Finite Group Databasebeta

← #13137407

Subgroups — #13137407

Order 1728.

528 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C2×#88435C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#11720323C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10590330—31nomaximal
9#10590326—31nomaximal
10C2×C4⋊#272C411yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14#88542C411yescharacteristic
15#88542C411yescharacteristic
16C4⋊#1084C411yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88435—22no—
22#88434—22no—
23C3×C4×D12—31no—
24C3×C4×D12—31no—
25C3×C4×D12—61no—
26C2×#3711—31no—
27C2×#3711—31no—
28C2×D3×C3⋊C8—31no—
29C2×D3×C3⋊C8—31no—
30C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
31#63630D311yescharacteristic, complemented
32C3×C4⋊(C3⋊C8)—31no—
33C3×#803—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36(C2×D3)⋊(C3⋊C8)—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38C22×#272C2×C411yescharacteristic
39C22×#272C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2×C411yescharacteristic
41C2×C4×H(3,𝔽3)C2×C411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2×C411yescharacteristic
44C2×#1084C2×C411yescharacteristic
45C4⋊#272—22no—
46C4×#272—21no—
47C4×#272—21no—
48C4×#272—21no—
49C4×#272—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5011—91no—
55#4836—91no—
56#4706—91no—
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—61no—
66C32×C42—21no—
67C32×C42C2×D311yescharacteristic
68C32×C42C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—31no—
74C3×(C2×D3)⋊C4—31no—
75C3×(C2×D3)⋊C4—31no—
76C3×(C2×D3)⋊C4—31no—
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—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86#3711—62no—
87#3711—62no—
88D3×C3⋊C8—62no—
89D3×C3⋊C8—62no—
90C22×H(3,𝔽3)C22×C411yescharacteristic
91C2×#272—21no—
92C2×#272—21no—
93C2×#272—21no—
94C2×#272—21no—
95C2×#272—21no—
96C2×#272M1612yescomplemented
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)#4111yescharacteristic
101#1084—21no—
102#1084—21no—
103C3×C4×D4—91no—
104C2×#208—91no—
105C22×C3⋊C8—91no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110C4×D12—31no—
111C4×D12—31no—
112C4×D12—61no—
113C3×C4⋊C8—91no—
114C3×#97—91no—
115#830—91no—
116#830—91no—
117(C2×D3)⋊C8—91no—
118(C2×D3)⋊C8—91no—
119(C2×D3)⋊C8—91no—
120(C2×D3)⋊C8—91no—
121C4⋊(C3⋊C8)—31no—
122C4⋊(C3⋊C8)—91no—
123#803—31no—
124#803—91no—
125C22×C3×D3—31no—
126C22×C3×D3—31no—
127C22×C3×D3—31no—
128C22×C3×D3—31no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C4×D311yescharacteristic
135C2×C32×C4C4×D311yescharacteristic
136C2×C32×C4C4×D311yescharacteristic
137C2×C32×C4C4×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C3×D12—62no—
147C3×D12—62no—
148C3×D12—64no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×C4×D3—61no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C3×C4×D3—121no—
158C3×C4×D3—121no—
159C3×C4×D3—121no—
160C3×C4×D3—121no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)—21no—
163C3⋊(C3⋊C8)—21no—
164C3⋊(C3⋊C8)D1212yescomplemented
165C3×C3⋊C8—32no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—61no—
168C3×C3⋊C8—61no—
169#435—271noSylow
170C2×H(3,𝔽3)C4×D411yescharacteristic
171C2×H(3,𝔽3)C2×M1611yescharacteristic
172C2×H(3,𝔽3)#13111yescharacteristic
173#272—22no—
174#272—41no—
175C2×C3×D4—91no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—61no—
181C2×C4×D3—31no—
182C2×C4×D3—31no—
183C2×C4×D3—31no—
184C2×C4×D3—31no—
185C2×C4×D3—61no—
186C2×C4×D3—61no—
187C2×C3×C8—91no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C2×C3×C8—91no—
191C3×C4⋊C4—91no—
192C3×C22⋊C4—91no—
193C3×C22⋊C4—91no—
194C3×C42—31no—
195C3×C42—31no—
196C3×C42—61no—
197C3×C42D3211yescharacteristic
198(C2×D3)⋊C4—31no—
199(C2×D3)⋊C4—31no—
200(C2×D3)⋊C4—31no—
201(C2×D3)⋊C4—31no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—61no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—61no—
207#208—182no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—91no—
214C2×C3⋊C8—91no—
215C2×C3⋊C8—91no—
216C2×C3⋊C8—182no—
217ΓL(1,𝔽25)—182no—
218ΓL(1,𝔽25)—182no—
219D3×C8—182no—
220D3×C8—182no—
221C22×C32—21no—
222C22×C32C2×C4×D311yescharacteristic
223C22×C32C2×C4×D311yescharacteristic
224C2×C3×D3—32no—
225C2×C3×D3—32no—
226C2×C3×D3—62no—
227C2×C3×D3—61no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—62no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—61no—
235C2×C3×D3—61no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C32×C4—22no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—41no—
250C32×C4C2×D1211yescharacteristic
251C32×C4#23511yescharacteristic
252C32×C4D3×D411yescharacteristic
253C32×C4#23911yescharacteristic
254C3×Dic3—61no—
255C3×Dic3—61no—
256C3×Dic3—61no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C3×Dic3—121no—
260C2×M16—271no—
261C22×C8—271no—
262C4×D4—91no—
263C4⋊C8—91no—
264C22⋊C8—271no—
265C22⋊C8—271no—
266#97—91no—
267H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
268C23×C3—91no—
269C23×C3—91no—
270C22×D3—31no—
271C22×D3—31no—
272C22×D3—31no—
273C22×D3—31no—
274C22×D3—61no—
275C22×D3—61no—
276C3×D4—182no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—91no—
287C2×C3×C4—91no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4#62411yescharacteristic
293C2×C3×C4#62411yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295C2×Dic3—31no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3—31no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301D12—62no—
302D12—62no—
303D12—64no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—121no—
313C4×D3—121no—
314C4×D3—121no—
315C4×D3—121no—
316C3×C8—92no—
317C3×C8—181no—
318C3×C8—181no—
319C3×C8—181no—
320C3⋊C8—32no—
321C3⋊C8—61no—
322C3⋊C8—61no—
323C3⋊C8—61no—
324C3⋊C8—92no—
325C3⋊C8—181no—
326C3⋊C8—181no—
327C3⋊C8—181no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32C4×D1211yescharacteristic
332C2×C32#89311yescharacteristic
333C2×C32#90111yescharacteristic
334C2×C32D3×M1611yescharacteristic
335C2×C32D3×M1611yescharacteristic
336C2×C32#90711yescharacteristic
337C3×D3—62no—
338C3×D3—62no—
339C3×D3—122no—
340C3×D3—122no—
341C3×D3—121no—
342C3×D3—121no—
343C2×D4—91no—
344C22×C4—91no—
345C22×C4—91no—
346M16—542no—
347C2×C8—91no—
348C2×C8—91no—
349C2×C8—91no—
350C2×C8—91no—
351C2×C8—542no—
352C4⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—61no—
359C22×C3—92no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3C2×#62411yescharacteristic
366C2×D3—32no—
367C2×D3—32no—
368C2×D3—62no—
369C2×D3—61no—
370C2×D3—61no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—62no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—61no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—62no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—121no—
398C3×C4—181no—
399C3×C4—181no—
400C3×C4#379911yescharacteristic
401C3×C4D3×D1211yescharacteristic
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—121no—
407Dic3—121no—
408C32—21no—
409C32#470611yescharacteristic, complemented
410C32#483611yescharacteristic, complemented
411C23—91no—
412C23—91no—
413D4—182no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4#619911yescharacteristic
421C2×C4#619911yescharacteristic
422C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
423C8—92no—
424C8—181no—
425C8—181no—
426C8—181no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—182no—
437C2×C3—361no—
438C2×C3#6381711yescharacteristic
439C2×C3#6381811yescharacteristic
440C2×C3#6389411yescharacteristic
441D3—62no—
442D3—62no—
443D3—122no—
444D3—121no—
445D3—121no—
446D3—122no—
447C22—92no—
448C22—181no—
449C22—181no—
450C22—181no—
451C22—181no—
452C22—181no—
453C22C2×#619911yescharacteristic
454C4—21no—
455C4—21no—
456C4—181no—
457C4—181no—
458C4#8865211yescharacteristic
459C4#8865411yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3#1059032811yescharacteristic
464C2—182no—
465C2—361no—
466C2#1172106411yescharacteristic
467C2#1172106511yescharacteristic
468C2#1172114911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC2×C4⋊#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88435C2semidirect
3. C2×#88434C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #63630⋊C3C2semidirect
6. #11720323C2semidirect
7. #11720323C2semidirect
30. C4⋊(C3⋊(C3⋊C8))D3semidirect
31. #63630D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
96. C2×#272M16semidirect
164. C3⋊(C3⋊C8)D12semidirect
267. H(3,𝔽3)#435semidirect
409. C32#4706semidirect
410. C32#4836semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C2×#884352
C2×#884342
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#63630⋊C32
#117203232
#105903303
#105903263