Finite Group Databasebeta

← C2×#11722889 #13170686

Subgroups — C2×#11722889 #13170686

Order 1728.

1354 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1354 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×#89014C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4C2×#88799C212yesmaximal, complemented
5C2×C3×#3715C212yesmaximal, complemented
6#11722889C218yesmaximal, direct product factor
7C2×#63946—32nomaximal
8C2×#63931—31nomaximal
9C2×#6308C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×#6290—21no—
14C2×D3×C3⋊Dic3—21no—
15#89014—22no—
16#89014C2214yes—
17C2×C3×#627—21no—
18C2×C3×#624—21no—
19C3×#3797—22no—
20C3×#3797C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yes—
22#88809C2214yescomplemented
23#88799C2218yescomplemented
24C3×#3715C2218yescomplemented
25C2×#3830—31no—
26C2×C3×D3×Q8—32no—
27C2×#3797D311yescharacteristic, complemented
28C2×#3795—32no—
29C2×#3795—61no—
30C2×#3756—32no—
31C2×C3×#215—32no—
32C2×C3×#214—32no—
33#63946—316no—
34#63931—38no—
35C2×#3719—32no—
36C2×#3719—32no—
37C2×#3719—62no—
38C2×#3715D312yescomplemented
39C2×#3712—31no—
40C2×#3711—32no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308C2312yes—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3D412yes—
49C3×C4×C3⋊D3C2312yes—
50C2×C32×Dic3—22no—
51C32×D12—21no—
52C32×D12C2312yes—
53C32×Dic6—22no—
54C32×Dic6C2314yes—
55#6290—24no—
56D3×C3⋊Dic3—24no—
57C3×#627—24no—
58C3×#624—24no—
59C3×C3⋊(C3⋊C8)C2314yes—
60C2×#942—91no—
61C2×#915—92no—
62C2×C32×Q8—32no—
63C2×C32×D4—31no—
64C2×#638—61no—
65C22×C3⋊Dic3—61no—
66#3830—34no—
67#3830—62no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—32no—
70C2×#634—31no—
71C2×#634—61no—
72C2×#634—62no—
73C2×#634—121no—
74C3×D3×Q8—38no—
75C3×D3×Q8—64no—
76C2×C3×D12—21no—
77C2×C3×D12C2×D312yes—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—62no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6—62no—
83C2×C3×Dic6C2×D312yes—
84C2×C3×Dic6C2×D312yes—
85C2×#627—21no—
86C2×#625—62no—
87C2×#625—62no—
88C2×#624—21no—
89C2×D3×Dic3—62no—
90C2×D3×Dic3—62no—
91#3797—22no—
92#3797C2×D314yescomplemented
93#3795—38no—
94#3795—64no—
95#3795—64no—
96#3795—64no—
97#3756—38no—
98C2×C3⋊(C3⋊C8)C2×D312yes—
99C3×#215—38no—
100C3×#214—38no—
101C2×C3×C3⋊C8—32no—
102C2×C3×C3⋊C8—32no—
103C2×C3×C3⋊C8—62no—
104#3719—38no—
105#3719—38no—
106#3719—68no—
107#3715C2×D318yescomplemented
108#3712—34no—
109#3711—38no—
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3C2×D412yes—
113C2×C32×D3—22no—
114C2×C32×D3—42no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2411yescharacteristic
117C3⋊(C3⋊Dic3)—22no—
118C3×C3⋊Dic3C2×D412yes—
119C32×Dic3—24no—
120C22×C3×Q8—91no—
121C2×D3×Q8—32no—
122C2×#237—92no—
123C2×#237—181no—
124C2×#235—91no—
125C2×C3×SD16—92no—
126C2×#216—92no—
127C2×#216—92no—
128C2×#216—182no—
129#942—98no—
130C2×#215—32no—
131C2×#214—32no—
132C2×#208—91no—
133#915—916no—
134C2×Dic12—92no—
135C2×ΓL(1,𝔽25)—92no—
136C23×C32—61no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—32no—
141C22×C3×D3—61no—
142C32×Q8—34no—
143C32×Q8—62no—
144C32×D4—32no—
145C32×D4—61no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—62no—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—64no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—64no—
155C2×C3⋊Dic3—62no—
156C2×C3⋊Dic3—62no—
157C2×C3⋊Dic3—122no—
158C2×C3⋊Dic3—122no—
159C2×C3⋊Dic3#6711yescharacteristic
160C4×C3⋊D3#6712yes—
161C4×C3⋊D3C22×D312yes—
162#634—32no—
163#634—34no—
164#634—62no—
165#634—64no—
166#634—61no—
167#634—62no—
168#634—62no—
169#634—122no—
170#634—122no—
171#634—122no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—42no—
176C2×C3×Dic3—61no—
177C3×D12—22no—
178C3×D12—22no—
179C3×D12—22no—
180C3×D12C22×D314yes—
181C3×C4×D3—34no—
182C3×C4×D3—34no—
183C3×C4×D3—62no—
184C3×C4×D3—62no—
185C3×C4×D3—68no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—24no—
189C3×Dic6—42no—
190C3×Dic6—68no—
191C3×Dic6C22×D314yes—
192C3×Dic6C22×D314yes—
193#627—24no—
194#625—68no—
195#625—68no—
196#624—24no—
197D3×Dic3—68no—
198D3×Dic3—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—22no—
207C32×D3—42no—
208C2×C3×Q8—32no—
209C2×C3×Q8—32no—
210C2×C3×Q8—62no—
211C2×C3×Q8—94no—
212C2×C3×Q8—181no—
213C2×C3×Q8—182no—
214C2×C3×D4—32no—
215C2×C3×D4—61no—
216C22×C3×C4—91no—
217C22×C3×C4—181no—
218C2×#67—181no—
219C2×#67—182no—
220C2×#67—182no—
221C22×Dic3—182no—
222C22×Dic3—182no—
223D3×Q8—38no—
224D3×Q8—64no—
225#237—98no—
226#237—184no—
227#237—184no—
228#237—184no—
229#235—94no—
230#235—182no—
231C2×D12D3211yescharacteristic
232C2×C4×D3—32no—
233C2×C4×D3—61no—
234C2×C4×D3—62no—
235C2×C4×D3—181no—
236C2×Dic6—62no—
237C2×Dic6—92no—
238C2×Dic6—91no—
239C2×Dic6—181no—
240C2×Dic6—182no—
241C2×Dic6—361no—
242C2×Dic6D3212yes—
243C3×SD16—98no—
244C2×C3×C8—92no—
245#216—98no—
246#216—98no—
247#216—188no—
248#215—38no—
249#214—38no—
250#208—94no—
251C2×C3⋊C8—32no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—62no—
254Dic12—98no—
255ΓL(1,𝔽25)—98no—
256C22×C32—22no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32—62no—
260C22×C32—122no—
261C22×C32D3×D412yes—
262C22×C32C2×#6711yescharacteristic
263C2×C3⋊D3—22no—
264C2×C3⋊D3C2×#6712yes—
265C2×C3×D3—24no—
266C2×C3×D3—24no—
267C2×C3×D3—34no—
268C2×C3×D3—44no—
269C2×C3×D3—44no—
270C2×C3×D3—62no—
271C2×C3×D3—64no—
272C2×C3×D3—122no—
273C32×C4—22no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—21no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—64no—
280C32×C4D3×D412yes—
281C32×C4C2×#6711yescharacteristic
282C32×C4C23×D312yes—
283C32×C4C23×D311yescharacteristic
284C3⋊Dic3—62no—
285C3⋊Dic3—64no—
286C3⋊Dic3—64no—
287C3⋊Dic3—124no—
288C3⋊Dic3—124no—
289C3⋊Dic3C2×#6712yes—
290C3×Dic3—24no—
291C3×Dic3—24no—
292C3×Dic3—34no—
293C3×Dic3—44no—
294C3×Dic3—62no—
295C2×#41—271no—
296C22×Q8—91no—
297#137—278no—
298C2×Q16—272no—
299C2×SD16—92no—
300C2×M16—271no—
301C33C2×#13711yescharacteristic, complemented, Sylow
302C23×C3—62no—
303C23×C3—62no—
304C23×C3—91no—
305C22×D3—21no—
306C22×D3—32no—
307C22×D3—61no—
308C3×Q8—34no—
309C3×Q8—34no—
310C3×Q8—64no—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—122no—
314C3×Q8—184no—
315C3×D4—34no—
316C3×D4—62no—
317C3×D4—62no—
318C3×D4—62no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—91no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4—122no—
328C2×C3×C4—184no—
329C2×C3×C4C2×D3212yes—
330C2×C3×C4C2×D3211yescharacteristic
331#67—184no—
332#67—188no—
333#67—188no—
334C2×Dic3—22no—
335C2×Dic3—32no—
336C2×Dic3—61no—
337C2×Dic3—188no—
338C2×Dic3—188no—
339C2×Dic3—182no—
340C2×Dic3—182no—
341C2×Dic3—181no—
342C2×Dic3—362no—
343C2×Dic3—362no—
344D12—21no—
345D12C2×D3212yes—
346C4×D3—34no—
347C4×D3—34no—
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—68no—
351C4×D3—184no—
352Dic6—22no—
353Dic6—68no—
354Dic6—94no—
355Dic6—92no—
356Dic6—182no—
357Dic6—184no—
358Dic6—182no—
359Dic6—182no—
360Dic6—181no—
361Dic6—362no—
362Dic6—362no—
363Dic6—362no—
364Dic6C2×D3214yes—
365C3×C8—94no—
366C3⋊C8—34no—
367C3⋊C8—34no—
368C3⋊C8—64no—
369C2×C32—22no—
370C2×C32—21no—
371C2×C32—22no—
372C2×C32—24no—
373C2×C32—42no—
374C2×C32—41no—
375C2×C32—122no—
376C2×C32#91514yescomplemented
377C2×C32#94212yescomplemented
378C2×C32C2×D3×D412yes—
379C2×C32C22×#6711yescharacteristic
380C3⋊D3—22no—
381C3×D3—44no—
382C3×D3—44no—
383C3×D3—64no—
384C3×D3—122no—
385#41—274no—
386#41—542no—
387C2×Q8—32no—
388C2×Q8—94no—
389C2×Q8—181no—
390C2×Q8—182no—
391C2×Q8—271no—
392C2×D4—31no—
393C2×D4—541no—
394C22×C4—91no—
395C22×C4—181no—
396C22×C4—541no—
397Q16—278no—
398SD16—98no—
399M16—274no—
400C2×C8—92no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—64no—
405C22×C3—64no—
406C22×C3—92no—
407C22×C3—124no—
408C22×C3—124no—
409C22×C3—182no—
410C22×C3D3×#6712yes—
411C22×C3#381111yescharacteristic
412C2×D3—22no—
413C2×D3—34no—
414C2×D3—42no—
415C2×D3—62no—
416C2×D3—64no—
417C2×D3—122no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—64no—
425C3×C4—64no—
426C3×C4—92no—
427C3×C4—124no—
428C3×C4D3×#6712yes—
429C3×C4#381111yescharacteristic
430C3×C4C22×D3212yes—
431C3×C4C22×D3211yescharacteristic
432Dic3—24no—
433Dic3—34no—
434Dic3—62no—
435Dic3—184no—
436Dic3—184no—
437Dic3—182no—
438Dic3—364no—
439Dic3—364no—
440C32—22no—
441C32—21no—
442C32—41no—
443C32C2×#91512yescomplemented
444C32C2×#94211yescharacteristic, complemented
445C23—61no—
446C23—91no—
447Q8—34no—
448Q8—62no—
449Q8—184no—
450Q8—272no—
451Q8—541no—
452D4—32no—
453D4—61no—
454D4—544no—
455C2×C4—62no—
456C2×C4—91no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—184no—
460C2×C4—544no—
461C2×C4—541no—
462C2×C4D3311yescharacteristic
463C8—94no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—24no—
468C2×C3—41no—
469C2×C3—42no—
470C2×C3—124no—
471C2×C3—124no—
472C2×C3—182no—
473C2×C3#6393112yescomplemented
474C2×C3#6394614yescomplemented
475C2×C3C2×D3×#6712yes—
476C2×C3C2×#381111yescharacteristic
477D3—42no—
478D3—64no—
479D3—122no—
480C22—62no—
481C22—92no—
482C22—122no—
483C22—182no—
484C22#8895211yescharacteristic
485C4—64no—
486C4—92no—
487C4—542no—
488C4#8895211yescharacteristic
489C4C2×D3311yescharacteristic
490C3—21no—
491C3—22no—
492C3—41no—
493C3C2×#6393111yescharacteristic, complemented
494C3C2×#6394612yescomplemented
495C2—122no—
496C2—182no—
497C2#1172288912yesdirect product factor
498C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#89014C2
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×#88799C2semidirect
5. C2×C3×#3715C2semidirect
6. #11722889C2direct (N × H)
20. C3×#3797C22semidirect
22. #88809C22semidirect
23. #88799C22semidirect
24. C3×#3715C22semidirect
27. C2×#3797D3semidirect
38. C2×#3715D3semidirect
92. #3797C2×D3semidirect
107. #3715C2×D3semidirect
204. C2×C33#137semidirect
301. C33C2×#137semidirect
376. C2×C32#915semidirect
377. C2×C32#942semidirect
443. C32C2×#915semidirect
444. C32C2×#942semidirect
473. C2×C3#63931semidirect
474. C2×C3#63946semidirect
493. C3C2×#63931semidirect
494. C3C2×#63946semidirect
497. C2#11722889direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890142
C2×C3×#37972
C2×#888092
C2×#887992
C2×C3×#37152
#117228892
C2×#639463
C2×#639313