Finite Group Databasebeta

← #13147901

Subgroups — #13147901

Order 1728.

1231 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1231 conjugacy classes of subgroups — too many to draw. Only its 171 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×#88654C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊Q8C211yesmaximal, characteristic, complemented
3#11721721C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
6#11721160C214yesmaximal, complemented
7#11721114C211yesmaximal, characteristic, complemented
8#11721110C211yesmaximal, characteristic, complemented
9#11721103C214yesmaximal, complemented
10#10594009—31nomaximal
11D3×#875—31nomaximal
12C22×((C3⋊D3)⋊C3)⋊C2C2212yes—
13C2×C4⋊#272C2211yescharacteristic
14C2×#6229C2211yescharacteristic, complemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic, complemented
19C2×#6200C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×#6196C2212yescomplemented
22#88654—24no—
23((C3⋊D3)⋊C3)⋊Q8—24no—
24C42×H(3,𝔽3)C2211yescharacteristic, complemented
25#88542C2214yescomplemented
26#3752⋊C3C2214yescomplemented
27C4×(C3⋊Dic3)⋊C3C2212yescomplemented
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
29#88451C2214yescomplemented
30#88450C2214yescomplemented
31C2×#3802—31no—
32C2×D3×D12—31no—
33C2×#3797—31no—
34C2×D3×Dic6—31no—
35#64084D311yescharacteristic, complemented
36C42×C3⋊D3D311yescharacteristic, complemented
37C3×#875—31no—
38C3×#875—31no—
39C3×#875—31no—
40C3×#875—61no—
41C3×C42×D3—31no—
42#63921—34no—
43D3×(C2×D3)⋊C4—34no—
44#63915—34no—
45#63864—31no—
46#63863—31no—
47#63863—31no—
48#63860—31no—
49#63852—34no—
50C22×#272C2312yes—
51C22×(C3⋊D3)⋊C3C2312yes—
52C22×(C3⋊D3)⋊C3C2311yescharacteristic
53C2×((C3⋊D3)⋊C3)⋊C2—24no—
54C2×((C3⋊D3)⋊C3)⋊C2—28no—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084C2312yes—
58C4⋊#272—22no—
59#6229—22no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
63C4⋊((C3⋊D3)⋊C3)—22no—
64C4×(C3⋊D3)⋊C3—24no—
65C4×(C3⋊D3)⋊C3D414yes—
66#634⋊C3—22no—
67#6200—24no—
68((C3⋊D3)⋊C3)⋊C4—24no—
69#6196—24no—
70D3×#124—91no—
71D3×#124—91no—
72C2×#875—91no—
73C22×D32—32no—
74C22×D32—32no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×C4×C3⋊D3C2×D312yes—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic, complemented
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#627—32no—
90C2×#627—32no—
91C2×#626—32no—
92C2×#625—32no—
93C2×#624—32no—
94C2×D3×Dic3—32no—
95#3802—64no—
96D3×D12—64no—
97#3797—64no—
98D3×Dic6—64no—
99C32×C42—21no—
100C32×C42C2×D311yescharacteristic, complemented
101C32×C42C2×D311yescharacteristic, complemented
102#3752C2×D314yescomplemented
103C4×C3⋊Dic3C2×D311yescharacteristic, complemented
104C4×C3⋊Dic3C2×D312yescomplemented
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×(C2×D3)⋊C4—34no—
108C3×(C2×D3)⋊C4—64no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—31no—
111#3722—34no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—34no—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×#272—22no—
117C2×#272—24no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—24no—
120C2×(C3⋊D3)⋊C3C2×D412yes—
121C2×(C3⋊D3)⋊C3#4114yes—
122((C3⋊D3)⋊C3)⋊C2—48no—
123C4×H(3,𝔽3)—22no—
124C4×H(3,𝔽3)C2×D412yes—
125#1084—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3C2×D412yes—
129C2×D3×Q8—91no—
130C2×D3×Q8—91no—
131C2×D3×D4—91no—
132C2×D3×D4—91no—
133C22×D12—91no—
134C22×Dic6—91no—
135C3×#124—91no—
136C3×#124—91no—
137C2×C3×C42—91no—
138#947—91no—
139#947—91no—
140#936—91no—
141#936—91no—
142C2×(C2×D3)⋊C4—94no—
143#885—94no—
144#885—94no—
145D3×C22⋊C4—94no—
146D3×C22⋊C4—94no—
147#875—31no—
148#875—31no—
149#875—31no—
150#875—61no—
151#875—91no—
152#875—94no—
153#875—91no—
154#875—91no—
155C42×D3—31no—
156C42×D3—91no—
157C22×C3⋊D3C22×D312yes—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—31no—
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—32no—
163C22×C3×D3—62no—
164C2×D32—68no—
165C2×D32—64no—
166C2×D32—64no—
167C2×D32—68no—
168C2×C32×C4—21no—
169C2×C32×C4—22no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D311yescharacteristic
177C4⋊(C3⋊D3)—22no—
178C4×C3⋊D3—24no—
179C4×C3⋊D3D1214yes—
180#634—22no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—62no—
187C3×D12—62no—
188C3×D12—62no—
189C3×D12—62no—
190C3×D12—122no—
191C3×C4×D3—34no—
192C3×C4×D3—64no—
193C3×Dic6—62no—
194C3×Dic6—62no—
195C3×Dic6—62no—
196C3×Dic6—122no—
197#627—64no—
198#627—64no—
199#626—64no—
200#625—64no—
201#624—64no—
202D3×Dic3—64no—
203C2×#124—271noSylow
204C2×H(3,𝔽3)C22×D411yescharacteristic
205C2×H(3,𝔽3)C2×#4112yes—
206#272—42no—
207(C3⋊D3)⋊C3—42no—
208(C3⋊D3)⋊C3#12414yescomplemented
209C23×D3—92no—
210C23×D3—92no—
211C23×D3—92no—
212C2×C3×Q8—91no—
213C2×C3×Q8—91no—
214C2×C3×D4—91no—
215C2×C3×D4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—92no—
218C2×#67—92no—
219C2×#67—92no—
220C22×Dic3—92no—
221D3×Q8—184no—
222D3×Q8—184no—
223D3×D4—184no—
224D3×D4—184no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—31no—
228C2×D12—61no—
229C2×D12—91no—
230C2×D12—92no—
231C2×D12—184no—
232C2×C4×D3—31no—
233C2×C4×D3—32no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6—91no—
243C2×Dic6—92no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C2×Dic6—184no—
247C3×C22⋊C4—94no—
248C3×C22⋊C4—94no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252C3×C42—91no—
253C3×C42—92no—
254C3×C42D3211yescharacteristic
255#217—94no—
256#217—94no—
257(C2×D3)⋊C4—34no—
258(C2×D3)⋊C4—34no—
259(C2×D3)⋊C4—34no—
260(C2×D3)⋊C4—64no—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—94no—
266C4×Dic3—32no—
267C4×Dic3—31no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C22×C32—21no—
271C22×C32C23×D311yescharacteristic
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3—24no—
275C2×C3⋊D3C2×D1212yes—
276C2×C3⋊D3#23514yes—
277C2×C3×D3—32no—
278C2×C3×D3—34no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—64no—
282C2×C3×D3—64no—
283C2×C3×D3—62no—
284C2×C3×D3—64no—
285C2×C3×D3—122no—
286C2×C3×D3—124no—
287D32—128no—
288D32—128no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—42no—
293C32×C4C2×D1212yes—
294C32×C4D3×D412yes—
295C3⋊Dic3—22no—
296C3⋊Dic3—22no—
297C3⋊Dic3C2×D1212yes—
298C3×Dic3—32no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—122no—
304C22×Q8—271no—
305C22×D4—271no—
306#124—91no—
307#124—91no—
308#124—274no—
309#124—271no—
310#124—271no—
311C2×C22⋊C4—274no—
312C2×C42—91no—
313H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
314C23×C3—91no—
315C23×C3—92no—
316C23×C3—92no—
317C22×D3—31no—
318C22×D3—32no—
319C22×D3—32no—
320C22×D3—32no—
321C22×D3—62no—
322C22×D3—91no—
323C22×D3—92no—
324C22×D3—184no—
325C22×D3—188no—
326C22×D3—184no—
327C22×D3—188no—
328C22×D3—184no—
329C22×D3—188no—
330C3×Q8—182no—
331C3×Q8—182no—
332C3×D4—182no—
333C3×D4—182no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—61no—
339C2×C3×C4—62no—
340C2×C3×C4—94no—
341C2×C3×C4—91no—
342C2×C3×C4—92no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—184no—
346C2×C3×C4C2×D3212yes—
347C2×C3×C4C2×D3211yescharacteristic
348#67—184no—
349#67—184no—
350C2×Dic3—31no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—62no—
356C2×Dic3—91no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—184no—
360D12—62no—
361D12—62no—
362D12—62no—
363D12—122no—
364D12—182no—
365D12—184no—
366C4×D3—34no—
367C4×D3—64no—
368C4×D3—94no—
369C4×D3—184no—
370C4×D3—184no—
371C4×D3—184no—
372Dic6—62no—
373Dic6—62no—
374Dic6—62no—
375Dic6—122no—
376Dic6—182no—
377Dic6—184no—
378Dic6—184no—
379Dic6—184no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32C22×D1211yescharacteristic
383C2×C32C2×#23512yes—
384C2×C32C2×D3×D411yescharacteristic
385C2×C32D3×#4112yes—
386C3⋊D3—42no—
387C3⋊D3#87514yescomplemented
388C3×D3—34no—
389C3×D3—122no—
390C3×D3—122no—
391C3×D3—122no—
392C3×D3—242no—
393C24—272no—
394C2×Q8—91no—
395C2×Q8—91no—
396C2×Q8—272no—
397C2×Q8—544no—
398C2×D4—91no—
399C2×D4—91no—
400C2×D4—272no—
401C2×D4—544no—
402C22×C4—91no—
403C22×C4—92no—
404C22×C4—272no—
405C22⋊C4—94no—
406C22⋊C4—94no—
407C22⋊C4—274no—
408C22⋊C4—274no—
409C42—92no—
410C42—91no—
411C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
412C22×C3—31no—
413C22×C3—31no—
414C22×C3—61no—
415C22×C3—92no—
416C22×C3—94no—
417C22×C3—182no—
418C22×C3—184no—
419C22×C3—182no—
420C22×C3—184no—
421C22×C3C22×D3211yescharacteristic
422C2×D3—32no—
423C2×D3—34no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—62no—
429C2×D3—64no—
430C2×D3—92no—
431C2×D3—94no—
432C2×D3—122no—
433C2×D3—124no—
434C2×D3—182no—
435C2×D3—184no—
436C2×D3—368no—
437C2×D3—368no—
438C2×D3—368no—
439C3×C4—22no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—92no—
446C3×C4—122no—
447C3×C4—182no—
448C3×C4—182no—
449C3×C4—182no—
450C3×C4D3×D1212yes—
451Dic3—32no—
452Dic3—62no—
453Dic3—62no—
454Dic3—62no—
455Dic3—62no—
456Dic3—92no—
457Dic3—122no—
458Dic3—182no—
459Dic3—182no—
460C32—21no—
461C32C2×#87511yescharacteristic, complemented
462C32D3×#12411yescharacteristic, complemented
463C23—91no—
464C23—92no—
465C23—92no—
466C23—544no—
467C23—548no—
468Q8—182no—
469Q8—182no—
470Q8—544no—
471D4—182no—
472D4—182no—
473D4—544no—
474C2×C4—91no—
475C2×C4—94no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—92no—
479C2×C4—184no—
480C2×C4—544no—
481C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
482C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—32no—
486C2×C3—32no—
487C2×C3—61no—
488C2×C3—62no—
489C2×C3—94no—
490C2×C3—362no—
491C2×C3—362no—
492C2×C3C2×D3×D1211yescharacteristic
493C2×C3D3×#23512yes—
494D3—34no—
495D3—94no—
496D3—122no—
497D3—122no—
498D3—122no—
499D3—242no—
500D3—362no—
501C22—92no—
502C22—94no—
503C22—182no—
504C22—182no—
505C22—184no—
506C22—184no—
507C22—1088no—
508C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C4—22no—
510C4—92no—
511C4—182no—
512C4—182no—
513C4—182no—
514C4#8865412yes—
515C3—31no—
516C3—31no—
517C3—61no—
518C3D3×#87511yescharacteristic
519C2—94no—
520C2—362no—
521C2—362no—
522C2C2×#8865411yescharacteristic
523C2#1172249012yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88654C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#124
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×((C3⋊D3)⋊C3)⋊Q8C2semidirect
3. #11721721C2semidirect
4. #64084⋊C3C2semidirect
5. C42×(C3⋊D3)⋊C3C2semidirect
6. #11721160C2semidirect
7. #11721114C2semidirect
8. #11721110C2semidirect
9. #11721103C2semidirect
14. C2×#6229C22semidirect
18. C2×#634⋊C3C22semidirect
21. C2×#6196C22semidirect
24. C42×H(3,𝔽3)C22semidirect
25. #88542C22semidirect
26. #3752⋊C3C22semidirect
27. C4×(C3⋊Dic3)⋊C3C22semidirect
28. C4×(C3⋊Dic3)⋊C3C22semidirect
29. #88451C22semidirect
30. #88450C22semidirect
35. #64084D3semidirect
36. C42×C3⋊D3D3semidirect
78. C2×#634C2×D3semidirect
100. C32×C42C2×D3semidirect
101. C32×C42C2×D3semidirect
102. #3752C2×D3semidirect
103. C4×C3⋊Dic3C2×D3semidirect
104. C4×C3⋊Dic3C2×D3semidirect
208. (C3⋊D3)⋊C3#124semidirect
313. H(3,𝔽3)C2×#124semidirect
387. C3⋊D3#875semidirect
411. C42((C3⋊D3)⋊C3)⋊C2semidirect
461. C32C2×#875semidirect
462. C32D3×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×((C3⋊D3)⋊C3)⋊Q82
#117217212
#64084⋊C32
C42×(C3⋊D3)⋊C32
#117211602
#117211142
#117211102
#117211032
#105940093
D3×#8753