C2×C3⋊Dic3C22×D311yescharacteristic
120C2×C3⋊Dic3C22×D312yes—
121C3×#67—64no—
122C3×#67—64no—
123C3×#67—124no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—42no—
129C2×C3×Dic3—44no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—121no—
141C3×Dic6—64no—
142C3×Dic6—64no—
143C3×Dic6—68no—
144#627—22no—
145#627—24no—
146#627—32no—
147#627—62no—
148#627—62no—
149#627—61no—
150#627C22×D314yes—
151#626—62no—
152#626—62no—
153#625—32no—
154#625—61no—
155#624—64no—
156D3×Dic3—64no—
157D3×Dic3—64no—
158#622—34no—
159C2×#137—271noSylow
160C2×C33—22no—
161C2×C33#13712yescomplemented
162C2×C33C22×D411yescharacteristic
163C3×C3⋊D3—42no—
164C2×C3×Q8—92no—
165C2×C3×Q8—94no—
166C2×C3×Q8—181no—
167C2×C3×Q8—182no—
168C2×C3×D4—181no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4—91no—
172C22×C3×C4—121no—
173C2×#67—61no—
174C2×#67—61no—
175C2×#67—121no—
176C2×#67—181no—
177C22×Dic3—21no—
178C22×Dic3—61no—
179C22×Dic3—61no—
180C22×Dic3—181no—
181#237—184no—
182#237—184no—
183#235—94no—
184#235—182no—
185C2×D12—91no—
186C2×C4×D3—181no—
187C2×C4×D3—181no—
188C2×Dic6—62no—
189C2×Dic6—64no—
190C2×Dic6—64no—
191C2×Dic6—62no—
192C2×Dic6—91no—
193C2×Dic6—181no—
194#216—98no—
195#215—98no—
196#208—94no—
197C2×C3⋊C8—92no—
198C22×C32—24no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—42no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—44no—
210C22×C32—82no—
211C22×C32C2×#6712yes—
212C22×C32C2×#6711yescharacteristic
213C2×C3⋊D3—22no—
214C2×C3⋊D3—42no—
215C2×C3×D3—62no—
216C2×C3×D3—62no—
217C2×C3×D3—122no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—242no—
221C32×C4—64no—
222C3⋊Dic3—22no—
223C3⋊Dic3C2×#6712yes—
224C3⋊Dic3C2×#6711yescharacteristic
225C3⋊Dic3C23×D311yescharacteristic
226C3×Dic3—24no—
227C3×Dic3—24no—
228C3×Dic3—44no—
229C3×Dic3—62no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—62no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—122no—
238C2×#41—271no—
239C22×Q8—91no—
240#137—278no—
241C2×Q16—272no—
242C2×SD16—272no—
243C2×M16—271no—
244C33C2×#13711yescharacteristic, complemented, Sylow
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—181no—
250C23×C3GO+(4,𝔽2)11yescharacteristic
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—121no—
254C3×Q8—94no—
255C3×Q8—182no—
256C3×Q8—184no—
257C3×D4—184no—
258C2×C3×C4—64no—
259C2×C3×C4—62no—
260C2×C3×C4—64no—
261C2×C3×C4—62no—
262C2×C3×C4—91no—
263C2×C3×C4—92no—
264C2×C3×C4—91no—
265C2×C3×C4—92no—
266C2×C3×C4—124no—
267C2×C3×C4—122no—
268C2×C3×C4—181no—
269#67—64no—
270#67—64no—
271#67—124no—
272#67—182no—
273#67—182no—
274C2×Dic3—24no—
275C2×Dic3—22no—
276C2×Dic3—62no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—62no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—121no—
287C2×Dic3—184no—
288D12—92no—
289D12—181no—
290C4×D3—184no—
291C4×D3—184no—
292Dic6—64no—
293Dic6—64no—
294Dic6—68no—
295Dic6—92no—
296Dic6—182no—
297Dic6—181no—
298Dic6—182no—
299C3⋊C8—94no—
300C2×C32—24no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—42no—
307C2×C32—42no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—42no—
311C2×C32—44no—
312C2×C32—82no—
313C2×C32#94212yescomplemented
314C2×C32C22×#6711yescharacteristic
315C3⋊D3—42no—
316C3×D3—122no—
317C3×D3—122no—
318C3×D3—242no—
319#41—274no—
320#41—542no—
321C2×Q8—92no—
322C2×Q8—94no—
323C2×Q8—182no—
324C2×Q8—181no—
325C2×Q8—271no—
326C2×D4—181no—
327C2×D4—271no—
328C22×C4—61no—
329C22×C4—91no—
330C22×C4—541no—
331Q16—278no—
332SD16—278no—
333M16—274no—
334C2×C8—272no—
335C22×C3—24no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—42no—
342C22×C3—44no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—82no—
348C22×C3—182no—
349C22×C3—362no—
350C22×C3C2×GO+(4,𝔽2)12yes—
351C22×C3C2×GO+(4,𝔽2)11yescharacteristic
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—242no—
358C3×C4—64no—
359C3×C4—64no—
360C3×C4—91no—
361C3×C4—91no—
362C3×C4—92no—
363C3×C4—124no—
364C3×C4—182no—
365Dic3—24no—
366Dic3—62no—
367Dic3—61no—
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—62no—
372Dic3—62no—
373Dic3—62no—
374Dic3—122no—
375C32—21no—
376C32—21no—
377C32—41no—
378C32—41no—
379C32C2×#94211yescharacteristic, complemented
380C23—181no—
381C23D32⋊D311yescharacteristic
382Q8—94no—
383Q8—184no—
384Q8—182no—
385Q8—272no—
386Q8—541no—
387D4—184no—
388D4—272no—
389D4—541no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—92no—
393C2×C4—92no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—181no—
397C2×C4—544no—
398C8—274no—
399C2×C3—24no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—42no—
406C2×C3—44no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—82no—
412C2×C3—362no—
413C2×C3#6423912yescomplemented
414C2×C3C22×GO+(4,𝔽2)11yescharacteristic
415D3—122no—
416D3—122no—
417D3—242no—
418C22—22no—
419C22—182no—
420C22—362no—
421C22C2×D32⋊D312yes—
422C22C2×D32⋊D311yescharacteristic
423C4—64no—
424C4—91no—
425C4—92no—
426C4—91no—
427C4—182no—
428C3—21no—
429C3—21no—
430C3—41no—
431C3—41no—
432C3C2×#6423911yescharacteristic, complemented
433C2—22no—
434C2—362no—
435C2#1172432612yesdirect product factor
436C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89046C2
hypercenterC23D32⋊D3
hypocenterC33C2×#137
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3×#627C2semidirect
3. C2×#88993C2semidirect
4. C2×#88938C2semidirect
5. #11724326C2direct (N × H)
6. C2×#88936C2semidirect
19. C2×C3×#627C22semidirect
20. #88993C22semidirect
22. #88938C22semidirect
23. #88936C22semidirect
25. C22×#627D3semidirect
72. C2×#627C2×D3semidirect
161. C2×C33#137semidirect
244. C33C2×#137semidirect
313. C2×C32#942semidirect
379. C32C2×#942semidirect
413. C2×C3#64239semidirect
432. C3C2×#64239semidirect
435. C2#11724326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890462
C22×C3×#6272
C2×#889932
C2×#889382
#117243262
C2×#889362
C2×#642393
C2×#9429