Finite Group Databasebeta

← #13141059

Subgroups — #13141059

Order 1728.

639 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#41×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
3#11721053C211yesmaximal, characteristic, complemented
4#11720586C211yesmaximal, characteristic, complemented
5#63664⋊C3C211yesmaximal, characteristic, complemented
6#11720322C211yesmaximal, characteristic, complemented
7#11720305C211yesmaximal, characteristic, complemented
8#10591671—31nomaximal
9D3×#837—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722C411yescharacteristic, complemented
12Q8×(C3⋊D3)⋊C3C411yescharacteristic, complemented
13#3830⋊C3—21no—
14#3830⋊C3C411yescharacteristic, complemented
15D4×(C3⋊D3)⋊C3—21no—
16D4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
17#3828⋊C3—21no—
18#3828⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C2×((C3⋊D3)⋊C3)⋊C4—21no—
22C4×#1084C2211yescharacteristic, complemented
23#88538C2211yescharacteristic, complemented
24C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
25#3748⋊C3C2211yescharacteristic, complemented
26#88447—21no—
27#88430—21no—
28((C3⋊D3)⋊C3)⋊C8—21no—
29#41×C3⋊D3D311yescharacteristic
30C3×D3×#41—31no—
31C4×#624—31no—
32C4×D3×Dic3—31no—
33#63818—31no—
34D3×#208—31no—
35#63664D311yescharacteristic, complemented
36C3×#837—31no—
37C3×#837—31no—
38C3×#837—31no—
39C3×#837—61no—
40#63570—31no—
41#63570—31no—
42#63569—31no—
43#63568—31no—
44C22×(C3⋊D3)⋊C3—21no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46Q8×H(3,𝔽3)C2×C411yescharacteristic, complemented
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)C2×C411yescharacteristic, complemented
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×#1084—21no—
52#638⋊C3—21no—
53#638⋊C3—21no—
54#638⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3D411yescharacteristic
58C4⋊((C3⋊D3)⋊C3)—21no—
59C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic, complemented
60C4×(C3⋊D3)⋊C3—21no—
61C4×(C3⋊D3)⋊C3—21no—
62C4×(C3⋊D3)⋊C3—21no—
63C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
64C4×(C3⋊D3)⋊C3C2×C411yescharacteristic
65C4×(C3⋊D3)⋊C3D411yescharacteristic
66C4×(C3⋊D3)⋊C3D411yescharacteristic
67#634⋊C3—21no—
68#634⋊C3C2×C411yescharacteristic, complemented
69((C3⋊D3)⋊C3)⋊C4—24no—
70#6180—21no—
71(C3⋊(C3⋊C8))⋊C3—21no—
72C2×#837—91no—
73D3×#104—91no—
74D3×#104—91no—
75C32×#41—21no—
76C32×#41C2×D311yescharacteristic
77C32×#41C2×D311yescharacteristic
78#3832Dic311yescharacteristic, complemented
79Q8×C3⋊D3Dic311yescharacteristic, complemented
80#3830—21no—
81#3830Dic311yescharacteristic, complemented
82D4×C3⋊D3—21no—
83D4×C3⋊D3Dic311yescharacteristic, complemented
84#3828—21no—
85#3828C2×D311yescharacteristic
86C2×C4×C3⋊D3—21no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C3×#239—31no—
89C3×D3×Q8—31no—
90C3×#237—31no—
91C3×#237—61no—
92C3×D3×D4—31no—
93C3×D3×D4—61no—
94C3×#235—31no—
95C3×#235—61no—
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3—61no—
98C2×#624—61no—
99C2×D3×Dic3—61no—
100C4×C3⋊Dic3C2×D311yescharacteristic, complemented
101#3748C2×D311yescharacteristic, complemented
102C3×C4×Dic3—31no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×#208—31no—
107C3×#208—31no—
108C3×#208—31no—
109C3×#208—61no—
110Dic32—61no—
111Dic32—61no—
112#3712—61no—
113#3711—61no—
114#3710—61no—
115D3×C3⋊C8—61no—
116C22×H(3,𝔽3)—21no—
117C22×H(3,𝔽3)C2×D411yescharacteristic
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—21no—
120C2×(C3⋊D3)⋊C3—42no—
121C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
122C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
123C4×H(3,𝔽3)—21no—
124C4×H(3,𝔽3)C22×C411yescharacteristic
125C4×H(3,𝔽3)C2×D411yescharacteristic
126#1084—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3—21no—
129(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
130(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
131C2×C3×#41—91no—
132D3×#41—31no—
133D3×#41—91no—
134C2×C4×Dic3—91no—
135C2×#208—91no—
136D3×M16—91no—
137D3×M16—91no—
138C42×D3—91no—
139C42×D3—91no—
140C3×#104—91no—
141C3×#104—91no—
142#837—31no—
143#837—31no—
144#837—31no—
145#837—61no—
146#837—91no—
147#837—91no—
148#837—91no—
149#825—91no—
150#825—91no—
151#805—91no—
152#805—91no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3#6711yescharacteristic
155C22×C3×D3—31no—
156C22×C3×D3—61no—
157C32×Q8—21no—
158C32×Q8C4×D311yescharacteristic, complemented
159C32×Q8C2×Dic311yescharacteristic, complemented
160C32×D4—21no—
161C32×D4—21no—
162C32×D4—21no—
163C32×D4—41no—
164C32×D4C4×D311yescharacteristic, complemented
165C32×D4C2×Dic311yescharacteristic, complemented
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—41no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172#638—21no—
173#638—21no—
174#638—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3#6711yescharacteristic
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×Dic311yescharacteristic, complemented
180C4×C3⋊D3—21no—
181C4×C3⋊D3—21no—
182C4×C3⋊D3—21no—
183C4×C3⋊D3C2×Dic311yescharacteristic
184C4×C3⋊D3C2×Dic311yescharacteristic
185C4×C3⋊D3#6711yescharacteristic
186C4×C3⋊D3#6711yescharacteristic
187#634—21no—
188#634C2×Dic311yescharacteristic, complemented
189C3×#67—61no—
190C3×#67—61no—
191C3×#67—61no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C3×D12—31no—
199C3×D12—61no—
200C3×C4×D3—31no—
201C3×C4×D3—31no—
202C3×C4×D3—31no—
203C3×C4×D3—31no—
204C3×C4×D3—61no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×Dic6—31no—
208C3×Dic6—61no—
209#624—64no—
210D3×Dic3—64no—
211C3⋊(C3⋊C8)—21no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—121no—
216C2×#104—271noSylow
217C2×H(3,𝔽3)—21no—
218C2×H(3,𝔽3)—41no—
219C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
220(C3⋊D3)⋊C3—21no—
221(C3⋊D3)⋊C3—41no—
222(C3⋊D3)⋊C3#10412yescomplemented
223C3×#41—31no—
224C3×#41—31no—
225C3×#41—61no—
226C3×#41—91no—
227C3×#41—91no—
228C3×#41—91no—
229C3×#41—181no—
230C3×#41—181no—
231C3×#41D3211yescharacteristic
232C2×C3×Q8—91no—
233C2×C3×D4—91no—
234C2×C3×D4—181no—
235C22×C3×C4—91no—
236C22×C3×C4—181no—
237C22×Dic3—181no—
238#239—31no—
239#239—91no—
240D3×Q8—31no—
241D3×Q8—91no—
242#237—31no—
243#237—61no—
244#237—91no—
245#237—181no—
246D3×D4—31no—
247D3×D4—61no—
248D3×D4—91no—
249D3×D4—181no—
250#235—31no—
251#235—61no—
252#235—91no—
253#235—181no—
254C2×C4×D3—31no—
255C2×C4×D3—61no—
256C2×C4×D3—91no—
257C2×C4×D3—181no—
258C2×C4×D3—181no—
259C2×C4×D3—181no—
260C3×M16—91no—
261C3×M16—91no—
262C3×C42—91no—
263C3×C42—91no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—61no—
268C4×Dic3—91no—
269C4×Dic3—181no—
270C4×Dic3—181no—
271C4×Dic3—181no—
272#208—31no—
273#208—31no—
274#208—31no—
275#208—61no—
276#208—91no—
277#208—181no—
278C2×C3⋊C8—181no—
279ΓL(1,𝔽25)—181no—
280ΓL(1,𝔽25)—181no—
281D3×C8—181no—
282D3×C8—181no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32D3×D411yescharacteristic
288C22×C32C2×#6711yescharacteristic
289C2×C3⋊D3—22no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—42no—
292C2×C3⋊D3#21711yescharacteristic
293C2×C3⋊D3#21711yescharacteristic
294C2×C3×D3—31no—
295C2×C3×D3—31no—
296C2×C3×D3—62no—
297C2×C3×D3—61no—
298C2×C3×D3—122no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—41no—
304C32×C4C2×C4×D311yescharacteristic
305C32×C4D3×D411yescharacteristic
306C32×C4C22×Dic311yescharacteristic
307C32×C4C2×#6711yescharacteristic
308C3⋊Dic3—22no—
309C3⋊Dic3—21no—
310C3⋊Dic3#21711yescharacteristic
311C3⋊Dic3#21711yescharacteristic
312C3×Dic3—31no—
313C3×Dic3—31no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—122no—
319C2×#41—91no—
320C2×M16—271no—
321C2×C42—271no—
322#104—91no—
323#104—91no—
324#104—271no—
325#104—271no—
326H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
327C23×C3—91no—
328C23×C3—181no—
329C22×D3—31no—
330C22×D3—61no—
331C22×D3—91no—
332C22×D3—181no—
333C3×Q8—31no—
334C3×Q8—31no—
335C3×Q8—61no—
336C3×Q8—91no—
337C3×Q8—181no—
338C3×Q8D3×Dic311yescharacteristic
339C3×D4—21no—
340C3×D4—31no—
341C3×D4—31no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—91no—
346C3×D4—121no—
347C3×D4—181no—
348C3×D4—181no—
349C3×D4—181no—
350C3×D4—181no—
351C3×D4D3×Dic311yescharacteristic
352C2×C3×C4—21no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—121no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371#67—61no—
372#67—61no—
373#67—61no—
374#67—181no—
375#67—181no—
376#67—181no—
377C2×Dic3—31no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—91no—
383C2×Dic3—121no—
384C2×Dic3—181no—
385C2×Dic3—181no—
386C2×Dic3—184no—
387D12—31no—
388D12—61no—
389D12—91no—
390D12—181no—
391C4×D3—31no—
392C4×D3—31no—
393C4×D3—31no—
394C4×D3—31no—
395C4×D3—61no—
396C4×D3—61no—
397C4×D3—61no—
398C4×D3—91no—
399C4×D3—91no—
400C4×D3—91no—
401C4×D3—91no—
402C4×D3—181no—
403C4×D3—181no—
404C4×D3—181no—
405C4×D3—184no—
406C4×D3—184no—
407Dic6—31no—
408Dic6—61no—
409Dic6—91no—
410Dic6—181no—
411C3×C8—181no—
412C3×C8—181no—
413C3⋊C8—61no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—121no—
417C3⋊C8—181no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32—41no—
424C2×C32—81no—
425C2×C32D3×C22⋊C411yescharacteristic
426C2×C32C2×#21711yescharacteristic
427C3⋊D3—21no—
428C3⋊D3—41no—
429C3⋊D3#83712yescomplemented
430C3×D3—32no—
431C3×D3—61no—
432C3×D3—121no—
433#41—91no—
434#41—91no—
435#41—91no—
436#41—181no—
437#41—181no—
438#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C2×Q8—91no—
440C2×D4—91no—
441C2×D4—181no—
442C22×C4—91no—
443C22×C4—181no—
444C22×C4—541no—
445M16—91no—
446M16—91no—
447M16—541no—
448C2×C8—541no—
449C42—91no—
450C42—91no—
451C42—541no—
452C22×C3—21no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—61no—
458C22×C3—91no—
459C22×C3—91no—
460C22×C3—121no—
461C22×C3—181no—
462C22×C3—182no—
463C22×C3—362no—
464C22×C3D3×#6711yescharacteristic
465C2×D3—31no—
466C2×D3—31no—
467C2×D3—62no—
468C2×D3—61no—
469C2×D3—91no—
470C2×D3—91no—
471C2×D3—122no—
472C2×D3—182no—
473C2×D3—181no—
474C2×D3—362no—
475C3×C4—21no—
476C3×C4—31no—
477C3×C4—31no—
478C3×C4—31no—
479C3×C4—31no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—61no—
483C3×C4—61no—
484C3×C4—91no—
485C3×C4—91no—
486C3×C4—121no—
487C3×C4—182no—
488C3×C4—181no—
489C3×C4—182no—
490C3×C4C2×D3×Dic311yescharacteristic
491C3×C4D3×#6711yescharacteristic
492Dic3—31no—
493Dic3—31no—
494Dic3—62no—
495Dic3—62no—
496Dic3—62no—
497Dic3—61no—
498Dic3—91no—
499Dic3—91no—
500Dic3—122no—
501Dic3—181no—
502Dic3—182no—
503C32—21no—
504C32D3×#10411yescharacteristic, complemented
505C32C2×#83711yescharacteristic, complemented
506C23—91no—
507C23—181no—
508Q8—91no—
509Q8—181no—
510Q8((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
511D4—21no—
512D4—91no—
513D4—181no—
514D4—181no—
515D4—181no—
516D4—181no—
517D4((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
518C2×C4—21no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—91no—
523C2×C4—91no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—544no—
531C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C8—181no—
533C8—181no—
534C2×C3—21no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—41no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—92no—
542C2×C3—121no—
543C2×C3—121no—
544C2×C3—121no—
545C2×C3—181no—
546C2×C3—241no—
547C2×C3—361no—
548C2×C3D3×#21711yescharacteristic
549D3—32no—
550D3—61no—
551D3—92no—
552D3—121no—
553D3—181no—
554D3—361no—
555C22—21no—
556C22—91no—
557C22—91no—
558C22—182no—
559C22—181no—
560C22—362no—
561C22#8867611yescharacteristic
562C4—21no—
563C4—91no—
564C4—91no—
565C4—182no—
566C4—182no—
567C4—181no—
568C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
569C4#8867611yescharacteristic
570C3—31no—
571C3—31no—
572C3—61no—
573C3D3×#83711yescharacteristic
574C2—21no—
575C2—41no—
576C2—92no—
577C2—181no—
578C2—361no—
579C2#1172131011yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radical#41×(C3⋊D3)⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. #11721053C2semidirect
4. #11720586C2semidirect
5. #63664⋊C3C2semidirect
6. #11720322C2semidirect
7. #11720305C2semidirect
11. #88722C4semidirect
12. Q8×(C3⋊D3)⋊C3C4semidirect
14. #3830⋊C3C4semidirect
16. D4×(C3⋊D3)⋊C3C4semidirect
22. C4×#1084C22semidirect
23. #88538C22semidirect
24. C4×(C3⋊Dic3)⋊C3C22semidirect
25. #3748⋊C3C22semidirect
35. #63664D3semidirect
46. Q8×H(3,𝔽3)C2×C4semidirect
48. D4×H(3,𝔽3)C2×C4semidirect
59. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
68. #634⋊C3C2×C4semidirect
78. #3832Dic3semidirect
79. Q8×C3⋊D3Dic3semidirect
81. #3830Dic3semidirect
83. D4×C3⋊D3Dic3semidirect
100. C4×C3⋊Dic3C2×D3semidirect
101. #3748C2×D3semidirect
158. C32×Q8C4×D3semidirect
159. C32×Q8C2×Dic3semidirect
164. C32×D4C4×D3semidirect
165. C32×D4C2×Dic3semidirect
179. C4⋊(C3⋊D3)C2×Dic3semidirect
188. #634C2×Dic3semidirect
222. (C3⋊D3)⋊C3#104semidirect
326. H(3,𝔽3)C2×#104semidirect
429. C3⋊D3#837semidirect
504. C32D3×#104semidirect
505. C32C2×#837semidirect
510. Q8((C3⋊D3)⋊C3)⋊C4semidirect
517. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
#41×(C3⋊D3)⋊C32
C4×((C3⋊D3)⋊C3)⋊C42
#117210532
#117205862
#63664⋊C32
#117203222
#117203052
#105916713
D3×#8373