Finite Group Databasebeta

← #13137415

Subgroups — #13137415

Order 1728.

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

normal subgroups only

This group has 556 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, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic
4#63630⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590346—31nomaximal
9#10590344—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88436C414yescomplemented
22C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
23C3×C4×D12—31no—
24C3×C4×D12—31no—
25C3×C4×D12—31no—
26C3×C4×D12—61no—
27C2×#3714—31no—
28C2×#3713—31no—
29#63630D311yescharacteristic, complemented
30C3×#803—31no—
31#63574—31no—
32#63573—31no—
33#63564—31no—
34#63563—31no—
35#63562—31no—
36#63562—31no—
37C22×#272—21no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272C2×C412yescomplemented
45C4×#272—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
49C4×(C3⋊D3)⋊C3—42no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52#5014—91no—
53#5014—91no—
54#4711—91no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68#3752—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—62no—
73C3×(C2×D3)⋊C4—61no—
74C3×(C2×D3)⋊C4—61no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82#3714—34no—
83#3713—34no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C2×#272—22no—
86C2×#272—41no—
87C2×#272—41no—
88C2×(C3⋊D3)⋊C3—22no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)C22×C411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97C3×C4×D4—91no—
98C3×C4×D4—91no—
99C2×#213—91no—
100C2×#213—91no—
101C2×D24—91no—
102C4×D12—31no—
103C4×D12—31no—
104C4×D12—31no—
105C4×D12—61no—
106C4×D12—91no—
107C3×#97—91no—
108#832—91no—
109#832—91no—
110#821—91no—
111#821—91no—
112#817—91no—
113#809—91no—
114#809—91no—
115#808—91no—
116#808—91no—
117#803—31no—
118#803—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—62no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4D1211yescharacteristic
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)C4×D312yescomplemented
136C4×C3⋊D3—42no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—32no—
142C3×D12—32no—
143C3×D12—32no—
144C3×D12—62no—
145C3×D12—62no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×C4×D3—122no—
150C3×C4×D3—124no—
151C3×C4×D3—122no—
152C3×C4×D3—122no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C4×D312yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157#442—271noSylow
158C2×H(3,𝔽3)C4×D411yescharacteristic
159C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
160C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
161#272—42no—
162(C3⋊D3)⋊C3—42no—
163C2×C3×D4—91no—
164C2×C3×D4—91no—
165C22×C3×C4—181no—
166C22×C3×C4—181no—
167C2×D12—31no—
168C2×D12—31no—
169C2×D12—31no—
170C2×D12—61no—
171C2×D12—91no—
172C2×C4×D3—61no—
173C2×C4×D3—61no—
174C2×C4×D3—62no—
175C2×C4×D3—61no—
176C2×C4×D3—181no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—91no—
181C3×C22⋊C4—181no—
182C3×C22⋊C4—181no—
183C3×C42—31no—
184C3×C42—31no—
185C3×C42—61no—
186C3×C42D3211yescharacteristic
187#213—94no—
188#213—94no—
189(C2×D3)⋊C4—61no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—62no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—181no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—61no—
198C4⋊Dic3—91no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203D24—94no—
204C22×C32—21no—
205C22×C32C2×D1211yescharacteristic
206C22×C32C2×#6711yescharacteristic
207C2×C3⋊D3—22no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—64no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—41no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4C2×C4×D311yescharacteristic
232C32×C4#23511yescharacteristic
233C32×C4#23511yescharacteristic
234C3⋊Dic3—41no—
235C3×Dic3—122no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×D8—271no—
240C4×D4—91no—
241C4×D4—91no—
242#106—271no—
243#102—271no—
244#102—271no—
245#97—91no—
246H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
247C23×C3—181no—
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—62no—
252C22×D3—61no—
253C22×D3—181no—
254C3×D4—92no—
255C3×D4—92no—
256C3×D4—181no—
257C3×D4—181no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—362no—
270C2×C3×C4—362no—
271C2×C3×C4#62611yescharacteristic
272C2×C3×C4#62611yescharacteristic
273C2×C3×C4C2×D3211yescharacteristic
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—62no—
277C2×Dic3—61no—
278C2×Dic3—181no—
279D12—32no—
280D12—32no—
281D12—32no—
282D12—62no—
283D12—61no—
284D12—61no—
285D12—62no—
286D12—61no—
287D12—92no—
288D12—181no—
289C4×D3—122no—
290C4×D3—122no—
291C4×D3—122no—
292C4×D3—124no—
293C4×D3—362no—
294C3×C8—92no—
295C3×C8—181no—
296C3⋊C8—32no—
297C3⋊C8—61no—
298C3⋊C8—92no—
299C3⋊C8—181no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32C4×D1211yescharacteristic
304C2×C32#90811yescharacteristic
305C2×C32#90811yescharacteristic
306C2×C32C4×#6711yescharacteristic
307C2×C32#93211yescharacteristic
308C2×C32#94911yescharacteristic
309C3⋊D3—42no—
310C3×D3—124no—
311C3×D3—122no—
312C3×D3—122no—
313C3×D3—122no—
314C2×D4—91no—
315C2×D4—91no—
316C22×C4—181no—
317C22×C4—181no—
318D8—274no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—91no—
322C4⋊C4—91no—
323C22⋊C4—181no—
324C22⋊C4—181no—
325C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—182no—
330C22×C3—182no—
331C22×C3—361no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3C2×#62611yescharacteristic
336C2×D3—62no—
337C2×D3—62no—
338C2×D3—64no—
339C2×D3—62no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—122no—
345C2×D3—122no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—182no—
349C2×D3—361no—
350C2×D3—361no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—62no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—121no—
365C3×C4—361no—
366C3×C4—361no—
367C3×C4#379811yescharacteristic
368C3×C4C4×D3211yescharacteristic
369Dic3—121no—
370Dic3—121no—
371Dic3—122no—
372Dic3—121no—
373Dic3—361no—
374C32—21no—
375C32#471111yescharacteristic, complemented
376C32#501411yescharacteristic, complemented
377C23—181no—
378C23—181no—
379D4—92no—
380D4—92no—
381D4—181no—
382D4—181no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—362no—
386C2×C4—362no—
387C2×C4#620011yescharacteristic
388C2×C4#620011yescharacteristic
389C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
390C8—92no—
391C8—181no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—362no—
402C2×C3—362no—
403C2×C3#6382511yescharacteristic
404C2×C3#6383011yescharacteristic
405C2×C3C4×#62611yescharacteristic
406D3—122no—
407D3—122no—
408D3—124no—
409D3—122no—
410D3—362no—
411C22—182no—
412C22—182no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22C2×#620011yescharacteristic
418C4—21no—
419C4—21no—
420C4—361no—
421C4—361no—
422C4#8865011yescharacteristic
423C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C3—31no—
425C3—31no—
426C3—61no—
427C3#1059034611yescharacteristic
428C2—362no—
429C2—362no—
430C2#1172106811yescharacteristic
431C2#1172107111yescharacteristic
432C2C4×#620011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#88436C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ 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. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #63630⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720308C2semidirect
7. #11720307C2semidirect
21. #88436C4semidirect
22. C4×C4⋊(C3⋊D3)D3semidirect
29. #63630D3semidirect
44. C4⋊#272C2×C4semidirect
48. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
135. C4⋊(C3⋊D3)C4×D3semidirect
154. C3⋊(C3⋊C8)C4×D3semidirect
246. H(3,𝔽3)#442semidirect
375. C32#4711semidirect
376. C32#5014semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884362
#63630⋊C32
#117203242
#117203082
#117203072
#105903463
#105903443