Finite Group Databasebeta

← C2×#11722848 #13170604

Subgroups — C2×#11722848 #13170604

Order 1728.

1474 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 196 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×#89019C212yesmaximal
2C2×#88812C211yesmaximal, characteristic, complemented
3C2×#88785C211yesmaximal, characteristic
4C2×C3×#3719C212yesmaximal, complemented
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6#11722848C218yesmaximal, direct product factor
7C2×#63944—32nomaximal
8C2×#63802—31nomaximal
9C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2212yes—
11C2×C32×Dic6C2212yes—
12C2×#6293—22no—
13C2×#6289—22no—
14#89019—24no—
15#89019C2218yes—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812C2214yescomplemented
19#88785C2214yes—
20C3×#3719C2218yescomplemented
21C3×#3718C2214yescomplemented
22C2×Q8×C3⋊D3—32no—
23C2×#3797—32no—
24C2×#3797—32no—
25C2×#3797—62no—
26C2×C8×C3⋊D3—31no—
27C2×C3×#216—32no—
28C2×C3×#216—32no—
29C2×C3×Dic12—32no—
30#63944—316no—
31C2×#3719—32no—
32C2×#3719D312yescomplemented
33C2×#3718—31no—
34C2×#3718D311yescharacteristic, complemented
35C2×#3710—32no—
36C2×#3710—61no—
37#63802—38no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4×C3⋊(C3⋊D3)D412yes—
42C4×C3⋊(C3⋊D3)C2312yes—
43C2×C3×C3⋊Dic3—22no—
44C3×#634—22no—
45C3×#634C2314yes—
46C2×C32×Dic3—22no—
47C32×Dic6—22no—
48C32×Dic6C2314yes—
49#6293—28no—
50#6289—28no—
51C3×C3⋊(C3⋊C8)C2312yes—
52C32×C3⋊C8C2312yes—
53C2×D3×Q16—92no—
54C2×D3×Q16—91no—
55C2×C32×Q8—32no—
56Q8×C3⋊D3—38no—
57Q8×C3⋊D3—64no—
58C2×C4×C3⋊D3—31no—
59C2×C4×C3⋊D3—32no—
60C2×C4×C3⋊D3—62no—
61C2×C4×C3⋊D3—62no—
62C2×C4×C3⋊D3—61no—
63C2×C4×C3⋊D3—121no—
64C2×#634—62no—
65C2×#634C2×D312yes—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—62no—
70C2×C3×Dic6C2×D312yes—
71C2×C3×Dic6C2×D312yes—
72C2×#627—62no—
73C2×#627—62no—
74C2×#627—64no—
75C2×#624—62no—
76C2×#624—62no—
77C2×#624—64no—
78#3797—38no—
79#3797—38no—
80#3797—64no—
81#3797—64no—
82#3797—68no—
83#3797—68no—
84C2×C32×C8—31no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C8×C3⋊D3—34no—
87C3×#216—38no—
88C3×#216—38no—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8C2×D312yes—
93C3×Dic12—38no—
94#3719—38no—
95#3719C2×D318yescomplemented
96#3718—34no—
97#3718C2×D314yescomplemented
98#3710—38no—
99#3710—64no—
100C22×C33C2×D411yescharacteristic
101C2×C3⋊(C3⋊D3)Q1614yescomplemented
102C2×C3⋊(C3⋊D3)C2×D412yes—
103C33×C4C2×D411yescharacteristic
104C33×C4C2411yescharacteristic
105C3⋊(C3⋊Dic3)C2×D412yes—
106C3×C3⋊Dic3—24no—
107C32×Dic3—24no—
108C2×D3×Q8—92no—
109C2×D3×Q8—92no—
110C2×D3×Q8—92no—
111C2×D3×Q8—182no—
112C2×C3×Q16—92no—
113C2×C3×Q16—91no—
114C2×#216—32no—
115C2×#216—32no—
116C2×#216—92no—
117D3×Q16—916no—
118D3×Q16—98no—
119C2×Dic12—32no—
120C2×Dic12—91no—
121C2×D3×C8—92no—
122C2×D3×C8—91no—
123C2×D3×C8—181no—
124C22×C3⋊D3—31no—
125C22×C3⋊D3—32no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—121no—
129C32×Q8—34no—
130C32×Q8—62no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—41no—
134C2×C32×C4—62no—
135C2×C32×C4C22×D312yes—
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—31no—
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—62no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—121no—
143C4×C3⋊D3—32no—
144C4×C3⋊D3—34no—
145C4×C3⋊D3—34no—
146C4×C3⋊D3—32no—
147C4×C3⋊D3—62no—
148C4×C3⋊D3—64no—
149C4×C3⋊D3—62no—
150C4×C3⋊D3—64no—
151C4×C3⋊D3—68no—
152C4×C3⋊D3—122no—
153C4×C3⋊D3—122no—
154#634—22no—
155#634—68no—
156#634C22×D314yes—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—64no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C3×Dic6—24no—
164C3×Dic6—22no—
165C3×Dic6—22no—
166C3×Dic6—24no—
167C3×Dic6—34no—
168C3×Dic6—34no—
169C3×Dic6—64no—
170C3×Dic6—64no—
171C3×Dic6—62no—
172C3×Dic6—62no—
173C3×Dic6C22×D314yes—
174C3×Dic6C22×D314yes—
175#627—68no—
176#627—68no—
177#627—616no—
178#624—68no—
179#624—68no—
180#624—616no—
181C32×C8—32no—
182C3⋊(C3⋊C8)C22×D312yes—
183C3×C3⋊C8—22no—
184C3×C3⋊C8—34no—
185C3×C3⋊C8—62no—
186C3×C3⋊C8C22×D314yes—
187C22×Q16—271noSylow
188C2×C33C2×Q1612yescomplemented
189C2×C33C22×D411yescharacteristic
190C3⋊(C3⋊D3)C2×Q1614yescomplemented
191C2×C3×Q8—32no—
192C2×C3×Q8—32no—
193C2×C3×Q8—62no—
194C2×C3×Q8—92no—
195D3×Q8—98no—
196D3×Q8—98no—
197D3×Q8—98no—
198D3×Q8—188no—
199D3×Q8—184no—
200D3×Q8—184no—
201D3×Q8—184no—
202D3×Q8—188no—
203C2×C4×D3—91no—
204C2×C4×D3—92no—
205C2×C4×D3—181no—
206C2×C4×D3—182no—
207C2×C4×D3—182no—
208C2×C4×D3—182no—
209C2×C4×D3—182no—
210C2×C4×D3—184no—
211C2×C4×D3—361no—
212C2×Dic6—32no—
213C2×Dic6—32no—
214C2×Dic6—62no—
215C2×Dic6—182no—
216C2×Dic6—182no—
217C2×Dic6—182no—
218C2×Dic6—184no—
219C2×Dic6D3212yes—
220C3×Q16—98no—
221C3×Q16—94no—
222C2×C3×C8—32no—
223C2×C3×C8—61no—
224C2×C3×C8—91no—
225#216—38no—
226#216—38no—
227#216—98no—
228C2×C3⋊C8—32no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8D3211yescharacteristic
231Dic12—38no—
232Dic12—94no—
233D3×C8—98no—
234D3×C8—94no—
235D3×C8—184no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32—41no—
239C22×C32D3×D412yes—
240C22×C32D3×D411yescharacteristic
241C2×C3⋊D3—38no—
242C2×C3⋊D3—34no—
243C2×C3⋊D3—32no—
244C2×C3⋊D3—34no—
245C2×C3⋊D3—64no—
246C2×C3⋊D3—68no—
247C2×C3⋊D3—64no—
248C2×C3⋊D3—62no—
249C2×C3⋊D3—124no—
250C2×C3⋊D3—122no—
251C32×C4—21no—
252C32×C4—22no—
253C32×C4—21no—
254C32×C4—22no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—64no—
258C32×C4D3×D412yes—
259C32×C4D3×D411yescharacteristic
260C32×C4C23×D312yes—
261C32×C4C23×D311yescharacteristic
262C3⋊Dic3—24no—
263C3⋊Dic3—32no—
264C3⋊Dic3—34no—
265C3⋊Dic3—64no—
266C3⋊Dic3—62no—
267C3⋊Dic3—122no—
268C3×Dic3—28no—
269C3×Dic3—24no—
270C3×Dic3—24no—
271C3×Dic3—68no—
272C3×Dic3—64no—
273C3×Dic3—64no—
274C22×Q8—272no—
275C2×Q16—92no—
276C2×Q16—91no—
277C2×Q16—271no—
278C2×Q16—278no—
279C22×C8—271no—
280C33C22×Q1611yescharacteristic, complemented, Sylow
281C22×D3—91no—
282C22×D3—92no—
283C22×D3—181no—
284C22×D3—182no—
285C22×D3—361no—
286C3×Q8—34no—
287C3×Q8—34no—
288C3×Q8—64no—
289C3×Q8—64no—
290C3×Q8—62no—
291C3×Q8—62no—
292C3×Q8—94no—
293C3×Q8—182no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—41no—
297C2×C3×C4—64no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—182no—
301C2×C3×C4C2×D3211yescharacteristic
302C2×C3×C4C2×D3212yes—
303C2×Dic3—22no—
304C2×Dic3—64no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—91no—
308C2×Dic3—92no—
309C2×Dic3—182no—
310C2×Dic3—181no—
311C2×Dic3—361no—
312C4×D3—92no—
313C4×D3—94no—
314C4×D3—92no—
315C4×D3—94no—
316C4×D3—182no—
317C4×D3—182no—
318C4×D3—184no—
319C4×D3—184no—
320C4×D3—188no—
321C4×D3—188no—
322C4×D3—188no—
323C4×D3—1816no—
324C4×D3—362no—
325C4×D3—362no—
326Dic6—22no—
327Dic6—34no—
328Dic6—34no—
329Dic6—64no—
330Dic6—64no—
331Dic6—62no—
332Dic6—62no—
333Dic6—188no—
334Dic6—188no—
335Dic6—188no—
336Dic6—1816no—
337Dic6C2×D3214yes—
338C3×C8—34no—
339C3×C8—62no—
340C3×C8—92no—
341C3⋊C8—34no—
342C3⋊C8—62no—
343C3⋊C8C2×D3212yes—
344C2×C32—21no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—24no—
348C2×C32—42no—
349C2×C32—41no—
350C2×C32D3×Q1614yescomplemented
351C2×C32D3×Q1612yescomplemented
352C2×C32C2×D3×D412yes—
353C2×C32C2×D3×D411yescharacteristic
354C3⋊D3—34no—
355C3⋊D3—38no—
356C3⋊D3—68no—
357C3⋊D3—64no—
358C3⋊D3—124no—
359C2×Q8—32no—
360C2×Q8—92no—
361C2×Q8—278no—
362C2×Q8—542no—
363C2×Q8—544no—
364C22×C4—271no—
365C22×C4—542no—
366Q16—98no—
367Q16—94no—
368Q16—274no—
369C2×C8—31no—
370C2×C8—91no—
371C2×C8—274no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—41no—
375C22×C3#380211yescharacteristic
376C22×C3#381112yes—
377C2×D3—92no—
378C2×D3—94no—
379C2×D3—94no—
380C2×D3—98no—
381C2×D3—184no—
382C2×D3—182no—
383C2×D3—184no—
384C2×D3—188no—
385C2×D3—362no—
386C2×D3—364no—
387C3×C4—21no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—41no—
392C3×C4—41no—
393C3×C4—68no—
394C3×C4—64no—
395C3×C4—64no—
396C3×C4—184no—
397C3×C4#380211yescharacteristic
398C3×C4#381112yes—
399C3×C4C22×D3211yescharacteristic
400C3×C4C22×D3212yes—
401Dic3—24no—
402Dic3—68no—
403Dic3—64no—
404Dic3—64no—
405Dic3—92no—
406Dic3—94no—
407Dic3—184no—
408Dic3—182no—
409Dic3—362no—
410C32—22no—
411C32—21no—
412C32—41no—
413C32C2×D3×Q1611yescharacteristic, complemented
414C32C2×D3×Q1612yescomplemented
415C23—271no—
416Q8—34no—
417Q8—62no—
418Q8—94no—
419Q8—182no—
420Q8—548no—
421C2×C4—62no—
422C2×C4—182no—
423C2×C4—271no—
424C2×C4—272no—
425C2×C4—272no—
426C2×C4—548no—
427C2×C4D3311yescharacteristic
428C8—32no—
429C8—92no—
430C2×C3—24no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—42no—
435C2×C3—41no—
436C2×C3#6380212yescomplemented
437C2×C3#6394414yescomplemented
438C2×C3C2×#380211yescharacteristic
439C2×C3C2×#381112yes—
440D3—98no—
441D3—94no—
442D3—188no—
443D3—184no—
444D3—364no—
445C22—274no—
446C22—272no—
447C22#8895411yescharacteristic
448C4—64no—
449C4—184no—
450C4—272no—
451C4#8895411yescharacteristic
452C4C2×D3311yescharacteristic
453C3—22no—
454C3—21no—
455C3—41no—
456C3C2×#6380211yescharacteristic, complemented
457C3C2×#6394412yescomplemented
458C2—274no—
459C2#1172284812yesdirect product factor
460C2C2×#8895411yescharacteristic
subgroupG/H
centerC22#88954
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
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
2. C2×#88812C2semidirect
4. C2×C3×#3719C2semidirect
5. C2×C3×#3718C2semidirect
6. #11722848C2direct (N × H)
18. #88812C22semidirect
20. C3×#3719C22semidirect
21. C3×#3718C22semidirect
32. C2×#3719D3semidirect
34. C2×#3718D3semidirect
95. #3719C2×D3semidirect
97. #3718C2×D3semidirect
101. C2×C3⋊(C3⋊D3)Q16semidirect
188. C2×C33C2×Q16semidirect
190. C3⋊(C3⋊D3)C2×Q16semidirect
280. C33C22×Q16semidirect
350. C2×C32D3×Q16semidirect
351. C2×C32D3×Q16semidirect
413. C32C2×D3×Q16semidirect
414. C32C2×D3×Q16semidirect
436. C2×C3#63802semidirect
437. C2×C3#63944semidirect
456. C3C2×#63802semidirect
457. C3C2×#63944semidirect
459. C2#11722848direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890192
C2×#888122
C2×#887852
C2×C3×#37192
C2×C3×#37182
#117228482
C2×#639443
C2×#638023