Finite Group Databasebeta

← #13140627

Subgroups — #13140627

Order 1728.

726 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 87 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×#88546C211yesmaximal, characteristic, complemented
3#11721107C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591467—31nomaximal
9#10591458—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6229C2211yescharacteristic, complemented
13C2×#638⋊C3—21no—
14D4×(C3⋊D3)⋊C3—22no—
15D4×(C3⋊D3)⋊C3—22no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18#88546—22no—
19C2×#6180C2211yescharacteristic, complemented
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
22#88450—21no—
23#88448—21no—
24#88438—22no—
25C2×D4×C3⋊D3D311yescharacteristic
26C2×C3×D3×D4—31no—
27C2×C3×#214—31no—
28C2×C3×#214—31no—
29C2×C3×#214—61no—
30#63871—31no—
31#63856—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63659D311yescharacteristic, complemented
35C3×#832—31no—
36#63564—31no—
37#63563—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3—42no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46D4×H(3,𝔽3)—21no—
47D4×H(3,𝔽3)D412yes—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×#1084—21no—
50#6229—21no—
51#6229—21no—
52#638⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3D411yescharacteristic
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D412yes—
57C4×(C3⋊D3)⋊C3—21no—
58C4×(C3⋊D3)⋊C3—21no—
59#6180—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61#5218—91no—
62#5169—91no—
63#4776—91no—
64C23×C3⋊D3—21no—
65C23×C3×D3—61no—
66C2×C32×D4—21no—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C2×D311yescharacteristic
69C2×#638—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C3×#67—61no—
75C3×D3×D4—62no—
76C3×D3×D4—62no—
77C2×C3×D12—31no—
78C2×C3×C4×D3—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
84C3×#214—64no—
85C3×#214—62no—
86C3×#214—62no—
87C3×C4⋊Dic3—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92#3724—61no—
93Dic3⋊Dic3—61no—
94#3722—61no—
95(C2×D3)⋊Dic3—61no—
96#3717—62no—
97#3715—62no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—48no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3SD1612yescomplemented
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111#1084—41no—
112(C3⋊Dic3)⋊C3—21no—
113(C3⋊Dic3)⋊C3—41no—
114C22×C3×D4—91no—
115C2×D3×D4—31no—
116C2×D3×D4—91no—
117C2×C3×SD16—91no—
118#946—91no—
119C2×#215—91no—
120C2×#214—31no—
121C2×#214—31no—
122C2×#214—61no—
123C2×#214—91no—
124#924—91no—
125C2×#204—91no—
126#897—91no—
127C3×#102—91no—
128#832—31no—
129#832—91no—
130#832—91no—
131#830—91no—
132#821—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135#809—91no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—41no—
139C22×C3⋊D3—24no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3—42no—
143C22×C3⋊D3#6711yescharacteristic
144C22×C3×D3—31no—
145C22×C3×D3—64no—
146C22×C3×D3—61no—
147C22×C3×D3—122no—
148C22×C3×D3—122no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—22no—
152C32×D4—41no—
153C32×D4D1212yes—
154C32×D4#6712yes—
155C2×C32×C4—21no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158#638—24no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3#6711yescharacteristic
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)#6712yes—
163C4×C3⋊D3—21no—
164C4×C3⋊D3—21no—
165C3×#67—64no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—121no—
171C3×D12—32no—
172C3×D12—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×Dic6—62no—
176C3×Dic6—61no—
177C3×Dic6—61no—
178C3×Dic6—61no—
179C3×Dic6—61no—
180C3×Dic6—121no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185C3×C3⋊C8—61no—
186#450—271noSylow
187C2×H(3,𝔽3)—42no—
188C2×H(3,𝔽3)#12011yescharacteristic
189C2×H(3,𝔽3)C2×SD1611yescharacteristic
190C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
191(C3⋊D3)⋊C3—22no—
192(C3⋊D3)⋊C3—42no—
193C24×C3—181no—
194C23×D3—61no—
195C23×D3—181no—
196C2×C3×Q8—91no—
197C2×C3×D4—31no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C2×C3×D4—91no—
201C2×C3×D4—181no—
202C2×C3×D4—182no—
203C2×C3×D4—182no—
204C2×C3×D4D3211yescharacteristic
205C22×C3×C4—91no—
206C2×#67—61no—
207C2×#67—181no—
208D3×D4—62no—
209D3×D4—62no—
210D3×D4—182no—
211D3×D4—182no—
212C2×D12—31no—
213C2×D12—91no—
214C2×C4×D3—31no—
215C2×C4×D3—91no—
216C2×Dic6—31no—
217C2×Dic6—31no—
218C2×Dic6—61no—
219C3×SD16—182no—
220C2×C3×C8—91no—
221C2×C3×C8—91no—
222C3×C4⋊C4—91no—
223#217—181no—
224#215—182no—
225#214—62no—
226#214—62no—
227#214—64no—
228#214—182no—
229(C2×D3)⋊C4—181no—
230(C2×D3)⋊C4—181no—
231C4⋊Dic3—31no—
232C4⋊Dic3—91no—
233C4⋊Dic3—181no—
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8—91no—
241#204—182no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—42no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—81no—
251C22×C32—81no—
252C22×C32D3×D411yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3⋊D3—22no—
257C2×C3⋊D3—48no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3⋊D3#21412yescomplemented
261C2×C3×D3—32no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—62no—
265C2×C3×D3—128no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4C2×D1211yescharacteristic
271C32×C4D3×D411yescharacteristic
272C32×C4C2×#6711yescharacteristic
273C32×C4C2×#6711yescharacteristic
274C3⋊Dic3—21no—
275C3⋊Dic3—41no—
276C3×Dic3—61no—
277C3×Dic3—122no—
278C3×Dic3—121no—
279C3×Dic3—121no—
280C3×Dic3—121no—
281C22×D4—91no—
282C2×SD16—91no—
283C2×SD16—271no—
284C22⋊Q8—271no—
285#102—91no—
286#102—271no—
287C22⋊C8—271no—
288H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—61no—
291C23×C3—61no—
292C23×C3—91no—
293C23×C3—121no—
294C23×C3—181no—
295C23×C3—184no—
296C23×C3—362no—
297C23×C3—362no—
298C22×D3—31no—
299C22×D3—64no—
300C22×D3—61no—
301C22×D3—91no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—181no—
305C22×D3—184no—
306C22×D3—362no—
307C22×D3—362no—
308C3×Q8—181no—
309C3×Q8—181no—
310C3×D4—21no—
311C3×D4—32no—
312C3×D4—32no—
313C3×D4—61no—
314C3×D4—61no—
315C3×D4—62no—
316C3×D4—92no—
317C3×D4—121no—
318C3×D4—181no—
319C3×D4—184no—
320C3×D4#62612yes—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—61no—
324C2×C3×C4—91no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4C2×D3211yescharacteristic
330#67—64no—
331#67—184no—
332C2×Dic3—31no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—91no—
337C2×Dic3—121no—
338C2×Dic3—181no—
339D12—32no—
340D12—61no—
341D12—92no—
342D12—181no—
343C4×D3—61no—
344C4×D3—61no—
345C4×D3—181no—
346C4×D3—181no—
347Dic6—61no—
348Dic6—61no—
349Dic6—61no—
350Dic6—61no—
351Dic6—62no—
352Dic6—121no—
353C3×C8—181no—
354C3×C8—181no—
355C3⋊C8—61no—
356C3⋊C8—61no—
357C3⋊C8—62no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—82no—
366C2×C32#88211yescharacteristic
367C2×C32#91111yescharacteristic
368C2×C32D3×SD1611yescharacteristic
369C2×C32#93211yescharacteristic
370C2×C32C2×#21411yescharacteristic
371C2×C32#95311yescharacteristic
372C3⋊D3—22no—
373C3⋊D3—42no—
374C3×D3—62no—
375C3×D3—122no—
376C24—181no—
377C2×Q8—91no—
378C2×D4—91no—
379C2×D4—181no—
380C2×D4—182no—
381C2×D4—182no—
382C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C22×C4—91no—
384SD16—182no—
385SD16—542no—
386C2×C8—91no—
387C2×C8—91no—
388C4⋊C4—91no—
389C4⋊C4—541no—
390C22⋊C4—541no—
391C22×C3—22no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—61no—
399C22×C3—92no—
400C22×C3—122no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3—182no—
408C22×C3—241no—
409C22×C3—241no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—368no—
413C22×C3D3×#6711yescharacteristic
414C2×D3—32no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—62no—
418C2×D3—92no—
419C2×D3—128no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—181no—
423C2×D3—181no—
424C2×D3—182no—
425C2×D3—368no—
426C2×D3—361no—
427C2×D3—361no—
428C3×C4—31no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—181no—
435C3×C4—361no—
436C3×C4—361no—
437C3×C4C2×#62611yescharacteristic
438C3×C4D3×#6711yescharacteristic
439Dic3—61no—
440Dic3—121no—
441Dic3—121no—
442Dic3—122no—
443Dic3—121no—
444Dic3—181no—
445Dic3—361no—
446C32—21no—
447C32#477611yescharacteristic, complemented
448C32#521811yescharacteristic, complemented
449C23—21no—
450C23—91no—
451C23—184no—
452C23—181no—
453C23—362no—
454C23—362no—
455Q8—181no—
456Q8—181no—
457D4—21no—
458D4—92no—
459D4—184no—
460D4—181no—
461D4#620012yes—
462C2×C4—91no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
468C8—181no—
469C8—181no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—42no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—122no—
481C2×C3—122no—
482C2×C3—182no—
483C2×C3—242no—
484C2×C3—362no—
485C2×C3D3×#21411yescharacteristic
486C2×C3#6393211yescharacteristic
487C2×C3#6397711yescharacteristic
488D3—62no—
489D3—122no—
490D3—182no—
491D3—362no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—92no—
496C22—181no—
497C22—181no—
498C22—182no—
499C22—368no—
500C22—361no—
501C22—361no—
502C22#8867611yescharacteristic
503C4—181no—
504C4—361no—
505C4—361no—
506C4C2×#620011yescharacteristic
507C4#8867611yescharacteristic
508C3—31no—
509C3—31no—
510C3—61no—
511C3#1059145911yescharacteristic
512C2—42no—
513C2—182no—
514C2—362no—
515C2#1172119811yescharacteristic
516C2#1172120011yescharacteristic
517C2#1172132111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×D4×(C3⋊D3)⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88546C2semidirect
3. #11721107C2semidirect
4. C2×#88438C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
12. C2×#6229C22semidirect
19. C2×#6180C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
34. #63659D3semidirect
82. C4⋊(C3⋊Dic3)C2×D3semidirect
83. C2×C3⋊(C3⋊C8)C2×D3semidirect
108. C2×(C3⋊D3)⋊C3SD16semidirect
260. C2×C3⋊D3#214semidirect
288. H(3,𝔽3)#450semidirect
447. C32#4776semidirect
448. C32#5218semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×#885462
#117211072
C2×#884382
#63659⋊C32
#117203182
#117203172
#105914673
#105914583