Finite Group Databasebeta

← #13140613

Subgroups — #13140613

Order 1728.

510 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721846C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3#11721136C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591432—31nomaximal
9#10591431—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14#88552—21no—
15#88552—21no—
16C4×#1084C2211yescharacteristic
17#3753⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88438—22no—
23C2×#3753D311yescharacteristic
24C3×#936—31no—
25C3×#936—31no—
26C3×#936—61no—
27C2×C3×#213—31no—
28#63894—31no—
29Dic3×D12—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63566—31no—
37#63565—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084D411yescharacteristic
45C2×#1084D411yescharacteristic
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)D412yes—
51((C3⋊D3)⋊C3)⋊C4—42no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5153—91no—
55#5057—91no—
56#4787—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×#624—61no—
66C2×D3×Dic3—61no—
67#3753—22no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#217—61no—
72C3×#217—61no—
73C3×#217—61no—
74C3×#217—61no—
75C3×#217—62no—
76C3×#217—62no—
77C3×#213—62no—
78C3×C4⋊Dic3—31no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3717—62no—
87#3715—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—22no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97#1084—21no—
98#1084—21no—
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C2×C3×D8—91no—
102C3×#124—91no—
103C2×#215—91no—
104#936—31no—
105#936—31no—
106#936—61no—
107D4×Dic3—91no—
108C2×#214—91no—
109C2×#213—31no—
110C2×#213—91no—
111C2×#204—91no—
112#893—91no—
113C4×D12—91no—
114C3×#102—91no—
115#835—91no—
116#832—31no—
117#832—91no—
118#816—91no—
119Dic3⋊C8—91no—
120Dic3⋊C8—91no—
121#810—91no—
122C4⋊(C3⋊C8)—91no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)#6712yes—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×D12—32no—
151C3×D12—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—121no—
157C3×Dic6—121no—
158#624—122no—
159D3×Dic3—122no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#463—271noSylow
165C2×H(3,𝔽3)—81no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169(C3⋊D3)⋊C3—42no—
170C2×C3×Q8—91no—
171C2×C3×D4—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4D3211yescharacteristic
176C22×Dic3—181no—
177C2×D12—31no—
178C2×D12—91no—
179C2×C4×D3—181no—
180C2×C4×D3—181no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C3×D8—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C3×C4⋊C4—91no—
188C3×C22⋊C4—181no—
189C3×C22⋊C4—181no—
190C3×C42—91no—
191#217—61no—
192#217—61no—
193#217—61no—
194#217—61no—
195#217—62no—
196#217—62no—
197#217—181no—
198#215—182no—
199#214—182no—
200#213—62no—
201#213—182no—
202(C2×D3)⋊C4—181no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3—31no—
205C4⋊Dic3—91no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—61no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213#204—182no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32D3×D411yescharacteristic
225C22×C32D3×D411yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—62no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4#23511yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4#23711yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—41no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C2×SD16—271no—
250C2×D8—91no—
251#124—91no—
252C4×D4—271no—
253C4⋊C8—271no—
254#103—271no—
255#102—91no—
256H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
257C23×C3—21no—
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—62no—
261C23×C3—181no—
262C22×D3—61no—
263C22×D3—181no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×D4—21no—
267C3×D4—21no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—92no—
273C3×D4—121no—
274C3×D4—121no—
275C3×D4—181no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—61no—
279C2×C3×C4—91no—
280C2×C3×C4—91no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4C2×D3211yescharacteristic
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—61no—
294C2×Dic3—181no—
295C2×Dic3—362no—
296D12—32no—
297D12—61no—
298D12—92no—
299D12—181no—
300C4×D3—362no—
301C4×D3—362no—
302Dic6—61no—
303Dic6—61no—
304Dic6—61no—
305Dic6—61no—
306Dic6—121no—
307Dic6—121no—
308C3×C8—181no—
309C3×C8—181no—
310C3⋊C8—61no—
311C3⋊C8—61no—
312C3⋊C8—181no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—82no—
318C2×C32—81no—
319C2×C32—81no—
320C2×C32#88411yescharacteristic
321C2×C32#91111yescharacteristic
322C2×C32#91111yescharacteristic
323C2×C32#91211yescharacteristic
324C2×C32#91611yescharacteristic
325C2×C32#93911yescharacteristic
326C3⋊D3—42no—
327C3×D3—122no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C22×C4—541no—
332SD16—542no—
333D8—182no—
334C2×C8—91no—
335C2×C8—91no—
336C4⋊C4—91no—
337C22⋊C4—181no—
338C22⋊C4—181no—
339C22⋊C4—541no—
340C42—91no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—122no—
354C22×C3—122no—
355C22×C3—122no—
356C22×C3—182no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3#381111yescharacteristic
360C2×D3—62no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—182no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—181no—
373C3×C4—181no—
374C3×C4—361no—
375C3×C4—361no—
376C3×C4#380411yescharacteristic
377C3×C4D3×#6711yescharacteristic
378Dic3—61no—
379Dic3—61no—
380Dic3—61no—
381Dic3—61no—
382Dic3—121no—
383Dic3—121no—
384Dic3—121no—
385Dic3—121no—
386Dic3—122no—
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#478711yescharacteristic, complemented
391C32#515311yescharacteristic, complemented
392C23—21no—
393C23—181no—
394Q8—181no—
395Q8—181no—
396D4—21no—
397D4—21no—
398D4—92no—
399D4—181no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—1082no—
405C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C8—181no—
407C8—181no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—81no—
418C2×C3—241no—
419C2×C3—241no—
420C2×C3—242no—
421C2×C3—362no—
422C2×C3#6392711yescharacteristic
423C2×C3#6393611yescharacteristic
424C2×C3#6397111yescharacteristic
425D3—122no—
426D3—362no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—182no—
431C22—361no—
432C22—361no—
433C22#8867711yescharacteristic
434C4—181no—
435C4—181no—
436C4—361no—
437C4—361no—
438C4#8867111yescharacteristic
439C4#8867611yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3#1059143411yescharacteristic
444C2—81no—
445C2—362no—
446C2#1172119111yescharacteristic
447C2#1172119211yescharacteristic
448C2#1172131211yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#3753⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721846C2semidirect
3. #11721136C2semidirect
4. C2×#88438C2semidirect
5. #63659⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720304C2semidirect
32. #63659D3semidirect
256. H(3,𝔽3)#463semidirect
390. C32#4787semidirect
391. C32#5153semidirect

Maximal subgroups

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

subgroupindex
#117218462
C2×#3753⋊C32
#117211362
C2×#884382
#63659⋊C32
#117203262
#117203042
#105914323
#105914313