Finite Group Databasebeta

← C2×#11722890 #13170687

Subgroups — C2×#11722890 #13170687

Order 1728.

1386 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1386 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×#89018C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C2×#88795C212yesmaximal, complemented
5C2×C3×#3718C212yesmaximal, complemented
6#11722890C218yesmaximal, direct product factor
7C2×#63948—32nomaximal
8C2×#63945—31nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×#6292—21no—
14C2×#6289—21no—
15#89018—22no—
16#89018C2214yes—
17C2×C3×#627—21no—
18C2×C3×#624—21no—
19C3×#3797—22no—
20C3×#3797C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yes—
22#88809C2214yescomplemented
23#88795C2218yescomplemented
24C3×#3718C2218yescomplemented
25C2×#3832—31no—
26C2×C3×D3×Q8—32no—
27C2×#3799—32no—
28C2×#3799—61no—
29C2×#3797D311yescharacteristic, complemented
30C2×#3755—32no—
31C2×C3×#216—32no—
32C2×C3×#216—32no—
33#63948—316no—
34#63945—38no—
35C2×#3718D312yescomplemented
36C2×#3717—32no—
37C2×#3717—32no—
38C2×#3717—62no—
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))C2312yes—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4×C3⋊D3D412yes—
48C3×C4×C3⋊D3C2312yes—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—22no—
51C32×Dic6—21no—
52C32×Dic6—22no—
53C32×Dic6C2314yes—
54C32×Dic6C2312yes—
55#6292—24no—
56#6289—24no—
57C3×#627—24no—
58C3×#624—24no—
59C3×C3⋊(C3⋊C8)C2314yes—
60C2×#942—91no—
61C2×#918—92no—
62C2×C32×Q8—31no—
63C2×C32×Q8—32no—
64#3832—34no—
65#3832—62no—
66C2×C4⋊(C3⋊D3)—32no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—62no—
71C2×C4⋊(C3⋊D3)—121no—
72C2×C4×C3⋊D3—61no—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C3×D3×Q8—38no—
75C3×D3×Q8—64no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—62no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—22no—
81C2×C3×Dic6—62no—
82C2×C3×Dic6C2×D312yes—
83C2×C3×Dic6C2×D312yes—
84C2×C3×Dic6C2×D312yes—
85C2×#627—21no—
86C2×#626—62no—
87C2×#626—62no—
88C2×#624—21no—
89C2×#624—62no—
90C2×#624—62no—
91#3799—38no—
92#3799—64no—
93#3799—64no—
94#3799—64no—
95#3797—22no—
96#3797C2×D314yescomplemented
97#3755—38no—
98C2×C3⋊(C3⋊C8)C2×D312yes—
99C3×#216—38no—
100C3×#216—38no—
101C2×C3×C3⋊C8—32no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—62no—
104#3718C2×D318yescomplemented
105#3717—38no—
106#3717—38no—
107#3717—68no—
108#3712—34no—
109#3711—38no—
110C22×C33C2×D411yescharacteristic
111C2×C3⋊(C3⋊D3)—22no—
112C2×C3⋊(C3⋊D3)—42no—
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3C2×D412yes—
115C33×C4C2×D411yescharacteristic
116C33×C4C2411yescharacteristic
117C3×C3⋊Dic3C2×D412yes—
118C32×Dic3—22no—
119C32×Dic3—24no—
120C22×C3×Q8—91no—
121C2×#239—92no—
122C2×#239—181no—
123C2×D3×Q8—32no—
124C2×#235—91no—
125C2×C3×Q16—92no—
126C2×#216—32no—
127C2×#216—32no—
128#942—98no—
129C2×#215—92no—
130C2×#215—92no—
131C2×#215—182no—
132C2×#208—91no—
133#918—916no—
134C2×#204—92no—
135C2×ΓL(1,𝔽25)—92no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—62no—
138C22×C3⋊D3—62no—
139C22×C3⋊D3—122no—
140C22×C3⋊D3—122no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C32×Q8—32no—
145C32×Q8—34no—
146C32×Q8—61no—
147C32×Q8—62no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—62no—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3#6711yescharacteristic
156C4⋊(C3⋊D3)—32no—
157C4⋊(C3⋊D3)—34no—
158C4⋊(C3⋊D3)—62no—
159C4⋊(C3⋊D3)—64no—
160C4⋊(C3⋊D3)—64no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—62no—
163C4⋊(C3⋊D3)—62no—
164C4⋊(C3⋊D3)—122no—
165C4⋊(C3⋊D3)—122no—
166C4⋊(C3⋊D3)—122no—
167C4×C3⋊D3—64no—
168C4×C3⋊D3#6712yes—
169C4×C3⋊D3C22×D312yes—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—42no—
176C2×C3×Dic3—61no—
177C3×C4×D3—34no—
178C3×C4×D3—34no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181C3×C4×D3—68no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184C3×Dic6—22no—
185C3×Dic6—22no—
186C3×Dic6—22no—
187C3×Dic6—24no—
188C3×Dic6—42no—
189C3×Dic6—68no—
190C3×Dic6C22×D314yes—
191C3×Dic6C22×D314yes—
192C3×Dic6C22×D314yes—
193#627—24no—
194#626—68no—
195#626—68no—
196#624—24no—
197#624—68no—
198#624—68no—
199C3⋊(C3⋊C8)C22×D314yes—
200C3×C3⋊C8—34no—
201C3×C3⋊C8—34no—
202C3×C3⋊C8—64no—
203C2×#137—271noSylow
204C2×C33#13712yescomplemented
205C2×C33C22×D411yescharacteristic
206C3⋊(C3⋊D3)—42no—
207C3×C3⋊D3—22no—
208C2×C3×Q8—32no—
209C2×C3×Q8—32no—
210C2×C3×Q8—32no—
211C2×C3×Q8—61no—
212C2×C3×Q8—62no—
213C2×C3×Q8—94no—
214C2×C3×Q8—181no—
215C2×C3×Q8—182no—
216C22×C3×C4—91no—
217C22×C3×C4—181no—
218C2×#67—181no—
219#239—98no—
220#239—184no—
221#239—184no—
222#239—184no—
223D3×Q8—38no—
224D3×Q8—64no—
225#235—94no—
226#235—182no—
227C2×D12—92no—
228C2×D12—91no—
229C2×D12—181no—
230C2×D12—182no—
231C2×D12—182no—
232C2×D12—182no—
233C2×D12—361no—
234C2×C4×D3—32no—
235C2×C4×D3—61no—
236C2×C4×D3—62no—
237C2×C4×D3—181no—
238C2×C4×D3—182no—
239C2×C4×D3—182no—
240C2×Dic6—62no—
241C2×Dic6D3212yes—
242C2×Dic6D3211yescharacteristic
243C3×Q16—98no—
244C2×C3×C8—92no—
245#216—38no—
246#216—38no—
247#215—98no—
248#215—98no—
249#215—188no—
250#208—94no—
251C2×C3⋊C8—32no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—62no—
254#204—98no—
255ΓL(1,𝔽25)—98no—
256C22×C32—22no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32D3×D412yes—
260C22×C32C2×#6711yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—64no—
265C2×C3⋊D3—124no—
266C2×C3⋊D3—124no—
267C2×C3⋊D3—122no—
268C2×C3⋊D3—124no—
269C2×C3⋊D3—124no—
270C2×C3⋊D3—244no—
271C2×C3⋊D3—244no—
272C2×C3⋊D3C2×#6712yes—
273C2×C3×D3—34no—
274C2×C3×D3—62no—
275C2×C3×D3—64no—
276C2×C3×D3—122no—
277C32×C4—22no—
278C32×C4—21no—
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—62no—
284C32×C4—64no—
285C32×C4D3×D412yes—
286C32×C4C2×#6711yescharacteristic
287C32×C4C23×D312yes—
288C32×C4C23×D311yescharacteristic
289C3⋊Dic3C2×#6712yes—
290C3×Dic3—24no—
291C3×Dic3—24no—
292C3×Dic3—24no—
293C3×Dic3—24no—
294C3×Dic3—34no—
295C3×Dic3—44no—
296C3×Dic3—62no—
297C2×#41—271no—
298C22×Q8—91no—
299#137—278no—
300C2×Q16—92no—
301C2×SD16—272no—
302C2×M16—271no—
303C33C2×#13711yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C22×D3—32no—
306C22×D3—61no—
307C22×D3—182no—
308C22×D3—182no—
309C22×D3—181no—
310C22×D3—362no—
311C22×D3—362no—
312C3×Q8—34no—
313C3×Q8—34no—
314C3×Q8—34no—
315C3×Q8—62no—
316C3×Q8—64no—
317C3×Q8—62no—
318C3×Q8—62no—
319C3×Q8—62no—
320C3×Q8—62no—
321C3×Q8—122no—
322C3×Q8—184no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—91no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—122no—
334C2×C3×C4—184no—
335C2×C3×C4C2×D3212yes—
336C2×C3×C4C2×D3211yescharacteristic
337#67—184no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—32no—
341C2×Dic3—61no—
342D12—94no—
343D12—92no—
344D12—182no—
345D12—184no—
346D12—182no—
347D12—188no—
348D12—188no—
349D12—182no—
350D12—181no—
351D12—362no—
352D12—362no—
353D12—362no—
354C4×D3—34no—
355C4×D3—34no—
356C4×D3—62no—
357C4×D3—62no—
358C4×D3—68no—
359C4×D3—184no—
360C4×D3—188no—
361C4×D3—188no—
362Dic6—22no—
363Dic6—21no—
364Dic6—68no—
365Dic6C2×D3214yes—
366Dic6C2×D3212yes—
367C3×C8—94no—
368C3⋊C8—34no—
369C3⋊C8—34no—
370C3⋊C8—64no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—22no—
374C2×C32—24no—
375C2×C32—42no—
376C2×C32—41no—
377C2×C32#91814yescomplemented
378C2×C32#94212yescomplemented
379C2×C32C2×D3×D412yes—
380C2×C32C22×#6711yescharacteristic
381C3⋊D3—22no—
382C3⋊D3—122no—
383C3⋊D3—124no—
384C3⋊D3—124no—
385C3⋊D3—244no—
386C3⋊D3—244no—
387C3×D3—64no—
388C3×D3—122no—
389#41—274no—
390#41—542no—
391C2×Q8—32no—
392C2×Q8—31no—
393C2×Q8—94no—
394C2×Q8—181no—
395C2×Q8—182no—
396C2×D4—271no—
397C2×D4—541no—
398C22×C4—91no—
399C22×C4—181no—
400C22×C4—541no—
401Q16—98no—
402SD16—278no—
403M16—274no—
404C2×C8—92no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—41no—
408C22×C3—92no—
409C22×C3—182no—
410C22×C3D3×#6712yes—
411C22×C3#381111yescharacteristic
412C2×D3—34no—
413C2×D3—62no—
414C2×D3—64no—
415C2×D3—122no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—182no—
419C2×D3—364no—
420C2×D3—364no—
421C2×D3—364no—
422C2×D3—362no—
423C2×D3—364no—
424C2×D3—724no—
425C2×D3—724no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—64no—
433C3×C4—64no—
434C3×C4—64no—
435C3×C4—64no—
436C3×C4—92no—
437C3×C4—124no—
438C3×C4D3×#6712yes—
439C3×C4#381111yescharacteristic
440C3×C4C22×D3212yes—
441C3×C4C22×D3211yescharacteristic
442Dic3—24no—
443Dic3—22no—
444Dic3—34no—
445Dic3—62no—
446C32—21no—
447C32—22no—
448C32—41no—
449C32C2×#91812yescomplemented
450C32C2×#94211yescharacteristic, complemented
451C23—91no—
452C23—541no—
453Q8—34no—
454Q8—32no—
455Q8—62no—
456Q8—61no—
457Q8—184no—
458D4—272no—
459D4—544no—
460D4—541no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—91no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—184no—
467C2×C4—544no—
468C2×C4D3311yescharacteristic
469C8—94no—
470C2×C3—21no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—24no—
474C2×C3—41no—
475C2×C3—42no—
476C2×C3—182no—
477C2×C3#6394512yescomplemented
478C2×C3#6394814yescomplemented
479C2×C3C2×D3×#6712yes—
480C2×C3C2×#381111yescharacteristic
481D3—64no—
482D3—122no—
483D3—364no—
484D3—362no—
485D3—364no—
486D3—724no—
487D3—724no—
488C22—92no—
489C22—182no—
490C22—542no—
491C22—1082no—
492C22#8895211yescharacteristic
493C4—64no—
494C4—62no—
495C4—92no—
496C4#8895211yescharacteristic
497C4C2×D3311yescharacteristic
498C3—21no—
499C3—22no—
500C3—41no—
501C3C2×#6394511yescharacteristic, complemented
502C3C2×#6394812yescomplemented
503C2—182no—
504C2—1082no—
505C2#1172289012yesdirect product factor
506C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#89018C2
hypercenterC2×C4D33
hypocenterC33C2×#137
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×C3×#3797C2semidirect
3. C2×#88809C2semidirect
4. C2×#88795C2semidirect
5. C2×C3×#3718C2semidirect
6. #11722890C2direct (N × H)
20. C3×#3797C22semidirect
22. #88809C22semidirect
23. #88795C22semidirect
24. C3×#3718C22semidirect
29. C2×#3797D3semidirect
35. C2×#3718D3semidirect
96. #3797C2×D3semidirect
104. #3718C2×D3semidirect
204. C2×C33#137semidirect
303. C33C2×#137semidirect
377. C2×C32#918semidirect
378. C2×C32#942semidirect
449. C32C2×#918semidirect
450. C32C2×#942semidirect
477. C2×C3#63945semidirect
478. C2×C3#63948semidirect
501. C3C2×#63945semidirect
502. C3C2×#63948semidirect
505. C2#11722890direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890182
C2×C3×#37972
C2×#888092
C2×#887952
C2×C3×#37182
#117228902
C2×#639483
C2×#639453