Finite Group Databasebeta

← #13156551

Subgroups — #13156551

Order 1728.

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

normal subgroups only

This group has 554 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#11723677C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722039C211yesmaximal, characteristic, complemented
4#11722033C211yesmaximal, characteristic
5#11722020C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590866—31nomaximal
9#10590535—31nomaximal
10#10590531—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#627—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88814—21no—
21C3×#3724—21no—
22C3×#888—31no—
23C3×#888—31no—
24#63919D311yescharacteristic, complemented
25#63864—31no—
26#63863—31no—
27#63860—31no—
28#63642—31no—
29#63637—31no—
30#63636—31no—
31C3×#821—31no—
32C3×#816—31no—
33C3×#810—31no—
34C3×#809—31no—
35#63567D311yescharacteristic, complemented
36#63565—31no—
37#63565—31no—
38#63565—61no—
39#63564D311yescharacteristic
40#63562—31no—
41#63562—31no—
42#63562—61no—
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C3×#627—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5089—91no—
61#4794—91no—
62#4765—91no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×#634C2×D311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6—61no—
72C2×C3×Dic6—61no—
73C2×#627—21no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77C4×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—121no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—61no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—21no—
100#3722—61no—
101#3722—61no—
102(C2×D3)⋊Dic3—61no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—21no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×C4⋊Q8—91no—
118#947—91no—
119#936—91no—
120#888—31no—
121#888—31no—
122#875—91no—
123C3×#103—91no—
124C3×#102—91no—
125#835—91no—
126#832—91no—
127#821—31no—
128#816—31no—
129#815—91no—
130#815—91no—
131#815—181no—
132#810—31no—
133#810—91no—
134#810—91no—
135#810—181no—
136#809—31no—
137#809—91no—
138#809—91no—
139#809—181no—
140#803—91no—
141C22×C3⋊D3—21no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—121no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3#6711yescharacteristic
156C2×C3⋊Dic3#6711yescharacteristic
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159#634—21no—
160#634—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—41no—
170C2×C3×Dic3—41no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C3×D12—62no—
177C3×D12—61no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—61no—
181C3×D12—121no—
182C3×Dic6—62no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C3×Dic6—61no—
186C3×Dic6—61no—
187C3×Dic6—121no—
188C3×Dic6—122no—
189C3×Dic6—122no—
190#627—42no—
191C32×C8—61no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—62no—
194C3⋊(C3⋊C8)—61no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—121no—
197C3⋊(C3⋊C8)—121no—
198C3⋊(C3⋊C8)—122no—
199C3×C3⋊C8—22no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202#489—271noSylow
203C2×C33#12411yescharacteristic
204C2×C33AGL(1,ℤ/8)11yescharacteristic
205C2×C33#13711yescharacteristic
206C3×C3⋊D3—81no—
207C2×C3×Q8—91no—
208C2×C3×Q8—181no—
209C2×C3×D4—91no—
210C2×D12—31no—
211C2×D12—31no—
212C2×D12—61no—
213C2×Dic6—31no—
214C2×Dic6—31no—
215C2×Dic6—61no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—61no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—181no—
228C3×C42—91no—
229#217—181no—
230#217—181no—
231(C2×D3)⋊C4—181no—
232(C2×D3)⋊C4—181no—
233(C2×D3)⋊C4—181no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3D3211yescharacteristic
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239C4×Dic3—31no—
240C4×Dic3—31no—
241C4×Dic3—61no—
242C2×C3⋊C8—91no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—181no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—361no—
249C2×C3⋊C8D3211yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32C2×#6711yescharacteristic
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3×D3—122no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—241no—
268C2×C3×D3—241no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—121no—
278C32×C4—121no—
279C32×C4#23511yescharacteristic
280C32×C4#23511yescharacteristic
281C32×C4#23711yescharacteristic
282C32×C4#23711yescharacteristic
283C32×C4#23911yescharacteristic
284C32×C4#23911yescharacteristic
285C3⋊Dic3—21no—
286C3⋊Dic3—21no—
287C3⋊Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—81no—
297C3×Dic3—81no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—122no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C4⋊Q8—91no—
304#124—271no—
305#103—91no—
306#103—271no—
307#102—91no—
308#102—271no—
309#97—271no—
310C33#48911yescharacteristic, complemented, Sylow
311C23×C3—181no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—121no—
315C3×Q8—181no—
316C3×Q8—181no—
317C3×Q8—362no—
318C3×D4—181no—
319C3×D4—181no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4—181no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—121no—
347D12—62no—
348D12—61no—
349D12—61no—
350D12—61no—
351D12—61no—
352D12—121no—
353Dic6—62no—
354Dic6—61no—
355Dic6—61no—
356Dic6—61no—
357Dic6—61no—
358Dic6—121no—
359Dic6—122no—
360Dic6—122no—
361C3×C8—62no—
362C3×C8—61no—
363C3×C8—61no—
364C3⋊C8—21no—
365C3⋊C8—182no—
366C3⋊C8—181no—
367C3⋊C8—181no—
368C3⋊C8—181no—
369C3⋊C8—361no—
370C3⋊C8—361no—
371C3⋊C8—362no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#88511yescharacteristic
385C2×C32#88511yescharacteristic
386C2×C32#91111yescharacteristic
387C2×C32#91411yescharacteristic
388C2×C32#91511yescharacteristic
389C2×C32#91811yescharacteristic
390C2×C32#93211yescharacteristic
391C2×C32#94211yescharacteristic
392C2×C32#94711yescharacteristic
393C3⋊D3—81no—
394C3×D3—242no—
395C3×D3—241no—
396C3×D3—241no—
397C2×Q8—91no—
398C2×Q8—181no—
399C2×D4—91no—
400C2×C8—31no—
401C2×C8—271no—
402C4⋊C4—31no—
403C4⋊C4—31no—
404C4⋊C4—181no—
405C22⋊C4—541no—
406C22⋊C4—541no—
407C42—91no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3#380211yescharacteristic
416C22×C3D3×#6711yescharacteristic
417C22×C3D3×#6711yescharacteristic
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—241no—
426C2×D3—241no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—181no—
440C3×C4—181no—
441C3×C4—241no—
442C3×C4—241no—
443C3×C4—361no—
444C3×C4#379611yescharacteristic
445C3×C4#379611yescharacteristic
446C3×C4#379811yescharacteristic
447C3×C4#379911yescharacteristic
448C3×C4#379911yescharacteristic
449C3×C4#380511yescharacteristic
450Dic3—41no—
451Dic3—41no—
452Dic3—61no—
453Dic3—61no—
454Dic3—61no—
455Dic3—61no—
456Dic3—121no—
457Dic3—121no—
458Dic3—122no—
459Dic3—121no—
460Dic3—121no—
461C32—21no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32#476511yescharacteristic, complemented
466C32#479411yescharacteristic, complemented
467C32#508911yescharacteristic, complemented
468C23—181no—
469Q8—181no—
470Q8—181no—
471Q8—362no—
472D4—181no—
473D4—181no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—91no—
477C2×C4—91no—
478C2×C4—181no—
479C2×C4D3311yescharacteristic
480C8—61no—
481C8—541no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—721no—
495C2×C3#6379811yescharacteristic
496C2×C3#6381111yescharacteristic
497C2×C3#6390711yescharacteristic
498C2×C3#6390711yescharacteristic
499C2×C3#6391511yescharacteristic
500C2×C3#6393211yescharacteristic
501C2×C3#6393411yescharacteristic
502C2×C3#6394611yescharacteristic
503C2×C3#6394811yescharacteristic
504D3—242no—
505D3—241no—
506D3—241no—
507C22—361no—
508C22—361no—
509C22—361no—
510C22#8895311yescharacteristic
511C4—121no—
512C4—121no—
513C4—181no—
514C4—181no—
515C4—361no—
516C4#8896411yescharacteristic
517C4#8896511yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059053111yescharacteristic, complemented
523C3#1059053511yescharacteristic, complemented
524C3#1059086611yescharacteristic, complemented
525C2—721no—
526C2#1172287511yescharacteristic
527C2#1172289211yescharacteristic
528C2#1172300711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#489
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
2. C3×#63919C2semidirect
3. #11722039C2semidirect
5. #11722020C2semidirect
6. C3×#63567C2semidirect
24. #63919D3semidirect
35. #63567D3semidirect
310. C33#489semidirect
465. C32#4765semidirect
466. C32#4794semidirect
467. C32#5089semidirect
522. C3#10590531semidirect
523. C3#10590535semidirect
524. C3#10590866semidirect

Maximal subgroups

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

subgroupindex
#117236772
C3×#639192
#117220392
#117220332
#117220202
C3×#635672
C3×#635642
#105908663
#105905353
#105905313