Finite Group Databasebeta

← #13167478

Subgroups — #13167478

Order 1728.

564 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 81 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×#88940C211yesmaximal, characteristic
2#11723682C211yesmaximal, characteristic
3C4×#6294C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5#11722791C211yesmaximal, characteristic
6#11722791C211yesmaximal, characteristic
7#11722791C211yesmaximal, characteristic
8#10593338—31nomaximal
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×#6294C2211yescharacteristic
15C2×#6294C2211yescharacteristic
16#88940C414yescomplemented
17C3×#3750—21no—
18C3×C4×C3⋊Dic3—21no—
19#88815—21no—
20#88813—21no—
21C2×#3773—31no—
22#63883—31no—
23C4×#624—31no—
24D3×Dic3⋊C4—61no—
25C4×D3×Dic3—61no—
26#63781D311yescharacteristic
27#63737—31no—
28#63737—31no—
29#63737—31no—
30C22×C3×C3⋊D3C2311yescharacteristic
31C2×C3×#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157C2×C412yes—
37C2×C33×C4D411yescharacteristic
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45#6294—21no—
46#6294—21no—
47#6294—21no—
48#6294—21no—
49#6294—21no—
50#6294C2×C412yes—
51#6294C2×C412yes—
52#6294Q812yes—
53#4653—91nomaximal
54C22×#157C2×D311yescharacteristic
55C22×#157C2×D311yescharacteristic
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×#624—31no—
60C2×#624—31no—
61C2×#624—31no—
62C2×#624—31no—
63C2×D3×Dic3—61no—
64C2×D3×Dic3—61no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67#3773—34no—
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—121no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C3×C4×Dic3—121no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78Dic3⋊Dic3—61no—
79Dic32—61no—
80Dic32—61no—
81Dic32—61no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3C22×C411yescharacteristic
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3#4111yescharacteristic
86C2×C3×C3⋊D3#4111yescharacteristic
87C2×C3×C3⋊D3#4111yescharacteristic
88C2×C3×C3⋊D3#4111yescharacteristic
89C3×#157—22no—
90C3×#157—41no—
91C33×C4—21no—
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—21no—
96C3×C3⋊Dic3—41no—
97C2×C4⋊Dic3—91no—
98C2×Dic3⋊C4—91no—
99C2×C4×Dic3—91no—
100D3×C4⋊C4—181no—
101C42×D3—181no—
102C3×#95—91no—
103C3⋊#95—91no—
104C3⋊#95—91no—
105C3⋊#95—91no—
106C22×C3⋊D3C22×D311yescharacteristic
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C2×#157—21no—
110C2×#157—21no—
111C2×#157—21no—
112C2×#157—21no—
113C2×#157—21no—
114C2×#157C4×D312yes—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4D1211yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3D1211yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—121no—
138C2×C3×Dic3—121no—
139C2×C3×Dic3—121no—
140C2×C3×Dic3—121no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—121no—
144C3×C4×D3—121no—
145#624—32no—
146#624—32no—
147#624—32no—
148#624—61no—
149#624—61no—
150#624—61no—
151#624—61no—
152#624—61no—
153D3×Dic3—62no—
154D3×Dic3—62no—
155D3×Dic3—62no—
156D3×Dic3—62no—
157D3×Dic3—62no—
158D3×Dic3—62no—
159D3×Dic3—121no—
160D3×Dic3—121no—
161#387—271noSylow
162C2×C33C4×D411yescharacteristic
163C2×C33C22⋊Q811yescharacteristic
164C2×C33#12311yescharacteristic
165C3×C3⋊D3#11711yescharacteristic
166C3×C3⋊D3C4×Q811yescharacteristic
167C3×C3⋊D3#12511yescharacteristic
168C3×C3⋊D3#12611yescharacteristic
169C22×C3×C4—91no—
170C22×C3×C4—91no—
171C22×C3×C4—91no—
172C22×Dic3—91no—
173C22×Dic3—91no—
174C22×Dic3—91no—
175C22×Dic3—91no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C3×C4⋊C4—181no—
183C3×C42—181no—
184C4⋊Dic3—94no—
185C4⋊Dic3—181no—
186Dic3⋊C4—61no—
187Dic3⋊C4—61no—
188Dic3⋊C4—121no—
189Dic3⋊C4—181no—
190Dic3⋊C4—181no—
191C4×Dic3—61no—
192C4×Dic3—61no—
193C4×Dic3—121no—
194C4×Dic3—181no—
195C4×Dic3—181no—
196C4×Dic3—181no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32C2×D1211yescharacteristic
202C2×C3⋊D3C2×C4×D311yescharacteristic
203C2×C3⋊D3#23511yescharacteristic
204C2×C3⋊D3#23711yescharacteristic
205C2×C3⋊D3#23711yescharacteristic
206C2×C3⋊D3#23711yescharacteristic
207C2×C3⋊D3D3×Q811yescharacteristic
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220#157—22no—
221#157—41no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—42no—
226C32×C4—81no—
227C3⋊Dic3—22no—
228C3⋊Dic3—22no—
229C3⋊Dic3—22no—
230C3⋊Dic3—21no—
231C3⋊Dic3—41no—
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—62no—
239C3×Dic3—121no—
240C3×Dic3—122no—
241C3×Dic3—122no—
242C3×Dic3—122no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—241no—
246C2×C4⋊C4—271no—
247C2×C4⋊C4—271no—
248C2×C42—271no—
249#95—91no—
250#95—271no—
251#95—271no—
252#95—271no—
253C33#38711yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C22×D3—61no—
256C22×D3—61no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—92no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4GO+(4,𝔽2)11yescharacteristic
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—92no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288C2×Dic3—121no—
289C2×Dic3—121no—
290C2×Dic3—121no—
291C2×Dic3—121no—
292C2×Dic3—181no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295C2×Dic3—181no—
296C2×Dic3—181no—
297C4×D3—62no—
298C4×D3—62no—
299C4×D3—121no—
300C4×D3—121no—
301C4×D3—182no—
302C4×D3—182no—
303C4×D3—182no—
304C4×D3—182no—
305C4×D3—182no—
306C4×D3—182no—
307C4×D3—361no—
308C4×D3—361no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32C4×D1211yescharacteristic
322C2×C32#88611yescharacteristic
323C2×C32#89711yescharacteristic
324C3⋊D3#87711yescharacteristic
325C3⋊D3#87911yescharacteristic
326C3⋊D3#88711yescharacteristic
327C3⋊D3#88911yescharacteristic
328C3×D3—61no—
329C3×D3—61no—
330C3×D3—61no—
331C3×D3—61no—
332C3×D3—61no—
333C3×D3—61no—
334C3×D3—61no—
335C3×D3—61no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4—271no—
340C22×C4—271no—
341C22×C4—271no—
342C22×C4—271no—
343C4⋊C4—181no—
344C4⋊C4—274no—
345C4⋊C4—541no—
346C42—181no—
347C42—541no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—91no—
353C22×C3—91no—
354C22×C3—91no—
355C22×C3—91no—
356C22×C3—91no—
357C22×C3—91no—
358C22×C3C2×GO+(4,𝔽2)11yescharacteristic
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—41no—
374C3×C4—42no—
375C3×C4—81no—
376C3×C4—182no—
377C3×C4—182no—
378C3×C4—182no—
379C3×C4—182no—
380C3×C4—181no—
381C3×C4—361no—
382C3×C4—361no—
383Dic3—62no—
384Dic3—62no—
385Dic3—62no—
386Dic3—62no—
387Dic3—62no—
388Dic3—62no—
389Dic3—62no—
390Dic3—121no—
391Dic3—122no—
392Dic3—122no—
393Dic3—122no—
394Dic3—121no—
395Dic3—121no—
396Dic3—241no—
397C32—21no—
398C32—21no—
399C32—41no—
400C32—41no—
401C32#465311yescharacteristic, complemented
402C23—91no—
403C2×C4—92no—
404C2×C4—91no—
405C2×C4—181no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4—272no—
417C2×C4—272no—
418C2×C4—272no—
419C2×C4—541no—
420C2×C4—541no—
421C2×C4—541no—
422C2×C4—541no—
423C2×C4—541no—
424C2×C4#157⋊D311yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—91no—
438C2×C3—91no—
439C2×C3—91no—
440C2×C3—91no—
441C2×C3D32⋊Q811yescharacteristic
442C2×C3C4×GO+(4,𝔽2)11yescharacteristic
443C2×C3#6423311yescharacteristic
444D3—61no—
445D3—61no—
446D3—61no—
447D3—61no—
448D3—61no—
449D3—61no—
450D3—61no—
451D3—61no—
452C22—91no—
453C22—91no—
454C22—91no—
455C22—91no—
456C22—91no—
457C22—91no—
458C22C2×#157⋊D311yescharacteristic
459C4—21no—
460C4—182no—
461C4—182no—
462C4—182no—
463C4—182no—
464C4—181no—
465C4—361no—
466C4—361no—
467C3—21no—
468C3—21no—
469C3—41no—
470C3—41no—
471C3#1059333811yescharacteristic, complemented
472C2—91no—
473C2—91no—
474C2—91no—
475C2—91no—
476C2#1172433311yescharacteristic
477C2C4×#157⋊D311yescharacteristic
478C2#1172434611yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#387
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
16. #88940C4semidirect
253. C33#387semidirect
401. C32#4653semidirect
471. C3#10593338semidirect

Maximal subgroups

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

subgroupindex
C2×#889402
#117236822
C4×#62942
C3×#637812
#117227912
#105933383
#46539