Finite Group Databasebeta

← C2×#11722847 #13170603

Subgroups — C2×#11722847 #13170603

Order 1728.

1250 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 212 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×#88798C212yesmaximal, complemented
4C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
5C2×C3×#3719C212yesmaximal, complemented
6#11722847C218yesmaximal, direct product factor
7C2×D3×#216—32nomaximal
8C2×#63802—31nomaximal
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634C2212yes—
11C2×C32×Dic6C2212yes—
12C2×#6296—21no—
13C2×#6294—21no—
14#89040—22no—
15#89040C2214yes—
16C2×C3×#627—21no—
17C2×C3×#624—21no—
18C3×#3797—22no—
19C3×#3797C2214yes—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88798C2218yescomplemented
23C3⋊D3×C3⋊C8C2214yes—
24C3×#3719C2218yescomplemented
25C2×C3×D3×Q8—32no—
26C2×#3797—31no—
27C2×#3797D311yescharacteristic
28C2×D3×Dic6—32no—
29C2×#3756—32no—
30C2×C8×C3⋊D3—31no—
31C2×C3×#216—32no—
32C2×C3×Dic12—32no—
33D3×#216—316no—
34C2×#3719D312yescomplemented
35C2×#3718—32no—
36C2×#3718—32no—
37C2×#3718—62no—
38C2×D3×C3⋊C8—32no—
39C2×D3×C3⋊C8—61no—
40#63802—38no—
41C22×C3×C3⋊D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3D412yes—
46C3×C4×C3⋊D3C2312yes—
47C3×#634—22no—
48C3×#634C2314yes—
49C2×C32×Dic3—22no—
50C32×Dic6—22no—
51C32×Dic6C2314yes—
52#6296—24no—
53#6294—24no—
54C3×#627—24no—
55C3×#624—24no—
56C3⋊(C3⋊(C3⋊C8))C2312yes—
57C32×C3⋊C8C2312yes—
58C22×#216—91no—
59C2×D3×Q16—92no—
60C2×C32×Q8—32no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634C2×D312yes—
63C3×D3×Q8—38no—
64C3×D3×Q8—64no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—62no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—32no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—62no—
72C2×C3×Dic6—62no—
73C2×C3×Dic6C2×D312yes—
74C2×C3×Dic6C2×D312yes—
75C2×#627—21no—
76C2×#627—62no—
77C2×#627—61no—
78C2×#624—21no—
79C2×#624—61no—
80C2×D3×Dic3—62no—
81#3797—22no—
82#3797—34no—
83#3797—62no—
84#3797C2×D314yes—
85D3×Dic6—38no—
86D3×Dic6—64no—
87C2×C32×C8—31no—
88#3756—38no—
89C2×C3⋊(C3⋊C8)—32no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—62no—
93C2×C3⋊(C3⋊C8)—121no—
94C8×C3⋊D3—34no—
95C3×#216—38no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D312yes—
98C3×Dic12—38no—
99#3719C2×D318yescomplemented
100#3718—38no—
101#3718—38no—
102#3718—68no—
103D3×C3⋊C8—38no—
104D3×C3⋊C8—64no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3Q1614yescomplemented
107C2×C3×C3⋊D3C2×D412yes—
108C33×C4C2×D411yescharacteristic
109C33×C4C2411yescharacteristic
110C3×C3⋊Dic3—24no—
111C3×C3⋊Dic3C2×D412yes—
112C32×Dic3—24no—
113C22×C3×Q8—91no—
114C2×D3×Q8—32no—
115C2×D3×Q8—92no—
116C22×Dic6—91no—
117C2×C3×Q16—92no—
118C2×#216—32no—
119C2×#216—98no—
120C2×#216—92no—
121C2×#216—92no—
122C2×#216—92no—
123C2×#216—182no—
124C22×C3⋊C8—91no—
125D3×Q16—916no—
126C2×Dic12—32no—
127C2×D3×C8—92no—
128C2×D3×C8—181no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—32no—
131C22×C3×D3—61no—
132C32×Q8—34no—
133C32×Q8—62no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—41no—
137C2×C32×C4—62no—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3#6711yescharacteristic
142C4×C3⋊D3#6712yes—
143C4×C3⋊D3C22×D312yes—
144#634—22no—
145#634C22×D314yes—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—42no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—122no—
154C3×C4×D3—34no—
155C3×C4×D3—34no—
156C3×C4×D3—62no—
157C3×C4×D3—62no—
158C3×C4×D3—68no—
159C3×Dic6—22no—
160C3×Dic6—22no—
161C3×Dic6—24no—
162C3×Dic6—34no—
163C3×Dic6—34no—
164C3×Dic6—42no—
165C3×Dic6—64no—
166C3×Dic6—68no—
167C3×Dic6—62no—
168C3×Dic6—62no—
169C3×Dic6—122no—
170C3×Dic6C22×D314yes—
171C3×Dic6C22×D314yes—
172#627—24no—
173#627—68no—
174#627—64no—
175#624—24no—
176#624—64no—
177D3×Dic3—68no—
178C32×C8—32no—
179C3⋊(C3⋊C8)—34no—
180C3⋊(C3⋊C8)—32no—
181C3⋊(C3⋊C8)—62no—
182C3⋊(C3⋊C8)—64no—
183C3⋊(C3⋊C8)—122no—
184C3×C3⋊C8—22no—
185C3×C3⋊C8C22×D314yes—
186C22×Q16—271noSylow
187C2×C33C2×Q1612yescomplemented
188C2×C33C22×D411yescharacteristic
189C3×C3⋊D3C2×Q1614yescomplemented
190C2×C3×Q8—32no—
191C2×C3×Q8—32no—
192C2×C3×Q8—62no—
193C2×C3×Q8—94no—
194C2×C3×Q8—92no—
195C2×C3×Q8—181no—
196C2×C3×Q8—182no—
197C22×C3×C4—91no—
198C22×C3×C4—181no—
199C22×Dic3—181no—
200D3×Q8—38no—
201D3×Q8—64no—
202D3×Q8—98no—
203D3×Q8—184no—
204C2×C4×D3—32no—
205C2×C4×D3—61no—
206C2×C4×D3—62no—
207C2×C4×D3—182no—
208C2×Dic6—32no—
209C2×Dic6—32no—
210C2×Dic6—62no—
211C2×Dic6—62no—
212C2×Dic6—94no—
213C2×Dic6—182no—
214C2×Dic6—182no—
215C2×Dic6—181no—
216C2×Dic6D3212yes—
217C3×Q16—98no—
218C2×C3×C8—32no—
219C2×C3×C8—61no—
220#216—38no—
221#216—98no—
222#216—98no—
223#216—98no—
224#216—188no—
225C2×C3⋊C8—92no—
226C2×C3⋊C8—91no—
227C2×C3⋊C8—94no—
228C2×C3⋊C8—181no—
229C2×C3⋊C8—182no—
230C2×C3⋊C8—361no—
231C2×C3⋊C8D3211yescharacteristic
232Dic12—38no—
233D3×C8—98no—
234D3×C8—184no—
235C22×C32—21no—
236C22×C32—22no—
237C22×C32—41no—
238C22×C32D3×D412yes—
239C22×C32C2×#6711yescharacteristic
240C2×C3⋊D3#21614yescomplemented
241C2×C3⋊D3C2×#6712yes—
242C2×C3×D3—38no—
243C2×C3×D3—34no—
244C2×C3×D3—64no—
245C2×C3×D3—62no—
246C32×C4—21no—
247C32×C4—22no—
248C32×C4—21no—
249C32×C4—22no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—64no—
253C32×C4D3×D412yes—
254C32×C4C2×#6711yescharacteristic
255C32×C4C23×D312yes—
256C32×C4C23×D311yescharacteristic
257C3⋊Dic3—24no—
258C3⋊Dic3C2×#6712yes—
259C3×Dic3—24no—
260C3×Dic3—24no—
261C3×Dic3—34no—
262C3×Dic3—44no—
263C3×Dic3—62no—
264C3×Dic3—64no—
265C3×Dic3—64no—
266C3×Dic3—124no—
267C22×Q8—91no—
268C22×Q8—271no—
269C2×Q16—92no—
270C2×Q16—278no—
271C2×Q16—272no—
272C22×C8—271no—
273C33C22×Q1611yescharacteristic, complemented, Sylow
274C23×C3—91no—
275C22×D3—32no—
276C22×D3—61no—
277C3×Q8—34no—
278C3×Q8—34no—
279C3×Q8—64no—
280C3×Q8—62no—
281C3×Q8—62no—
282C3×Q8—94no—
283C3×Q8—122no—
284C3×Q8—184no—
285C3×Q8—182no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—91no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—122no—
295C2×C3×C4—184no—
296C2×C3×C4—182no—
297C2×C3×C4C2×D3211yescharacteristic
298C2×C3×C4C2×D3212yes—
299C2×Dic3—22no—
300C2×Dic3—32no—
301C2×Dic3—61no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—122no—
305C2×Dic3—184no—
306C4×D3—34no—
307C4×D3—34no—
308C4×D3—62no—
309C4×D3—62no—
310C4×D3—68no—
311C4×D3—188no—
312Dic6—22no—
313Dic6—34no—
314Dic6—34no—
315Dic6—64no—
316Dic6—68no—
317Dic6—62no—
318Dic6—62no—
319Dic6—122no—
320Dic6—188no—
321Dic6—184no—
322Dic6C2×D3214yes—
323C3×C8—34no—
324C3×C8—62no—
325C3⋊C8—94no—
326C3⋊C8—92no—
327C3⋊C8—182no—
328C3⋊C8—184no—
329C3⋊C8—362no—
330C3⋊C8C2×D3212yes—
331C2×C32—21no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—24no—
335C2×C32—42no—
336C2×C32—41no—
337C2×C32D3×Q1614yescomplemented
338C2×C32C2×#21612yescomplemented
339C2×C32C2×D3×D412yes—
340C2×C32C22×#6711yescharacteristic
341C3⋊D3C2×#21614yescomplemented
342C3×D3—38no—
343C3×D3—64no—
344C2×Q8—32no—
345C2×Q8—94no—
346C2×Q8—92no—
347C2×Q8—181no—
348C2×Q8—182no—
349C2×Q8—274no—
350C2×Q8—541no—
351C2×Q8—542no—
352C22×C4—91no—
353C22×C4—181no—
354C22×C4—541no—
355Q16—98no—
356Q16—278no—
357C2×C8—31no—
358C2×C8—274no—
359C2×C8—271no—
360C22×C3—21no—
361C22×C3—22no—
362C22×C3—41no—
363C22×C3—92no—
364C22×C3—94no—
365C22×C3#380211yescharacteristic
366C22×C3D3×#6712yes—
367C2×D3—34no—
368C2×D3—38no—
369C2×D3—62no—
370C2×D3—64no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—64no—
378C3×C4—64no—
379C3×C4—92no—
380C3×C4—124no—
381C3×C4—184no—
382C3×C4#380211yescharacteristic
383C3×C4D3×#6712yes—
384C3×C4C22×D3211yescharacteristic
385C3×C4C22×D3212yes—
386Dic3—24no—
387Dic3—34no—
388Dic3—62no—
389Dic3—64no—
390Dic3—64no—
391Dic3—124no—
392C32—21no—
393C32—22no—
394C32—41no—
395C32C2×D3×Q1612yescomplemented
396C32C22×#21611yescharacteristic, complemented
397C23—91no—
398Q8—34no—
399Q8—62no—
400Q8—94no—
401Q8—184no—
402Q8—182no—
403Q8—544no—
404C2×C4—62no—
405C2×C4—91no—
406C2×C4—92no—
407C2×C4—92no—
408C2×C4—184no—
409C2×C4—182no—
410C2×C4—544no—
411C2×C4D3311yescharacteristic
412C8—32no—
413C8—272no—
414C2×C3—22no—
415C2×C3—24no—
416C2×C3—21no—
417C2×C3—22no—
418C2×C3—42no—
419C2×C3—41no—
420C2×C3—94no—
421C2×C3#6380212yescomplemented
422C2×C3D3×#21614yescomplemented
423C2×C3C2×#380211yescharacteristic
424C2×C3C2×D3×#6712yes—
425D3—38no—
426D3—64no—
427C22—94no—
428C22—92no—
429C22#8895311yescharacteristic
430C4—64no—
431C4—92no—
432C4—184no—
433C4#8895311yescharacteristic
434C4C2×D3311yescharacteristic
435C3—21no—
436C3—22no—
437C3—41no—
438C3C2×#6380211yescharacteristic, complemented
439C3C2×D3×#21612yescomplemented
440C2—94no—
441C2#1172284712yesdirect product factor
442C2C2×#8895311yescharacteristic
subgroupG/H
centerC22#88953
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4D33
hypocenterC33C22×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88798C2semidirect
5. C2×C3×#3719C2semidirect
6. #11722847C2direct (N × H)
22. #88798C22semidirect
24. C3×#3719C22semidirect
34. C2×#3719D3semidirect
99. #3719C2×D3semidirect
106. C2×C3×C3⋊D3Q16semidirect
187. C2×C33C2×Q16semidirect
189. C3×C3⋊D3C2×Q16semidirect
240. C2×C3⋊D3#216semidirect
273. C33C22×Q16semidirect
337. C2×C32D3×Q16semidirect
338. C2×C32C2×#216semidirect
341. C3⋊D3C2×#216semidirect
395. C32C2×D3×Q16semidirect
396. C32C22×#216semidirect
421. C2×C3#63802semidirect
422. C2×C3D3×#216semidirect
438. C3C2×#63802semidirect
439. C3C2×D3×#216semidirect
441. C2#11722847direct (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×#887982
C2×C3⋊D3×C3⋊C82
C2×C3×#37192
#117228472
C2×D3×#2163
C2×#638023