Finite Group Databasebeta

← #13150862

Subgroups — #13150862

Order 1728.

1333 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1333 conjugacy classes of subgroups — too many to draw. Only its 179 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×#88740C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721321C214yesmaximal, complemented
5#11721159C214yesmaximal, complemented
6#11721158C212yesmaximal, complemented
7#11721110C212yesmaximal, complemented
8#10595045—31nomaximal
9#10595040—31nomaximal
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235C2212yes—
13#88740—24no—
14C2×C4⋊#272C2211yescharacteristic, complemented
15C2×C4×#272C2212yescomplemented
16C2×#6229C2211yescharacteristic, complemented
17C2×C4⋊((C3⋊D3)⋊C3)—24no—
18C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
19C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
20C2×#6200C2218yescomplemented
21#3757⋊C3C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
23C4×(C3⋊Dic3)⋊C3C2212yescomplemented
24#88450C2218yescomplemented
25C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C2×C3×#235—31no—
28C2×C3×#235—61no—
29C22×C3×D12—31no—
30#64135D311yescharacteristic, complemented
31C3×#926—31no—
32#63981—34no—
33#63976—34no—
34#63914—32no—
35#63911—32no—
36#63909—34no—
37#63908—34no—
38#63864—32no—
39#63863—32no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272C2312yes—
42C22×(C3⋊D3)⋊C3—24no—
43C22×(C3⋊D3)⋊C3—44no—
44C22×(C3⋊D3)⋊C3C2314yes—
45C2×C4×H(3,𝔽3)D414yescomplemented
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#6235—24no—
49C2×#1084C2312yes—
50C4⋊#272—22no—
51C4×#272—24no—
52#6229—22no—
53C2×(C3⋊Dic3)⋊C3C2314yes—
54C4⋊((C3⋊D3)⋊C3)—24no—
55C4⋊((C3⋊D3)⋊C3)—24no—
56#6200—48no—
57#5830—91no—
58#5805—91no—
59#5525—91no—
60C23×C3⋊D3C2×D312yes—
61C23×C3×D3—32no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C2×C4⋊(C3⋊D3)—24no—
66C2×C4⋊(C3⋊D3)C2×D312yes—
67C2×C4⋊(C3⋊D3)C2×D312yes—
68C2×C3×#67—32no—
69C2×C3×#67—32no—
70C2×C3×#67—62no—
71C3×#235—68no—
72C3×#235—64no—
73C3×#235—64no—
74C2×C3×D12—32no—
75C2×C3×D12—32no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×D12—64no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—62no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×#626—38no—
87C2×#625—38no—
88#3757C2×D312yes—
89C4⋊(C3⋊Dic3)C2×D312yescomplemented
90C4×C3⋊Dic3C2×D312yescomplemented
91C3×#217—32no—
92C3×C4⋊Dic3—32no—
93C3×C4×Dic3—32no—
94#3722—38no—
95(C2×D3)⋊Dic3—38no—
96C22×H(3,𝔽3)—22no—
97C22×H(3,𝔽3)C2×D412yes—
98C22×H(3,𝔽3)C2411yescharacteristic
99C2×#272—22no—
100C2×#272—24no—
101C2×(C3⋊D3)⋊C3—24no—
102C2×(C3⋊D3)⋊C3—28no—
103C2×(C3⋊D3)⋊C3—48no—
104C4×H(3,𝔽3)C2×D412yes—
105C4×H(3,𝔽3)C2×D412yes—
106#1084—22no—
107(C3⋊Dic3)⋊C3—24no—
108C2×C3×#41—91no—
109C22×C3×D4—91no—
110C2×#235—31no—
111C2×#235—31no—
112C2×#235—61no—
113C22×D12—31no—
114C22×D12—91no—
115C3×#117—91no—
116#953—94no—
117#947—92no—
118#940—92no—
119#940—92no—
120#938—94no—
121#937—94no—
122#936—92no—
123#930—94no—
124#926—31no—
125#926—91no—
126#895—94no—
127#882—94no—
128#875—92no—
129#874—92no—
130C23×C32—21no—
131C23×C32C22×D311yescharacteristic
132C23×C32C22×D311yescharacteristic
133C22×C3⋊D3—24no—
134C22×C3⋊D3—44no—
135C22×C3⋊D3C22×D314yes—
136C22×C3×D3—34no—
137C22×C3×D3—32no—
138C22×C3×D3—32no—
139C22×C3×D3—62no—
140C22×C3×D3—64no—
141C22×C3×D3—124no—
142C2×C32×C4—24no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4D1214yescomplemented
146C2×C32×C4#6714yescomplemented
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3C22×D314yes—
152C4⋊(C3⋊D3)—24no—
153C4⋊(C3⋊D3)—24no—
154C3×#67—68no—
155C3×#67—64no—
156C3×#67—64no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—62no—
161C3×D12—64no—
162C3×D12—64no—
163C3×D12—64no—
164C3×D12—62no—
165C3×D12—62no—
166C3×C4×D3—68no—
167C3×C4×D3—64no—
168C3×C4×D3—64no—
169C3×Dic6—64no—
170C3×Dic6—62no—
171C3×Dic6—62no—
172#626—128no—
173#625—128no—
174#535—271noSylow
175C2×H(3,𝔽3)—22no—
176C2×H(3,𝔽3)C22×D411yescharacteristic
177C2×H(3,𝔽3)H(5,𝔽2)12yes—
178#272—42no—
179(C3⋊D3)⋊C3—44no—
180C24×C3—92no—
181C23×D3—32no—
182C23×D3—92no—
183C3×#41—184no—
184C2×C3×Q8—91no—
185C2×C3×D4—92no—
186C2×C3×D4—92no—
187C2×C3×D4—92no—
188C2×C3×D4—91no—
189C2×C3×D4—184no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—61no—
193C22×C3×C4—92no—
194C22×C3×C4D3211yescharacteristic
195C2×#67—32no—
196C2×#67—32no—
197C2×#67—62no—
198C2×#67—98no—
199C2×#67—98no—
200#235—64no—
201#235—64no—
202#235—68no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—32no—
206C2×D12—32no—
207C2×D12—61no—
208C2×D12—64no—
209C2×D12—92no—
210C2×D12—92no—
211C2×D12—98no—
212C2×D12—184no—
213C2×C4×D3—32no—
214C2×C4×D3—32no—
215C2×C4×D3—62no—
216C2×Dic6—31no—
217C2×Dic6—31no—
218C2×Dic6—61no—
219C3×C4⋊C4—92no—
220C3×C22⋊C4—92no—
221C3×C42—92no—
222#217—32no—
223#217—92no—
224#217—98no—
225(C2×D3)⋊C4—98no—
226(C2×D3)⋊C4—98no—
227C4⋊Dic3—32no—
228C4⋊Dic3—92no—
229C4×Dic3—32no—
230C4×Dic3—92no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—22no—
235C22×C32—42no—
236C22×C32C2×D1212yes—
237C22×C32C2×#6712yes—
238C22×C32C23×D311yescharacteristic
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—28no—
241C2×C3⋊D3—24no—
242C2×C3⋊D3—48no—
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—62no—
246C2×C3×D3—64no—
247C2×C3×D3—68no—
248C2×C3×D3—64no—
249C2×C3×D3—62no—
250C2×C3×D3—124no—
251C2×C3×D3—128no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4C2×D1212yes—
255C32×C4C2×D1212yes—
256C32×C4C2×#6712yes—
257C32×C4C2×#6712yes—
258C3⋊Dic3—24no—
259C3×Dic3—64no—
260C3×Dic3—62no—
261C3×Dic3—64no—
262C3×Dic3—62no—
263C2×#41—91no—
264C22×D4—91no—
265#127—272no—
266#124—272no—
267#121—274no—
268#120—274no—
269#117—91no—
270H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
271C23×C3—31no—
272C23×C3—31no—
273C23×C3—61no—
274C23×C3—94no—
275C23×C3—92no—
276C23×C3—184no—
277C23×C3—364no—
278C23×C3C2×D3211yescharacteristic
279C22×D3—32no—
280C22×D3—32no—
281C22×D3—34no—
282C22×D3—62no—
283C22×D3—64no—
284C22×D3—94no—
285C22×D3—124no—
286C22×D3—184no—
287C22×D3—364no—
288C3×Q8—182no—
289C3×D4—184no—
290C3×D4—184no—
291C3×D4—184no—
292C3×D4—182no—
293C2×C3×C4—34no—
294C2×C3×C4—34no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—64no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—184no—
305C2×C3×C4#62614yes—
306C2×C3×C4C2×D3211yescharacteristic
307C2×C3×C4C2×D3211yescharacteristic
308#67—64no—
309#67—64no—
310#67—68no—
311#67—368no—
312#67—368no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—34no—
316C2×Dic3—62no—
317C2×Dic3—94no—
318D12—62no—
319D12—62no—
320D12—64no—
321D12—64no—
322D12—64no—
323D12—184no—
324D12—184no—
325D12—368no—
326C4×D3—64no—
327C4×D3—64no—
328C4×D3—68no—
329Dic6—62no—
330Dic6—62no—
331Dic6—64no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—21no—
336C2×C32—42no—
337C2×C32C22×D1211yescharacteristic
338C2×C32#100412yes—
339C2×C32#100912yes—
340C2×C32C22×#6711yescharacteristic
341C3⋊D3—44no—
342C3×D3—124no—
343C3×D3—124no—
344C3×D3—122no—
345C3×D3—122no—
346C24—92no—
347#41—184no—
348C2×Q8—91no—
349C2×D4—92no—
350C2×D4—92no—
351C2×D4—92no—
352C2×D4—91no—
353C2×D4—184no—
354C2×D4—278no—
355C22×C4—92no—
356C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C4⋊C4—92no—
358C22⋊C4—92no—
359C22⋊C4—278no—
360C42—92no—
361C22×C3—22no—
362C22×C3—31no—
363C22×C3—31no—
364C22×C3—32no—
365C22×C3—32no—
366C22×C3—62no—
367C22×C3—62no—
368C22×C3—61no—
369C22×C3—62no—
370C22×C3—122no—
371C22×C3—188no—
372C22×C3—184no—
373C22×C3—184no—
374C22×C3—182no—
375C22×C3—368no—
376C22×C3C2×#62612yes—
377C22×C3C22×D3211yescharacteristic
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3—68no—
384C2×D3—64no—
385C2×D3—124no—
386C2×D3—128no—
387C2×D3—184no—
388C2×D3—188no—
389C2×D3—368no—
390C3×C4—32no—
391C3×C4—32no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—184no—
397C3×C4—182no—
398C3×C4C2×#62612yes—
399C3×C4C2×#62612yes—
400Dic3—62no—
401Dic3—62no—
402Dic3—64no—
403Dic3—64no—
404Dic3—184no—
405C32—21no—
406C32#552511yescharacteristic, complemented
407C32#580511yescharacteristic, complemented
408C23—94no—
409C23—92no—
410C23—184no—
411C23—364no—
412C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413Q8—182no—
414D4—184no—
415D4—184no—
416D4—184no—
417D4—182no—
418D4—1088no—
419C2×C4—94no—
420C2×C4—92no—
421C2×C4—184no—
422C2×C4#620014yescomplemented
423C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C2×C3—22no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—32no—
429C2×C3—32no—
430C2×C3—62no—
431C2×C3—62no—
432C2×C3—61no—
433C2×C3—62no—
434C2×C3—122no—
435C2×C3—364no—
436C2×C3—362no—
437C2×C3#6430712yes—
438C2×C3C22×#62611yescharacteristic
439D3—122no—
440D3—122no—
441D3—124no—
442D3—124no—
443D3—364no—
444C22—22no—
445C22—182no—
446C22—184no—
447C22—184no—
448C22—188no—
449C22—368no—
450C22C2×#620012yes—
451C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
452C4—182no—
453C4—184no—
454C4C2×#620012yes—
455C4C2×#620012yes—
456C3—31no—
457C3—31no—
458C3—61no—
459C3#1059504211yescharacteristic
460C2—22no—
461C2—364no—
462C2—362no—
463C2#1172249212yes—
464C2C22×#620011yescharacteristic
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)#535
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×#88740C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64135⋊C3C2semidirect
4. #11721321C2semidirect
5. #11721159C2semidirect
6. #11721158C2semidirect
7. #11721110C2semidirect
14. C2×C4⋊#272C22semidirect
15. C2×C4×#272C22semidirect
16. C2×#6229C22semidirect
20. C2×#6200C22semidirect
22. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. C4×(C3⋊Dic3)⋊C3C22semidirect
24. #88450C22semidirect
25. C22×C4⋊(C3⋊D3)D3semidirect
30. #64135D3semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
89. C4⋊(C3⋊Dic3)C2×D3semidirect
90. C4×C3⋊Dic3C2×D3semidirect
145. C2×C32×C4D12semidirect
146. C2×C32×C4#67semidirect
270. H(3,𝔽3)#535semidirect
406. C32#5525semidirect
407. C32#5805semidirect
422. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C22×C4⋊((C3⋊D3)⋊C3)2
#64135⋊C32
#117213212
#117211592
#117211582
#117211102
#105950453
#105950403