Finite Group Databasebeta

← #13165401

Subgroups — #13165401

Order 1728.

684 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 80 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×#88934C211yesmaximal, characteristic, complemented
2C2×D32⋊Dic3C211yesmaximal, characteristic
3C2×C3×D32⋊C4C211yesmaximal, characteristic
4#11722902C211yesmaximal, characteristic
5Dic3×#624C211yesmaximal, characteristic, complemented
6#11722791C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593399—31nomaximal
9C22×C3×D32C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×C3×#624C2211yescharacteristic
16#88934—22no—
17D32⋊Dic3C414yescomplemented
18C3×D32⋊C4—22no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×Dic32—21no—
23C2×#3773—31no—
24C2×D32⋊C4—31no—
25C2×D32⋊C4D311yescharacteristic
26D3×#217—61no—
27#63921—31no—
28C4×#624—31no—
29C4×D3×Dic3—61no—
30#63781—31no—
31#63737—31no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—21no—
34C2×C3×D32—21no—
35C2×C3×D32—21no—
36C2×C3×D32—21no—
37C2×C3×D32—21no—
38C2×C3×D32—21no—
39C2×C3×D32C2×C412yes—
40C2×C3⋊#157—21no—
41C2×C3⋊#157—21no—
42C2×C3⋊#157C2×C412yes—
43C2×C3×#157—21no—
44C2×C3×#157—21no—
45C2×C3×#157—21no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3D411yescharacteristic
51#6294—21no—
52#6294—21no—
53#6294—21no—
54#6294C2×C412yes—
55Dic3×C3⋊D3—21no—
56Dic3×C3⋊D3—21no—
57C3×#624—21no—
58C3×#624D412yes—
59#4945—91nomaximal
60C23×C3×D3—61no—
61C22×D32C2×D311yescharacteristic
62C22×#157—31no—
63C22×#157C2×D311yescharacteristic
64C2×C4×C3⋊D3—31no—
65C2×C3×C4×D3—61no—
66C2×#624—31no—
67C2×#624—31no—
68C2×#624C2×D311yescharacteristic
69C2×D3×Dic3—61no—
70C2×D3×Dic3—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73#3773—62no—
74D32⋊C4—22no—
75D32⋊C4—34no—
76#3757—61no—
77C4×C3⋊Dic3—61no—
78C3×#217—61no—
79C3×(C2×D3)⋊C4—61no—
80C3×C4×Dic3—61no—
81C3×C4×Dic3—61no—
82(C2×D3)⋊Dic3—21no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—121no—
86Dic32—21no—
87Dic32—61no—
88Dic32—61no—
89Dic32—121no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3C22×C411yescharacteristic
92C2×C3×C3⋊D3C2×D411yescharacteristic
93C2×C3×C3⋊D3#4111yescharacteristic
94C2×C3×C3⋊D3#4111yescharacteristic
95C2×C3×C3⋊D3#4111yescharacteristic
96C2×C3×C3⋊D3#4111yescharacteristic
97C2×C32×D3—22no—
98C2×C32×D3—41no—
99C2×C32×D3—41no—
100C3×D32—22no—
101C3×D32—22no—
102C3×D32—41no—
103C3×D32—41no—
104C3⋊#157—22no—
105C3×#157—41no—
106C3⋊(C3⋊Dic3)—21no—
107C3×C3⋊Dic3—22no—
108C3×C3⋊Dic3—41no—
109C32×Dic3—22no—
110C32×Dic3—21no—
111C2×C3×C22⋊C4—91no—
112C2×#217—91no—
113C2×#217—91no—
114C2×Dic3⋊C4—91no—
115C2×C4×Dic3—91no—
116D3×C22⋊C4—181no—
117C42×D3—181no—
118C3⋊#95—91no—
119C3⋊#95—91no—
120C23×C32—61no—
121C22×C3⋊D3C22×D311yescharacteristic
122C22×C3×D3—21no—
123C22×C3×D3—21no—
124C22×C3×D3—41no—
125C22×C3×D3—62no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—62no—
129C22×C3×D3—121no—
130C22×C3×D3—121no—
131C22×C3×D3—121no—
132C22×C3×D3—121no—
133C2×D32—21no—
134C2×D32—21no—
135C2×D32—21no—
136C2×D32—21no—
137C2×D32—21no—
138C2×D32C4×D312yes—
139C2×#157—21no—
140C2×#157—21no—
141C2×#157—21no—
142C2×#157—32no—
143C2×#157—61no—
144C2×#157—61no—
145C2×C32×C4—31no—
146C2×C32×C4—61no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—31no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—121no—
153C2×C3⋊Dic3—121no—
154C4×C3⋊D3—61no—
155C4×C3⋊D3—61no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—121no—
166C2×C3×Dic3—121no—
167C3×C4×D3—62no—
168C3×C4×D3—62no—
169#624—21no—
170#624—32no—
171#624—61no—
172#624—61no—
173#624—61no—
174#624#6712yes—
175D3×Dic3—62no—
176D3×Dic3—62no—
177D3×Dic3—62no—
178D3×Dic3—62no—
179D3×Dic3—121no—
180D3×Dic3—121no—
181D3×Dic3—121no—
182D3×Dic3—121no—
183#388—271noSylow
184C2×C33C4×D411yescharacteristic
185C2×C33#12111yescharacteristic
186C2×C33#12311yescharacteristic
187C3×C3⋊D3#11711yescharacteristic
188C3×C3⋊D3C4×D411yescharacteristic
189C3×C3⋊D3#12411yescharacteristic
190C3×C3⋊D3#12611yescharacteristic
191C32×D3—42no—
192C24×C3—91no—
193C23×D3—61no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—91no—
199C22×Dic3—91no—
200C2×C4×D3—61no—
201C2×C4×D3—181no—
202C2×C4×D3—181no—
203C2×C4×D3—181no—
204C2×C4×D3—181no—
205C3×C22⋊C4—181no—
206C3×C22⋊C4—182no—
207C3×C42—181no—
208#217—61no—
209#217—94no—
210#217—181no—
211#217—181no—
212#217—361no—
213(C2×D3)⋊C4—61no—
214(C2×D3)⋊C4—181no—
215Dic3⋊C4—182no—
216C4×Dic3—61no—
217C4×Dic3—61no—
218C4×Dic3—181no—
219C4×Dic3—181no—
220C4×Dic3—181no—
221C4×Dic3—361no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—62no—
227C22×C32—121no—
228C22×C32—121no—
229C22×C32D3×D411yescharacteristic
230C2×C3⋊D3C2×C4×D311yescharacteristic
231C2×C3⋊D3#23511yescharacteristic
232C2×C3⋊D3#23711yescharacteristic
233C2×C3⋊D3#23711yescharacteristic
234C2×C3⋊D3#23711yescharacteristic
235C2×C3⋊D3C2×#6711yescharacteristic
236C2×C3×D3—22no—
237C2×C3×D3—22no—
238C2×C3×D3—42no—
239C2×C3×D3—41no—
240C2×C3×D3—41no—
241C2×C3×D3—41no—
242C2×C3×D3—41no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C2×C3×D3—61no—
246C2×C3×D3—61no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—61no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—81no—
256C2×C3×D3—81no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—122no—
261D32—22no—
262D32—22no—
263D32—41no—
264D32—41no—
265#157—41no—
266#157—62no—
267C32×C4—62no—
268C32×C4—61no—
269C3⋊Dic3—22no—
270C3⋊Dic3—41no—
271C3⋊Dic3—62no—
272C3⋊Dic3—61no—
273C3⋊Dic3—121no—
274C3⋊Dic3—122no—
275C3⋊Dic3—241no—
276C3×Dic3—22no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—41no—
280C3×Dic3—42no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—122no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—241no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—271no—
290C2×C22⋊C4—271no—
291C2×C42—271no—
292#95—271no—
293#95—271no—
294C33#38811yescharacteristic, complemented, Sylow
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—92no—
298C23×C3—91no—
299C23×C3—121no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C22×D3—21no—
306C22×D3—62no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—62no—
310C22×D3—121no—
311C22×D3—121no—
312C22×D3—121no—
313C22×D3—121no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—92no—
319C2×C3×C4—121no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×Dic3—21no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—91no—
334C2×Dic3—121no—
335C2×Dic3—121no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—361no—
346C2×Dic3—361no—
347C2×Dic3GO+(4,𝔽2)11yescharacteristic
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—182no—
351C4×D3—182no—
352C4×D3—182no—
353C4×D3—182no—
354C4×D3—361no—
355C4×D3—361no—
356C4×D3—361no—
357C4×D3—361no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—122no—
371C2×C32#67⋊C411yescharacteristic
372C2×C32#88311yescharacteristic
373C2×C32#93911yescharacteristic
374C3⋊D3#87711yescharacteristic
375C3⋊D3#87911yescharacteristic
376C3⋊D3C4×#6711yescharacteristic
377C3⋊D3#93611yescharacteristic
378C3×D3—42no—
379C3×D3—42no—
380C3×D3—61no—
381C3×D3—61no—
382C3×D3—61no—
383C3×D3—61no—
384C3×D3—61no—
385C3×D3—61no—
386C3×D3—61no—
387C3×D3—61no—
388C3×D3—82no—
389C24—91no—
390C22×C4—91no—
391C22×C4—91no—
392C22×C4—271no—
393C22×C4—271no—
394C22×C4—271no—
395C22×C4—271no—
396C4⋊C4—542no—
397C22⋊C4—181no—
398C22⋊C4—182no—
399C22⋊C4—274no—
400C22⋊C4—541no—
401C42—181no—
402C42—541no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—91no—
410C22×C3—91no—
411C22×C3—91no—
412C22×C3—91no—
413C22×C3—91no—
414C22×C3—91no—
415C22×C3—122no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—121no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—241no—
423C22×C3—241no—
424C22×C3—361no—
425C22×C3—361no—
426C22×C3C2×GO+(4,𝔽2)11yescharacteristic
427C2×D3—22no—
428C2×D3—41no—
429C2×D3—41no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—61no—
440C2×D3—61no—
441C2×D3—61no—
442C2×D3—122no—
443C2×D3—122no—
444C2×D3—122no—
445C2×D3—122no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—121no—
450C3×C4—122no—
451C3×C4—182no—
452C3×C4—361no—
453C3×C4—361no—
454Dic3—22no—
455Dic3—21no—
456Dic3—62no—
457Dic3—62no—
458Dic3—122no—
459Dic3—121no—
460Dic3—121no—
461Dic3—182no—
462Dic3—182no—
463Dic3—181no—
464Dic3—241no—
465Dic3—361no—
466Dic3—362no—
467Dic3—721no—
468C32—21no—
469C32—21no—
470C32—41no—
471C32—41no—
472C32#494511yescharacteristic, complemented
473C23—61no—
474C23—92no—
475C23—91no—
476C23—181no—
477C23—181no—
478C23—181no—
479C23—181no—
480C23—181no—
481C2×C4—31no—
482C2×C4—61no—
483C2×C4—92no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—272no—
491C2×C4—272no—
492C2×C4—271no—
493C2×C4—541no—
494C2×C4—541no—
495C2×C4—541no—
496C2×C4—541no—
497C2×C4—541no—
498C2×C4—541no—
499C2×C4—541no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—91no—
513C2×C3—91no—
514C2×C3—91no—
515C2×C3—91no—
516C2×C3—122no—
517C2×C3—122no—
518C2×C3—242no—
519C2×C3C4×GO+(4,𝔽2)11yescharacteristic
520C2×C3#6423011yescharacteristic
521C2×C3#6423311yescharacteristic
522D3—42no—
523D3—61no—
524D3—61no—
525D3—61no—
526D3—61no—
527D3—61no—
528D3—61no—
529D3—61no—
530D3—61no—
531C22—62no—
532C22—91no—
533C22—91no—
534C22—91no—
535C22—91no—
536C22—91no—
537C22—91no—
538C22—121no—
539C22—121no—
540C22—182no—
541C22—182no—
542C22—361no—
543C22—361no—
544C22D3×GO+(4,𝔽2)11yescharacteristic
545C4—62no—
546C4—61no—
547C4—182no—
548C4—361no—
549C4—361no—
550C4—542no—
551C4—541no—
552C3—21no—
553C3—21no—
554C3—41no—
555C3—41no—
556C3#1059339911yescharacteristic, complemented
557C2—91no—
558C2—91no—
559C2—91no—
560C2—91no—
561C2—122no—
562C2(D32⋊D3)⋊C411yescharacteristic
563C2#1172411711yescharacteristic
564C2#1172412111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×C3×D32C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33#388
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88934C2semidirect
5. Dic3×#624C2semidirect
6. #11722791C2semidirect
7. #11722722C2semidirect
17. D32⋊Dic3C4semidirect
294. C33#388semidirect
472. C32#4945semidirect
556. C3#10593399semidirect

Maximal subgroups

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

subgroupindex
C2×#889342
C2×D32⋊Dic32
C2×C3×D32⋊C42
#117229022
Dic3×#6242
#117227912
#117227222
#105933993
#49459