Finite Group Databasebeta

← #13156497

Subgroups — #13156497

Order 1728.

534 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723667C211yesmaximal, characteristic
2C3×#63871C211yesmaximal, characteristic
3#11722053C211yesmaximal, characteristic, complemented
4#11722037C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591558—31nomaximal
9#10590715—31nomaximal
10#10590448—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—21no—
19#88813—21no—
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×#946—31no—
23C3×#897—31no—
24#63871D311yescharacteristic
25#63849—31no—
26#63841—31no—
27#63841—31no—
28#63662—31no—
29#63647—31no—
30#63635—31no—
31C3×#835—31no—
32C3×#818—31no—
33C3×#816—31no—
34C3×#807—31no—
35#63575D311yescharacteristic, complemented
36#63574—31no—
37#63574—31no—
38#63574—61no—
39#63566D311yescharacteristic, complemented
40#63565—31no—
41#63565—31no—
42#63565—61no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46C22×C3×C3⋊D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5045—91no—
60#4888—91no—
61#4814—91no—
62C2×C32×Q8—31no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—121no—
81C3×(C2×D3)⋊C4—61no—
82C3×(C2×D3)⋊C4—61no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—61no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97Dic3⋊Dic3—21no—
98#3722—21no—
99#3722—61no—
100(C2×D3)⋊Dic3—61no—
101(C2×D3)⋊Dic3—61no—
102Dic32—61no—
103Dic32—61no—
104Dic32—61no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C3×C22⋊Q8—91no—
117#946—31no—
118#926—91no—
119#897—31no—
120#892—91no—
121#892—91no—
122C3×#107—91no—
123C3×#103—91no—
124#835—31no—
125#835—91no—
126#835—91no—
127#835—181no—
128#830—91no—
129(C2×D3)⋊C8—91no—
130(C2×D3)⋊C8—91no—
131(C2×D3)⋊C8—181no—
132#818—31no—
133#816—31no—
134#810—91no—
135#808—91no—
136#808—91no—
137#808—91no—
138#808—181no—
139#807—31no—
140C22×C3⋊D3#6711yescharacteristic
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C32×Q8—61no—
145C32×Q8—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3#6711yescharacteristic
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3—121no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179C3×C4×D3—121no—
180C3×C4×D3—121no—
181C3×Dic6—22no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—41no—
187C32×C8—61no—
188C3⋊(C3⋊C8)—61no—
189C3⋊(C3⋊C8)—62no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—61no—
192C3⋊(C3⋊C8)—121no—
193C3⋊(C3⋊C8)—121no—
194C3⋊(C3⋊C8)—122no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—41no—
198#485—271noSylow
199C2×C33#12311yescharacteristic
200C2×C33#13511yescharacteristic
201C2×C33#13711yescharacteristic
202C3×C3⋊D3—41no—
203C2×C3×Q8—31no—
204C2×C3×Q8—31no—
205C2×C3×Q8—61no—
206C22×C3×C4—91no—
207C2×C4×D3—31no—
208C2×C4×D3—31no—
209C2×C4×D3—61no—
210C2×Dic6D3211yescharacteristic
211C2×C3×C8—31no—
212C2×C3×C8—31no—
213C2×C3×C8—61no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—181no—
220C3×C22⋊C4—181no—
221#217—181no—
222(C2×D3)⋊C4—61no—
223(C2×D3)⋊C4—61no—
224(C2×D3)⋊C4—181no—
225(C2×D3)⋊C4—181no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3—61no—
233C4⋊Dic3D3211yescharacteristic
234Dic3⋊C4—61no—
235C4×Dic3—181no—
236C4×Dic3—181no—
237C4×Dic3—181no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—181no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—361no—
245C2×C3⋊C8D3211yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32D3×D411yescharacteristic
251C22×C32D3×D411yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3×D3—62no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—121no—
274C32×C4—121no—
275C32×C4#23511yescharacteristic
276C32×C4#23511yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4#23711yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23911yescharacteristic
281C3⋊Dic3—21no—
282C3⋊Dic3—41no—
283C3⋊Dic3—41no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—81no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—241no—
298C3×Dic3—241no—
299C22⋊Q8—91no—
300#117—271no—
301#107—91no—
302#106—271no—
303#103—91no—
304#103—271no—
305C22⋊C8—271no—
306C33#48511yescharacteristic, complemented, Sylow
307C23×C3—91no—
308C22×D3—31no—
309C22×D3—31no—
310C22×D3—61no—
311C3×Q8—62no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—121no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—91no—
326C2×C3×C4—121no—
327C2×C3×C4—121no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3—121no—
346C4×D3—61no—
347C4×D3—61no—
348C4×D3—61no—
349C4×D3—61no—
350C4×D3—121no—
351C4×D3—121no—
352Dic6—21no—
353Dic6—21no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—121no—
357C3⋊C8—21no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32#88311yescharacteristic
378C2×C32#89411yescharacteristic
379C2×C32#91211yescharacteristic
380C2×C32#91511yescharacteristic
381C2×C32#91811yescharacteristic
382C2×C32#91911yescharacteristic
383C2×C32#93511yescharacteristic
384C2×C32#94211yescharacteristic
385C2×C32#95011yescharacteristic
386C3⋊D3—41no—
387C3×D3—122no—
388C3×D3—121no—
389C3×D3—121no—
390C2×Q8—31no—
391C22×C4—91no—
392C2×C8—31no—
393C2×C8—271no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—91no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C22⋊C4—541no—
400C42—541no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3—181no—
408C22×C3#380211yescharacteristic
409C22×C3D3×#6711yescharacteristic
410C22×C3D3×#6711yescharacteristic
411C2×D3—62no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—121no—
419C2×D3—121no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—122no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4—181no—
434C3×C4—241no—
435C3×C4—361no—
436C3×C4—361no—
437C3×C4#379511yescharacteristic
438C3×C4#379611yescharacteristic
439C3×C4#379911yescharacteristic
440C3×C4#380411yescharacteristic
441C3×C4#380411yescharacteristic
442C3×C4#380511yescharacteristic
443Dic3—41no—
444Dic3—41no—
445Dic3—61no—
446Dic3—61no—
447Dic3—121no—
448Dic3—121no—
449Dic3—121no—
450Dic3—121no—
451Dic3—121no—
452Dic3—241no—
453Dic3—241no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#481411yescharacteristic, complemented
459C32#488811yescharacteristic, complemented
460C32#504511yescharacteristic, complemented
461C23—91no—
462Q8—61no—
463Q8—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—91no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4D3311yescharacteristic
472C8—61no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—361no—
487C2×C3#6381011yescharacteristic
488C2×C3#6381111yescharacteristic
489C2×C3#6385411yescharacteristic
490C2×C3#6388411yescharacteristic
491C2×C3#6393311yescharacteristic
492C2×C3#6394611yescharacteristic
493C2×C3#6394811yescharacteristic
494C2×C3#6394911yescharacteristic
495C2×C3#6396911yescharacteristic
496D3—122no—
497D3—121no—
498D3—121no—
499C22—181no—
500C22—181no—
501C22—181no—
502C22#8895311yescharacteristic
503C4—121no—
504C4—121no—
505C4—181no—
506C4—361no—
507C4—361no—
508C4#8896111yescharacteristic
509C4#8896211yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059044811yescharacteristic, complemented
515C3#1059071511yescharacteristic, complemented
516C3#1059155811yescharacteristic, complemented
517C2—361no—
518C2#1172286011yescharacteristic
519C2#1172289211yescharacteristic
520C2#1172299411yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722053C2semidirect
4. #11722037C2semidirect
6. C3×#63575C2semidirect
7. C3×#63566C2semidirect
35. #63575D3semidirect
39. #63566D3semidirect
306. C33#485semidirect
458. C32#4814semidirect
459. C32#4888semidirect
460. C32#5045semidirect
514. C3#10590448semidirect
515. C3#10590715semidirect
516. C3#10591558semidirect

Maximal subgroups

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

subgroupindex
#117236672
C3×#638712
#117220532
#117220372
#117220232
C3×#635752
C3×#635662
#105915583
#105907153
#105904483