Finite Group Databasebeta

← #13167393

Subgroups — #13167393

Order 1728.

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

normal subgroups only

This group has 594 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×#64283C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4#11722782C212yesmaximal, complemented
5#11722781C212yesmaximal, complemented
6#10593311—31nomaximal
7C3×#3828—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9#89049C412yes—
10#89046—21no—
11C4⋊#1113C2211yescharacteristic
12C4×#1113C412yes—
13#89042—21no—
14#89041—21no—
15#89040C2211yescharacteristic
16C3×#3792—21no—
17C3×#3788C2212yes—
18C3×#3787—21no—
19C2×C3×C3⋊(C3⋊C8)—21no—
20#88809—21no—
21#88809C2212yes—
22#64306—31no—
23D3×#235—61no—
24#64283D311yescharacteristic, complemented
25C2×#3712—31no—
26C2×#3711—61no—
27#63728—32no—
28#63727—32no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C2×#1113C2×C412yes—
31C2×C33×C4D411yescharacteristic
32C3×#638—22no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4⋊(C3⋊D3)—21no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3C2311yescharacteristic
40C3×#634—21no—
41#6296—21no—
42#6295—22no—
43#6295—21no—
44#6294—22no—
45#6294C2×C412yes—
46C3×#622—22no—
47C3×C3⋊(C3⋊C8)—22no—
48#5132—91nomaximal
49#3828—21no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C3×#235—61no—
52C3×#235—61no—
53C3×#235—121no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56#3811—32no—
57D3×#67—62no—
58#3805—61no—
59#3804—62no—
60#3802—31no—
61D3×D12—61no—
62C4×D32—32no—
63C4×D32—62no—
64#3799—61no—
65#3798—62no—
66#3798—61no—
67#3797—31no—
68#3796—61no—
69#3795—61no—
70D3×Dic6—61no—
71#3792—21no—
72#3788C2×D312yes—
73#3787—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C2×C3×C3⋊C8—121no—
78#3712—32no—
79#3712—61no—
80#3711—62no—
81#3711—62no—
82C22×C33C22⋊C411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—41no—
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86C2×C3×C3⋊D3C22×C411yescharacteristic
87#1113—42no—
88C33×C4C22⋊C411yescharacteristic
89C33×C4C2×D411yescharacteristic
90C3×C3⋊Dic3—22no—
91C3×C3⋊Dic3C22⋊C411yescharacteristic
92C3×C3⋊Dic3C2×D411yescharacteristic
93D3×#41—181no—
94C2×#235—91no—
95C2×C3×M16—91no—
96C2×#208—91no—
97C2×ΓL(1,𝔽25)—181no—
98#824—92no—
99#823—92no—
100C22×C3⋊D3C4×D311yescharacteristic
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C2×D32—32no—
104C2×D32—62no—
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#6711yescharacteristic
112C4⋊(C3⋊D3)—21no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3C4×D311yescharacteristic
115C4×C3⋊D3D1211yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117C4×C3⋊D3C22×D311yescharacteristic
118#634—21no—
119C3×#67—62no—
120C3×#67—62no—
121C3×#67—122no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C3×D12—61no—
125C3×D12—61no—
126C3×D12—121no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—62no—
136C3×C4×D3—62no—
137C3×C4×D3—122no—
138C3×Dic6—61no—
139C3×Dic6—61no—
140C3×Dic6—121no—
141#627—62no—
142#627—61no—
143#626—62no—
144#626—62no—
145#626—62no—
146#625—62no—
147#625—61no—
148#624—32no—
149#624—62no—
150D3×Dic3—62no—
151D3×Dic3—62no—
152D3×Dic3—62no—
153#622—22no—
154C3⋊(C3⋊C8)—22no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—122no—
158#413—271noSylow
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—21no—
162C3×C3⋊D3—21no—
163C3×C3⋊D3—42no—
164C3×#41—181no—
165C22×C3×C4—91no—
166C2×#67—92no—
167#239—181no—
168D3×Q8—181no—
169#237—181no—
170#237—182no—
171D3×D4—181no—
172D3×D4—182no—
173#235—61no—
174#235—61no—
175#235—121no—
176#235—181no—
177#235—181no—
178#235—182no—
179#235—181no—
180C2×D12—91no—
181C2×C4×D3—61no—
182C2×C4×D3—61no—
183C2×C4×D3—92no—
184C2×C4×D3—182no—
185C2×Dic6—91no—
186C3×M16—92no—
187C3×M16—181no—
188C2×C3×C8—181no—
189C2×C3×C8—181no—
190#208—92no—
191#208—181no—
192C2×C3⋊C8—61no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—121no—
195ΓL(1,𝔽25)—182no—
196ΓL(1,𝔽25)—182no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32(C2×D3)⋊C411yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
205C2×C3⋊D3C2×C4×D311yescharacteristic
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C2×C3×D3—122no—
213C2×C3×D3—122no—
214C2×C3×D3—122no—
215D32—124no—
216D32—122no—
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×C4C2×D1211yescharacteristic
227C3⋊Dic3—22no—
228C3⋊Dic3(C2×D3)⋊C411yescharacteristic
229C3⋊Dic3C2×#6711yescharacteristic
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—122no—
237C2×#41—271no—
238C2×M16—91no—
239C2×M16—271no—
240#101—272no—
241#100—272no—
242C33#41311yescharacteristic, complemented, Sylow
243C23×C3—91no—
244C22×D3—61no—
245C22×D3—61no—
246C22×D3—92no—
247C22×D3—182no—
248C3×Q8—181no—
249C3×D4—181no—
250C3×D4—182no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—182no—
261C2×C3×C4GO+(4,𝔽2)11yescharacteristic
262#67—62no—
263#67—62no—
264#67—122no—
265#67—182no—
266#67—182no—
267#67—182no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—92no—
271C2×Dic3—182no—
272D12—61no—
273D12—61no—
274D12—121no—
275D12—181no—
276D12—182no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—122no—
288C4×D3—182no—
289C4×D3—182no—
290C4×D3—182no—
291Dic6—61no—
292Dic6—61no—
293Dic6—121no—
294Dic6—181no—
295Dic6—182no—
296C3×C8—182no—
297C3×C8—182no—
298C3⋊C8—62no—
299C3⋊C8—62no—
300C3⋊C8—122no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—81no—
310C2×C32C2×(C2×D3)⋊C411yescharacteristic
311C3⋊D3—21no—
312C3⋊D3—21no—
313C3⋊D3—42no—
314C3×D3—62no—
315C3×D3—121no—
316C3×D3—121no—
317C3×D3—121no—
318C3×D3—122no—
319C3×D3—122no—
320C3×D3—242no—
321#41—181no—
322#41—541no—
323#41—541no—
324#41—541no—
325C2×Q8—271no—
326C2×D4—271no—
327C2×D4—272no—
328C22×C4—91no—
329C22×C4—272no—
330M16—92no—
331M16—181no—
332M16—272no—
333M16—541no—
334C2×C8—181no—
335C2×C8—181no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—91no—
341C22×C3—91no—
342C22×C3—182no—
343C22×C3—361no—
344C22×C3D32⋊C411yescharacteristic
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—122no—
352C2×D3—122no—
353C2×D3—122no—
354C2×D3—364no—
355C2×D3—362no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—91no—
365C3×C4—91no—
366C3×C4—182no—
367C3×C4D32⋊C411yescharacteristic
368C3×C4C2×GO+(4,𝔽2)11yescharacteristic
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—62no—
374Dic3—62no—
375Dic3—122no—
376C32—21no—
377C32—21no—
378C32—41no—
379C32—41no—
380C32#513211yescharacteristic, complemented
381C23—91no—
382C23—272no—
383Q8—181no—
384Q8—541no—
385D4—182no—
386D4—181no—
387D4—541no—
388D4—542no—
389C2×C4—91no—
390C2×C4—91no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—182no—
395C2×C4—272no—
396C2×C4—542no—
397C2×C4#157⋊D311yescharacteristic
398C8—182no—
399C8—182no—
400C2×C3—22no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—41no—
405C2×C3—42no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—81no—
409C2×C3—181no—
410C2×C3—181no—
411C2×C3—362no—
412C2×C3C2×D32⋊C411yescharacteristic
413D3—62no—
414D3—121no—
415D3—121no—
416D3—121no—
417D3—122no—
418D3—122no—
419D3—242no—
420C22—91no—
421C22—91no—
422C22—182no—
423C22—361no—
424C22—1082no—
425C22#8893911yescharacteristic
426C4—91no—
427C4—91no—
428C4—182no—
429C4#8893911yescharacteristic
430C4C2×#157⋊D311yescharacteristic
431C3—21no—
432C3—21no—
433C3—41no—
434C3—41no—
435C3#1059331111yescharacteristic, complemented
436C2—21no—
437C2—181no—
438C2—181no—
439C2—362no—
440C2C2×#8893911yescharacteristic
subgroupG/H
centerC4#88939
commutatorC2×C3×C3⋊D3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#413
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×#64283C2semidirect
3. C2×#88809C2semidirect
4. #11722782C2semidirect
5. #11722781C2semidirect
24. #64283D3semidirect
242. C33#413semidirect
380. C32#5132semidirect
435. C3#10593311semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#642832
C2×#888092
#117227822
#117227812
#105933113
#51329