Finite Group Databasebeta

← #13150860

Subgroups — #13150860

Order 1728.

1007 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 139 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×#88542C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C212yesmaximal, complemented
3#11721323C212yesmaximal, complemented
4#11721318C211yesmaximal, characteristic, complemented
5#11721309C211yesmaximal, characteristic, complemented
6#11721140C212yesmaximal, complemented
7#11721103C212yesmaximal, complemented
8#11721100C212yesmaximal, complemented
9#11721092C212yesmaximal, complemented
10#10595036—32nomaximal
11C23×#272C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
16C2×#6200C2212yes—
17C2×#6200C2212yes—
18C2×#6198C2211yescharacteristic, complemented
19C2×#6196C2211yescharacteristic, complemented
20#88542C2212yes—
21#88542C2212yes—
22#3757⋊C3C2212yes—
23#3752⋊C3C2212yescomplemented
24#3750⋊C3C2212yescomplemented
25C4×(C3⋊Dic3)⋊C3C2212yescomplemented
26#88451C2212yes—
27#88451C2212yes—
28#88450C2212yescomplemented
29#88449C2211yescharacteristic, complemented
30#88449C2211yescharacteristic, complemented
31#88448C2212yescomplemented
32C4×#638D312yescomplemented
33C3×C4×#67—32no—
34C2×C3×(C2×D3)⋊C4—32no—
35C2×C3×(C2×D3)⋊C4—61no—
36#63978—32no—
37#63977—32no—
38#63976—32no—
39#63964—32no—
40#63915—32no—
41#63912—32no—
42#63906—32no—
43#63905—32no—
44#63862—32no—
45#63859—32no—
46#63853—32no—
47#63852—32no—
48C23×H(3,𝔽3)C2311yescharacteristic
49C22×#272—22no—
50C22×#272—41no—
51C22×#272—41no—
52C22×#272C2312yes—
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)—41no—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×#1084—41no—
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59#638⋊C3—24no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4×(C3⋊D3)⋊C3—42no—
64#6200—42no—
65#6200—42no—
66#6198—41no—
67#6196—41no—
68#5560—92no—
69#5494—91no—
70C23×C3×D3—32no—
71C23×C3×D3—61no—
72C22×C32×C4—21no—
73C22×C32×C4C2×D312yes—
74C2×#638C2×D312yes—
75C2×C4×C3⋊D3C2×D312yescomplemented
76C2×C3×#67—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—61no—
79C2×C3×C4×D3—32no—
80C2×#627—32no—
81C2×#626—32no—
82C2×#626—32no—
83C2×#626—32no—
84C2×#625—32no—
85C2×#625—32no—
86#3757C2×D312yes—
87#3752C2×D312yescomplemented
88#3750C2×D312yescomplemented
89C4×C3⋊Dic3C2×D312yescomplemented
90C3×#217—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100#3724—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104#3722—32no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)#4112yes—
113C22×H(3,𝔽3)C2411yescharacteristic
114C2×#272—22no—
115C2×#272—22no—
116C2×#272—22no—
117C2×#272—44no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—22no—
121C4×H(3,𝔽3)—22no—
122#1084—21no—
123#1084—21no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127C3×C4×D4—92no—
128C2×C3×C22⋊C4—91no—
129#953—92no—
130#939—92no—
131#937—92no—
132#930—92no—
133#930—91no—
134#929—92no—
135C4×#67—32no—
136C4×#67—92no—
137C2×(C2×D3)⋊C4—32no—
138C2×(C2×D3)⋊C4—61no—
139#924—91no—
140#898—92no—
141#894—92no—
142#885—92no—
143#885—92no—
144#884—92no—
145#884—92no—
146#883—92no—
147#882—92no—
148#879—92no—
149#878—92no—
150#876—92no—
151#875—92no—
152C23×C32—21no—
153C23×C32C22×D312yes—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—62no—
159C22×C3×D3—64no—
160C22×C3×D3—64no—
161C22×C3×D3—122no—
162C22×C3×D3—122no—
163C22×C3×D3—122no—
164C22×C3×D3—122no—
165C2×C32×C4—22no—
166C2×C32×C4—42no—
167C2×C32×C4—42no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D312yes—
170#638—24no—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D312yes—
174C4×C3⋊D3—42no—
175C3×#67—64no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—122no—
184C2×C3×Dic3—122no—
185C3×C4×D3—122no—
186#627—122no—
187#626—122no—
188#626—122no—
189#626—122no—
190#625—122no—
191#625—122no—
192#538—271noSylow
193C2×H(3,𝔽3)—22no—
194C2×H(3,𝔽3)C2×#4111yescharacteristic
195C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
196C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
197#272—42no—
198(C3⋊D3)⋊C3—42no—
199C24×C3—91no—
200C23×D3—32no—
201C23×D3—61no—
202C2×C3×D4—92no—
203C22×C3×C4—32no—
204C22×C3×C4—61no—
205C22×C3×C4—92no—
206C22×C3×C4—91no—
207C22×C3×C4D3211yescharacteristic
208C2×#67—32no—
209C2×#67—92no—
210C2×#67—92no—
211C2×#67—92no—
212C2×#67—92no—
213C2×#67—92no—
214C2×#67—92no—
215C22×Dic3—32no—
216C22×Dic3—61no—
217C2×D12—91no—
218C2×D12—92no—
219C2×D12—92no—
220C2×C4×D3—32no—
221C2×C4×D3—92no—
222C2×Dic6—91no—
223C2×Dic6—92no—
224C3×C4⋊C4—92no—
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4—92no—
229C3×C42—92no—
230#217—32no—
231#217—92no—
232#217—92no—
233#217—92no—
234#217—92no—
235(C2×D3)⋊C4—32no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—32no—
238(C2×D3)⋊C4—32no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—62no—
241(C2×D3)⋊C4—62no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252Dic3⋊C4—32no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258C4×Dic3—32no—
259C4×Dic3—92no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—22no—
266C22×C32#23514yes—
267C22×C32C23×D312yes—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3×D3—64no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—122no—
282C2×C3×D3—122no—
283C2×C3×D3—128no—
284C2×C3×D3—128no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—42no—
288C3⋊Dic3—22no—
289C3⋊Dic3—22no—
290C3⋊Dic3—22no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298#126—272no—
299#124—272no—
300#123—272no—
301C22⋊Q8—271no—
302#121—272no—
303#121—271no—
304#120—272no—
305C4×D4—92no—
306C2×C22⋊C4—91no—
307H(3,𝔽3)#53811yescharacteristic, complemented, Sylow
308C23×C3—32no—
309C23×C3—61no—
310C23×C3—92no—
311C23×C3—92no—
312C23×C3—182no—
313C23×C3—361no—
314C23×C3—361no—
315C23×C3C2×D3211yescharacteristic
316C22×D3—32no—
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—62no—
320C22×D3—64no—
321C22×D3—64no—
322C22×D3—92no—
323C22×D3—122no—
324C22×D3—122no—
325C22×D3—122no—
326C22×D3—122no—
327C3×D4—184no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—41no—
331C2×C3×C4—62no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—122no—
338C2×C3×C4—122no—
339C2×C3×C4—362no—
340C2×C3×C4—361no—
341C2×C3×C4C2×D3212yes—
342#67—64no—
343#67—184no—
344#67—362no—
345#67—362no—
346#67—362no—
347#67—362no—
348#67—362no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—92no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—122no—
360C2×Dic3—122no—
361D12—361no—
362D12—362no—
363D12—362no—
364C4×D3—122no—
365C4×D3—362no—
366Dic6—361no—
367Dic6—362no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32C2×#23512yes—
374C2×C32#100412yes—
375C2×C32#100912yes—
376C3⋊D3—42no—
377C3×D3—124no—
378C3×D3—122no—
379C3×D3—122no—
380C3×D3—122no—
381C24—91no—
382C2×Q8—271no—
383C2×D4—92no—
384C2×D4—272no—
385C2×D4—272no—
386C2×D4—271no—
387C22×C4—91no—
388C22×C4—92no—
389C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
390C4⋊C4—92no—
391C4⋊C4—272no—
392C4⋊C4—271no—
393C4⋊C4—271no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C42—92no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—32no—
405C22×C3—34no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—61no—
411C22×C3—62no—
412C22×C3—182no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—364no—
419C22×C3#379812yes—
420C22×C3C22×D3211yescharacteristic
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—64no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—122no—
432C2×D3—122no—
433C2×D3—128no—
434C2×D3—128no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C3×C4—22no—
439C3×C4—62no—
440C3×C4—62no—
441C3×C4—122no—
442C3×C4—182no—
443C3×C4—182no—
444C3×C4—182no—
445C3×C4—181no—
446C3×C4—181no—
447Dic3—62no—
448Dic3—62no—
449Dic3—62no—
450Dic3—62no—
451Dic3—62no—
452Dic3—121no—
453Dic3—121no—
454Dic3—182no—
455Dic3—182no—
456Dic3—182no—
457C32—21no—
458C32#556012yescomplemented
459C23—92no—
460C23—92no—
461C23—182no—
462C23—361no—
463C23—361no—
464C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
465Q8—1081no—
466D4—184no—
467D4—1082no—
468D4—1081no—
469D4—1082no—
470C2×C4—41no—
471C2×C4—92no—
472C2×C4—92no—
473C2×C4—91no—
474C2×C4—91no—
475C2×C4—92no—
476C2×C4—362no—
477C2×C4—361no—
478C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
479C2×C3—22no—
480C2×C3—32no—
481C2×C3—32no—
482C2×C3—32no—
483C2×C3—64no—
484C2×C3—64no—
485C2×C3—61no—
486C2×C3—61no—
487C2×C3—61no—
488C2×C3—362no—
489C2×C3—362no—
490C2×C3C2×#379811yescharacteristic
491C2×C3#6430811yescharacteristic
492C2×C3#6433011yescharacteristic
493D3—122no—
494D3—122no—
495D3—124no—
496D3—122no—
497D3—362no—
498C22—21no—
499C22—21no—
500C22—182no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22—364no—
507C22#8865012yes—
508C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C4—22no—
510C4—182no—
511C4—182no—
512C4—181no—
513C4—181no—
514C4—182no—
515C3—32no—
516C3—61no—
517C3#1059503911yescharacteristic
518C2—22no—
519C2—362no—
520C2—362no—
521C2C2×#8865011yescharacteristic
522C2#1172249311yescharacteristic
523C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#538
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88542C2semidirect
2. C4×#638⋊C3C2semidirect
3. #11721323C2semidirect
4. #11721318C2semidirect
5. #11721309C2semidirect
6. #11721140C2semidirect
7. #11721103C2semidirect
8. #11721100C2semidirect
9. #11721092C2semidirect
15. C2×C4×(C3⋊D3)⋊C3C22semidirect
18. C2×#6198C22semidirect
19. C2×#6196C22semidirect
23. #3752⋊C3C22semidirect
24. #3750⋊C3C22semidirect
25. C4×(C3⋊Dic3)⋊C3C22semidirect
28. #88450C22semidirect
29. #88449C22semidirect
30. #88449C22semidirect
31. #88448C22semidirect
32. C4×#638D3semidirect
75. C2×C4×C3⋊D3C2×D3semidirect
87. #3752C2×D3semidirect
88. #3750C2×D3semidirect
89. C4×C3⋊Dic3C2×D3semidirect
307. H(3,𝔽3)#538semidirect
458. C32#5560semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C4×#638⋊C32
#117213232
#117213182
#117213092
#117211402
#117211032
#117211002
#117210922
#105950363