Finite Group Databasebeta

← #13137422

Subgroups — #13137422

Order 1728.

528 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C4×C4⋊#272C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590366—31nomaximal
9#10590348—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17#3752⋊C3—21no—
18#3752⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88437—22no—
22#88436—22no—
23#64084D311yescharacteristic, complemented
24C3×#875—31no—
25C3×C4×D12—31no—
26C3×C4×D12—31no—
27C3×C4×D12—61no—
28C2×#3716—31no—
29C2×#3715—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
33C3×C4⋊(C3⋊C8)—31no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C4⋊#272—21no—
45C4⋊#272D412yes—
46C4×#272—42no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5057—91no—
55#5016—91no—
56#4844—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C32×C42—21no—
68C32×C42C2×D311yescharacteristic
69C32×C42C2×D311yescharacteristic
70#3752—21no—
71#3752—21no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×(C2×D3)⋊C4—61no—
78C3×(C2×D3)⋊C4—61no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84#3716—62no—
85#3715—62no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—22no—
90C2×#272—41no—
91C2×#272—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)C2×D411yescharacteristic
98C4×H(3,𝔽3)#4111yescharacteristic
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C3×#124—91no—
102C3×C4×D4—91no—
103C2×#215—91no—
104C2×#214—91no—
105C2×#213—91no—
106C2×#213—91no—
107C2×D24—91no—
108C2×#204—91no—
109#875—31no—
110#875—91no—
111C4×D12—31no—
112C4×D12—31no—
113C4×D12—61no—
114C3×C4⋊C8—91no—
115#835—91no—
116#832—91no—
117#821—91no—
118#816—91no—
119#810—91no—
120#809—91no—
121C4⋊(C3⋊C8)—31no—
122C4⋊(C3⋊C8)—91no—
123C22×C3⋊D3—21no—
124C22×C3×D3—62no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4D1211yescharacteristic
132C2×C32×C4D1211yescharacteristic
133C2×C32×C4#6711yescharacteristic
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140#634—21no—
141#634—21no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C3×D12—32no—
147C3×D12—32no—
148C3×D12—62no—
149C3×D12—62no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×C4×D3—124no—
155C3×C4×D3—122no—
156C3×C4×D3—122no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163#463—271noSylow
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167#272—42no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—91no—
170C2×C3×D4—91no—
171C2×C3×D4—91no—
172C22×C3×C4—181no—
173C2×D12—31no—
174C2×D12—31no—
175C2×D12—31no—
176C2×D12—61no—
177C2×D12—91no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—62no—
181C2×Dic6—31no—
182C2×Dic6—91no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186C3×C22⋊C4—181no—
187C3×C22⋊C4—181no—
188C3×C22⋊C4—181no—
189C3×C42—31no—
190C3×C42—31no—
191C3×C42—61no—
192C3×C42D3211yescharacteristic
193#215—182no—
194#214—182no—
195#213—182no—
196#213—182no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—61no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—61no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—181no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—61no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—91no—
211D24—182no—
212#204—182no—
213C22×C32—21no—
214C22×C32C2×D1211yescharacteristic
215C22×C32C2×#6711yescharacteristic
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3×D3—64no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—122no—
223C2×C3×D3—122no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—41no—
239C32×C4D3×D411yescharacteristic
240C32×C4D3×D411yescharacteristic
241C32×C4#23711yescharacteristic
242C32×C4#23911yescharacteristic
243C3⋊Dic3—41no—
244C3×Dic3—122no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C2×SD16—271no—
249C2×D8—271no—
250#124—91no—
251C4×D4—91no—
252C4⋊C8—91no—
253#103—271no—
254#102—271no—
255H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
256C23×C3—181no—
257C23×C3—181no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—62no—
261C22×D3—61no—
262C22×D3—181no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×D4—92no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4—181no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—362no—
281C2×C3×C4#62611yescharacteristic
282C2×C3×C4#62611yescharacteristic
283C2×C3×C4C2×D3211yescharacteristic
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—62no—
287C2×Dic3—61no—
288C2×Dic3—181no—
289D12—32no—
290D12—32no—
291D12—62no—
292D12—61no—
293D12—61no—
294D12—62no—
295D12—61no—
296D12—61no—
297D12—181no—
298D12—181no—
299C4×D3—122no—
300C4×D3—122no—
301C4×D3—124no—
302Dic6—61no—
303Dic6—61no—
304Dic6—181no—
305Dic6—181no—
306C3×C8—181no—
307C3×C8—181no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—181no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32#89511yescharacteristic
316C2×C32#90411yescharacteristic
317C2×C32#90811yescharacteristic
318C2×C32#93811yescharacteristic
319C2×C32#94911yescharacteristic
320C2×C32#95011yescharacteristic
321C3⋊D3—81no—
322C3×D3—124no—
323C3×D3—122no—
324C3×D3—122no—
325C3×D3—241no—
326C2×Q8—91no—
327C2×D4—91no—
328C2×D4—91no—
329C22×C4—181no—
330SD16—542no—
331D8—542no—
332C2×C8—91no—
333C2×C8—91no—
334C4⋊C4—91no—
335C22⋊C4—181no—
336C22⋊C4—181no—
337C22⋊C4—181no—
338C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—61no—
342C22×C3—182no—
343C22×C3—361no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3C2×#62611yescharacteristic
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—64no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—61no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—62no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—121no—
378C3×C4—361no—
379C3×C4—361no—
380C3×C4#379611yescharacteristic
381C3×C4#380211yescharacteristic
382Dic3—121no—
383Dic3—121no—
384Dic3—122no—
385Dic3—121no—
386Dic3—361no—
387C32—21no—
388C32#484411yescharacteristic, complemented
389C32#505711yescharacteristic, complemented
390C23—181no—
391C23—181no—
392Q8—181no—
393Q8—181no—
394D4—92no—
395D4—181no—
396D4—181no—
397D4—181no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—362no—
401C2×C4#620011yescharacteristic
402C2×C4#620011yescharacteristic
403C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
404C8—181no—
405C8—181no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—61no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—362no—
416C2×C3—721no—
417C2×C3#6382511yescharacteristic
418C2×C3#6382911yescharacteristic
419C2×C3#6391611yescharacteristic
420D3—122no—
421D3—122no—
422D3—124no—
423D3—241no—
424D3—721no—
425C22—182no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22C2×#620011yescharacteristic
432C4—21no—
433C4—21no—
434C4—361no—
435C4—361no—
436C4#8864911yescharacteristic
437C4#8865511yescharacteristic
438C3—31no—
439C3—31no—
440C3—61no—
441C3#1059035611yescharacteristic
442C2—362no—
443C2—721no—
444C2#1172106811yescharacteristic
445C2#1172107311yescharacteristic
446C2#1172115911yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#88436C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊#272C2semidirect
2. #64084⋊C3C2semidirect
3. C2×#88437C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
23. #64084D3semidirect
32. C4⋊(C3⋊(C3⋊C8))D3semidirect
255. H(3,𝔽3)#463semidirect
388. C32#4844semidirect
389. C32#5057semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64084⋊C32
C2×#884372
C2×#884362
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#117203082
#105903663
#105903483