Finite Group Databasebeta

← C2×#11722795 #13167575

Subgroups — C2×#11722795 #13167575

Order 1728.

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

normal subgroups only

This group has 720 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×#89046C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C2×#88813C211yesmaximal, characteristic, complemented
4#11722795C214yesmaximal, direct product factor
5C2×#63741—31nomaximal
6C2×C3×#638—21no—
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6296C411yescharacteristic, complemented
10C2×#6295C411yescharacteristic, complemented
11#89046—21no—
12#89046—21no—
13#89046C412yescomplemented
14#89046C2212yes—
15C2×#6294—21no—
16C3×#3792—21no—
17C3×#3792C2212yescomplemented
18C2×C3×#622—21no—
19#88813—21no—
20#88813C2212yescomplemented
21C2×#3805—31no—
22C2×#3804—61no—
23C2×#3792D311yescharacteristic, complemented
24C2×Dic32—31no—
25C2×Dic32—61no—
26#63741—34no—
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3—21no—
29C3×#638—22no—
30C3×#638—22no—
31C2×C3×C3⋊Dic3—24no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3C2311yescharacteristic
40#6296—21no—
41#6296C2×C412yescomplemented
42#6295—21no—
43#6295C2×C412yescomplemented
44#6294—22no—
45#6294—22no—
46C3×#622—22no—
47C2×#825—91nomaximal
48C2×#638—21no—
49C22×C3⋊Dic3—21no—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C2×C3×#67—61no—
52C2×C3×#67—61no—
53C2×C3×#67—121no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3—61no—
57C22×C3×Dic3—61no—
58C22×C3×Dic3—121no—
59C2×#627—31no—
60C2×#627—61no—
61C2×#626—61no—
62C2×#625—31no—
63C2×#624—61no—
64#3805—32no—
65#3805—32no—
66#3805—61no—
67#3805—61no—
68#3804—62no—
69#3804—62no—
70#3804—62no—
71#3804—62no—
72C2×D3×Dic3—61no—
73C2×D3×Dic3—61no—
74#3792—21no—
75#3792C2×D312yescomplemented
76C2×#622—21no—
77Dic32—32no—
78Dic32—62no—
79Dic32—62no—
80Dic32—61no—
81C22×C33—22no—
82C22×C33C22⋊C411yescharacteristic
83C22×C33C22⋊C411yescharacteristic
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—42no—
87C3×C3⋊Dic3—24no—
88C3×C3⋊Dic3—22no—
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C22⋊C411yescharacteristic
91C3×C3⋊Dic3C22×C411yescharacteristic
92C3×C3⋊Dic3C2×D411yescharacteristic
93C2×#237—181no—
94C2×#235—91no—
95C2×C3×M16—91no—
96C2×C4×Dic3—91no—
97C2×C4×Dic3—181no—
98#825—94no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—41no—
102C23×C32—41no—
103C23×C32D1211yescharacteristic
104C22×C3⋊D3—21no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—121no—
108#638—22no—
109#638—22no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—24no—
113C2×C3⋊Dic3C4×D311yescharacteristic
114C2×C3⋊Dic3C4×D311yescharacteristic
115C2×C3⋊Dic3D1211yescharacteristic
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3C22×D311yescharacteristic
119C3×#67—62no—
120C3×#67—62no—
121C3×#67—62no—
122C3×#67—62no—
123C3×#67—122no—
124C3×#67—122no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—122no—
144C2×C3×Dic3—124no—
145C2×C3×Dic3—121no—
146#627—32no—
147#627—62no—
148#627—62no—
149#627—61no—
150#626—62no—
151#626—62no—
152#625—32no—
153#625—61no—
154#624—62no—
155#624—62no—
156D3×Dic3—62no—
157D3×Dic3—62no—
158D3×Dic3—62no—
159D3×Dic3—62no—
160#622—22no—
161C2×#104—271noSylow
162C2×C33—21no—
163C2×C33—21no—
164C2×C33#10412yescomplemented
165C2×C33C2×C22⋊C411yescharacteristic
166C3×C3⋊D3—42no—
167C2×C3×D4—181no—
168C22×C3×C4—91no—
169C22×C3×C4—181no—
170C2×#67—61no—
171C2×#67—61no—
172C2×#67—121no—
173C2×#67—181no—
174C22×Dic3—61no—
175C22×Dic3—61no—
176C22×Dic3—61no—
177C22×Dic3—61no—
178C22×Dic3—121no—
179C22×Dic3—181no—
180#237—182no—
181#237—182no—
182#237—182no—
183#237—182no—
184#235—92no—
185#235—92no—
186#235—181no—
187#235—181no—
188C2×D12—91no—
189C2×C4×D3—181no—
190C2×C4×D3—181no—
191C2×Dic6—91no—
192C2×Dic6—181no—
193C3×M16—92no—
194C3×M16—181no—
195C2×C3×C8—181no—
196C4×Dic3—92no—
197C4×Dic3—182no—
198C4×Dic3—181no—
199C4×Dic3—182no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—82no—
219C22×C32(C2×D3)⋊C411yescharacteristic
220C22×C32(C2×D3)⋊C411yescharacteristic
221C22×C32C2×D1211yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—42no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—242no—
230C3⋊Dic3—22no—
231C3⋊Dic3—24no—
232C3⋊Dic3(C2×D3)⋊C411yescharacteristic
233C3⋊Dic3(C2×D3)⋊C411yescharacteristic
234C3⋊Dic3C2×C4×D311yescharacteristic
235C3⋊Dic3C2×#6711yescharacteristic
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—62no—
245C3×Dic3—64no—
246C3×Dic3—64no—
247C3×Dic3—62no—
248C3×Dic3—124no—
249C3×Dic3—122no—
250C2×#41—271no—
251C2×M16—91no—
252C2×C42—271no—
253#104—274no—
254C33C2×#10411yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—181no—
260C23×C3GO+(4,𝔽2)11yescharacteristic
261C22×D3—61no—
262C22×D3—61no—
263C22×D3—121no—
264C3×D4—182no—
265C3×D4—182no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—182no—
273C2×C3×C4—181no—
274C2×C3×C4—184no—
275#67—62no—
276#67—62no—
277#67—62no—
278#67—62no—
279#67—122no—
280#67—122no—
281#67—182no—
282#67—182no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—62no—
296C2×Dic3—61no—
297C2×Dic3—64no—
298C2×Dic3—64no—
299C2×Dic3—62no—
300C2×Dic3—61no—
301C2×Dic3—124no—
302C2×Dic3—122no—
303C2×Dic3—121no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306D12—92no—
307D12—181no—
308C4×D3—182no—
309C4×D3—182no—
310C4×D3—182no—
311C4×D3—182no—
312Dic6—92no—
313Dic6—182no—
314Dic6—181no—
315Dic6—182no—
316C3×C8—182no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—42no—
327C2×C32—42no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—41no—
334C2×C32—42no—
335C2×C32—81no—
336C2×C32—81no—
337C2×C32#82512yescomplemented
338C2×C32C2×(C2×D3)⋊C411yescharacteristic
339C3⋊D3—42no—
340C3×D3—122no—
341C3×D3—122no—
342C3×D3—242no—
343#41—272no—
344#41—272no—
345#41—541no—
346#41—541no—
347C2×Q8—271no—
348C2×D4—181no—
349C2×D4—271no—
350C22×C4—91no—
351C22×C4—181no—
352C22×C4—541no—
353M16—92no—
354M16—181no—
355C2×C8—181no—
356C42—272no—
357C42—541no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—42no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—82no—
377C22×C3—182no—
378C22×C3—362no—
379C22×C3D32⋊C411yescharacteristic
380C22×C3D32⋊C411yescharacteristic
381C22×C3C2×GO+(4,𝔽2)11yescharacteristic
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—242no—
388C3×C4—91no—
389C3×C4—91no—
390C3×C4—91no—
391C3×C4—91no—
392C3×C4—182no—
393C3×C4—184no—
394Dic3—61no—
395Dic3—61no—
396Dic3—61no—
397Dic3—61no—
398Dic3—61no—
399Dic3—61no—
400Dic3—61no—
401Dic3—61no—
402Dic3—62no—
403Dic3—64no—
404Dic3—64no—
405Dic3—62no—
406Dic3—124no—
407Dic3—122no—
408C32—21no—
409C32—21no—
410C32—41no—
411C32—41no—
412C32C2×#82511yescharacteristic, complemented
413C23—181no—
414C23#157⋊D311yescharacteristic
415Q8—272no—
416Q8—541no—
417D4—182no—
418D4—182no—
419D4—272no—
420D4—541no—
421C2×C4—91no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—184no—
428C2×C4—182no—
429C2×C4—181no—
430C2×C4—542no—
431C2×C4—542no—
432C8—182no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—42no—
447C2×C3—42no—
448C2×C3—42no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—81no—
452C2×C3—81no—
453C2×C3—362no—
454C2×C3#6374112yescomplemented
455C2×C3C2×D32⋊C411yescharacteristic
456D3—122no—
457D3—122no—
458D3—242no—
459C22—22no—
460C22—182no—
461C22—362no—
462C22#8893911yescharacteristic
463C22#8893911yescharacteristic
464C22C2×#157⋊D311yescharacteristic
465C4—91no—
466C4—91no—
467C4—91no—
468C4—91no—
469C4—184no—
470C4—182no—
471C3—21no—
472C3—21no—
473C3—41no—
474C3—41no—
475C3C2×#6374111yescharacteristic, complemented
476C2—21no—
477C2—21no—
478C2—362no—
479C2#1172279512yesdirect product factor
480C2C2×#8893911yescharacteristic
subgroupG/H
centerC22#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
FittingC23×C33D4
socleC22×C33C22⋊C4
prime radicalC2×#89046C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23#157⋊D3
hypocenterC33C2×#104
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. C2×C3×#3792C2semidirect
3. C2×#88813C2semidirect
4. #11722795C2direct (N × H)
9. C2×#6296C4semidirect
10. C2×#6295C4semidirect
13. #89046C4semidirect
17. C3×#3792C22semidirect
20. #88813C22semidirect
23. C2×#3792D3semidirect
41. #6296C2×C4semidirect
43. #6295C2×C4semidirect
75. #3792C2×D3semidirect
164. C2×C33#104semidirect
254. C33C2×#104semidirect
337. C2×C32#825semidirect
412. C32C2×#825semidirect
454. C2×C3#63741semidirect
475. C3C2×#63741semidirect
479. C2#11722795direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890462
C2×C3×#37922
C2×#888132
#117227952
C2×#637413
C2×#8259