Finite Group Databasebeta

← #13167390

Subgroups — #13167390

Order 1728.

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

normal subgroups only

This group has 626 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#11724618C211yesmaximal, characteristic
2C3×#64286C211yesmaximal, characteristic, complemented
3#11723673C211yesmaximal, characteristic, complemented
4#11722792C214yesmaximal, complemented
5#10593345—31nomaximal
6C3×#3828—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#89049C2212yes—
9#89046—21no—
10C4⋊#1113C411yescharacteristic, complemented
11C4×#1113C412yescomplemented
12#89042—21no—
13#89041—21no—
14#89040C411yescharacteristic, complemented
15C3×C22⋊#157C2212yes—
16C3×C4⋊#157—21no—
17C3×C4×#157—21no—
18C3×C4×C3⋊Dic3—21no—
19#88815—21no—
20#88814C2212yes—
21#64306—31no—
22D3×#235—61no—
23#64286D311yescharacteristic, complemented
24#63874—31no—
25#63855—61no—
26#63738—34no—
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×#1113C2×C412yes—
29C2×C3×#157—22no—
30C2×C33×C4D411yescharacteristic
31C3×#638—22no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4⋊(C3⋊D3)—21no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×#634—21no—
41#6296—21no—
42#6295—22no—
43#6295—21no—
44#6294—22no—
45#6294C2×C412yes—
46#5006—91nomaximal
47#3828—21no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C3×#235—61no—
50C3×#235—61no—
51C3×#235—121no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54#3811—32no—
55D3×#67—62no—
56#3805—61no—
57#3804—62no—
58#3802—31no—
59D3×D12—61no—
60C4×D32—32no—
61C4×D32—62no—
62#3799—61no—
63#3798—62no—
64#3798—61no—
65#3797—31no—
66#3796—61no—
67#3795—61no—
68D3×Dic6—61no—
69C22⋊#157C2×D312yes—
70C4⋊#157—21no—
71C4×#157—21no—
72C4×C3⋊Dic3—21no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—61no—
75C3×C4×Dic3—121no—
76#3724—61no—
77Dic3⋊Dic3—62no—
78#3722—32no—
79(C2×D3)⋊Dic3—62no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3C22×C411yescharacteristic
84C2×C3×C3⋊D3C2×D411yescharacteristic
85#1113—42no—
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22⋊C411yescharacteristic
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3—42no—
91C3×C3⋊Dic3C22⋊C411yescharacteristic
92C3×C3⋊Dic3C22⋊C411yescharacteristic
93D3×#41—181no—
94C2×#235—91no—
95C3×#117—91no—
96#926—91no—
97#872—181no—
98#806—94no—
99C22×C3⋊D3C22×D311yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C2×D32—32no—
103C2×D32—62no—
104C2×#157—22no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4D1211yescharacteristic
110#638—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3D1211yescharacteristic
113C4⋊(C3⋊D3)—21no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3C4×D311yescharacteristic
116C4×C3⋊D3C4×D311yescharacteristic
117C4×C3⋊D3#6711yescharacteristic
118C4×C3⋊D3#6711yescharacteristic
119#634—21no—
120C3×#67—62no—
121C3×#67—62no—
122C3×#67—122no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—122no—
128C3×D12—61no—
129C3×D12—61no—
130C3×D12—121no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—122no—
142C3×Dic6—61no—
143C3×Dic6—61no—
144C3×Dic6—121no—
145#627—62no—
146#627—61no—
147#626—62no—
148#626—62no—
149#626—62no—
150#625—62no—
151#625—61no—
152#624—32no—
153#624—62no—
154D3×Dic3—62no—
155D3×Dic3—62no—
156D3×Dic3—62no—
157#410—271noSylow
158C2×C33—21no—
159C2×C33C2×C22⋊C411yescharacteristic
160C3×C3⋊D3—21no—
161C3×C3⋊D3—21no—
162C3×C3⋊D3—42no—
163C3×#41—181no—
164C22×C3×C4—91no—
165C2×#67—92no—
166#239—181no—
167D3×Q8—181no—
168#237—181no—
169#237—182no—
170D3×D4—181no—
171D3×D4—182no—
172#235—61no—
173#235—61no—
174#235—121no—
175#235—181no—
176#235—181no—
177#235—182no—
178#235—181no—
179C2×D12—91no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C2×C4×D3—92no—
183C2×C4×D3—182no—
184C2×Dic6—91no—
185C3×C4⋊C4—181no—
186C3×C22⋊C4—92no—
187C3×C42—181no—
188C3×C42—181no—
189#217—92no—
190(C2×D3)⋊C4—182no—
191C4⋊Dic3—181no—
192Dic3⋊C4—182no—
193C4×Dic3—61no—
194C4×Dic3—61no—
195C4×Dic3—121no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32C2×D1211yescharacteristic
201C2×C3⋊D3—22no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3C2×C4×D311yescharacteristic
204C2×C3⋊D3C2×#6711yescharacteristic
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214D32—124no—
215D32—122no—
216#157—42no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4(C2×D3)⋊C411yescharacteristic
226C32×C4(C2×D3)⋊C411yescharacteristic
227C3⋊Dic3—22no—
228C3⋊Dic3—42no—
229C3⋊Dic3(C2×D3)⋊C411yescharacteristic
230C3⋊Dic3(C2×D3)⋊C411yescharacteristic
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—122no—
238C3×Dic3—122no—
239C3×Dic3—122no—
240C3×Dic3—124no—
241C2×#41—271no—
242#117—91no—
243#117—271no—
244#99—274no—
245C33#41011yescharacteristic, complemented, Sylow
246C23×C3—91no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—92no—
250C22×D3—182no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—182no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—182no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4GO+(4,𝔽2)11yescharacteristic
267#67—62no—
268#67—62no—
269#67—122no—
270#67—182no—
271#67—182no—
272#67—182no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—62no—
276C2×Dic3—62no—
277C2×Dic3—92no—
278C2×Dic3—122no—
279C2×Dic3—182no—
280D12—61no—
281D12—61no—
282D12—121no—
283D12—181no—
284D12—182no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—62no—
294C4×D3—62no—
295C4×D3—122no—
296C4×D3—182no—
297C4×D3—182no—
298C4×D3—182no—
299Dic6—61no—
300Dic6—61no—
301Dic6—121no—
302Dic6—181no—
303Dic6—182no—
304C2×C32—22no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—42no—
312C2×C32—81no—
313C2×C32C2×(C2×D3)⋊C411yescharacteristic
314C3⋊D3—21no—
315C3⋊D3—21no—
316C3⋊D3—42no—
317C3×D3—62no—
318C3×D3—121no—
319C3×D3—121no—
320C3×D3—121no—
321C3×D3—122no—
322C3×D3—122no—
323C3×D3—242no—
324#41—181no—
325#41—541no—
326#41—541no—
327#41—541no—
328C2×Q8—271no—
329C2×D4—271no—
330C2×D4—272no—
331C22×C4—91no—
332C22×C4—272no—
333C4⋊C4—181no—
334C4⋊C4—541no—
335C22⋊C4—92no—
336C22⋊C4—272no—
337C42—181no—
338C42—181no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—91no—
344C22×C3—91no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3C2×GO+(4,𝔽2)11yescharacteristic
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—364no—
358C2×D3—362no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—91no—
368C3×C4—91no—
369C3×C4—182no—
370C3×C4—362no—
371C3×C4—362no—
372C3×C4D32⋊C411yescharacteristic
373C3×C4D32⋊C411yescharacteristic
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378Dic3—62no—
379Dic3—62no—
380Dic3—122no—
381Dic3—122no—
382Dic3—122no—
383Dic3—124no—
384C32—21no—
385C32—21no—
386C32—41no—
387C32—41no—
388C32#500611yescharacteristic, complemented
389C23—91no—
390C23—272no—
391Q8—181no—
392Q8—541no—
393D4—182no—
394D4—181no—
395D4—541no—
396D4—542no—
397C2×C4—91no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4—272no—
406C2×C4—542no—
407C2×C4#157⋊D311yescharacteristic
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—42no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—81no—
417C2×C3—181no—
418C2×C3—181no—
419C2×C3—362no—
420C2×C3C2×D32⋊C411yescharacteristic
421D3—62no—
422D3—121no—
423D3—121no—
424D3—121no—
425D3—122no—
426D3—122no—
427D3—242no—
428C22—91no—
429C22—91no—
430C22—182no—
431C22—361no—
432C22—1082no—
433C22C2×#157⋊D311yescharacteristic
434C4—91no—
435C4—91no—
436C4—182no—
437C4—362no—
438C4—362no—
439C4#8893911yescharacteristic
440C4#8893911yescharacteristic
441C3—21no—
442C3—21no—
443C3—41no—
444C3—41no—
445C3#1059334511yescharacteristic, complemented
446C2—21no—
447C2—181no—
448C2—181no—
449C2—362no—
450C2C2×#8893911yescharacteristic
subgroupG/H
centerC4#88939
commutatorC2×C3×C3⋊D3C22×C4
FrattiniC22C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#410
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64286C2semidirect
3. #11723673C2semidirect
4. #11722792C2semidirect
10. C4⋊#1113C4semidirect
11. C4×#1113C4semidirect
14. #89040C4semidirect
23. #64286D3semidirect
245. C33#410semidirect
388. C32#5006semidirect
445. C3#10593345semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#642862
#117236732
#117227922
#105933453
#50069