Finite Group Databasebeta

← #13137416

Subgroups — #13137416

Order 1728.

584 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721385C211yesmaximal, characteristic, complemented
3C2×#88436C212yesmaximal, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590349—31nomaximal
7#10590347—31nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×C4×#272—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88436—24no—
18#64083D311yescharacteristic, complemented
19C3×#874—31no—
20C3×C4×D12—31no—
21C3×C4×D12—31no—
22C3×C4×D12—61no—
23C2×#3714—32no—
24C2×#3713—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
26C3×C4⋊(C3⋊C8)—31no—
27#63564—32no—
28#63562—32no—
29C22×#272—21no—
30C22×(C3⋊D3)⋊C3—22no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C4⋊#272—21no—
36C4⋊#272D412yescomplemented
37C4×#272—42no—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)—42no—
41C4⋊((C3⋊D3)⋊C3)—42no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43#5072—91no—
44#5015—91no—
45#4843—91no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C2×C3×D12—31no—
50C2×C3×D12—31no—
51C2×C3×D12—32no—
52C2×C3×D12—61no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—62no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C32×C42—21no—
59C32×C42C2×D311yescharacteristic
60C32×C42C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D312yes—
62C3×(C2×D3)⋊C4—62no—
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C2×C3×C3⋊C8—32no—
69#3714—64no—
70#3713—64no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×#272—22no—
73C2×#272—41no—
74C2×#272—41no—
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—42no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)D812yescomplemented
80C4×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83C3×#127—91no—
84C3×C4×D4—91no—
85C2×#213—92no—
86C2×#213—92no—
87C2×D24—92no—
88#874—31no—
89#874—91no—
90C4×D12—31no—
91C4×D12—31no—
92C4×D12—61no—
93C3×C4⋊C8—91no—
94#832—92no—
95#821—92no—
96#809—92no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—91no—
99C22×C3⋊D3—22no—
100C22×C3×D3—62no—
101C22×C3×D3—61no—
102C22×C3×D3—61no—
103C22×C3×D3—62no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4D1211yescharacteristic
108C2×C32×C4D1211yescharacteristic
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C4⋊(C3⋊D3)—22no—
114C4⋊(C3⋊D3)—22no—
115C4⋊(C3⋊D3)—42no—
116C4⋊(C3⋊D3)—42no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C3×D12—32no—
121C3×D12—32no—
122C3×D12—62no—
123C3×D12—62no—
124C3×D12—62no—
125C3×D12—61no—
126C3×D12—61no—
127C3×D12—62no—
128C3×D12—122no—
129C3×D12—122no—
130C3×C4×D3—124no—
131C3×C4×D3—122no—
132C3×C4×D3—122no—
133C3⋊(C3⋊C8)—22no—
134C3×C3⋊C8—62no—
135#459—271noSylow
136C2×H(3,𝔽3)#12111yescharacteristic
137C2×H(3,𝔽3)C2×D811yescharacteristic
138C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
139#272—42no—
140(C3⋊D3)⋊C3—82no—
141C2×C3×D4—92no—
142C2×C3×D4—91no—
143C2×C3×D4—181no—
144C2×C3×D4—181no—
145C22×C3×C4—181no—
146C2×D12—31no—
147C2×D12—31no—
148C2×D12—32no—
149C2×D12—61no—
150C2×D12—61no—
151C2×D12—61no—
152C2×D12—92no—
153C2×D12—181no—
154C2×D12—181no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—62no—
158C2×C3×C8—92no—
159C3×C4⋊C4—91no—
160C3×C22⋊C4—181no—
161C3×C42—31no—
162C3×C42—31no—
163C3×C42—61no—
164C3×C42D3211yescharacteristic
165#213—184no—
166#213—184no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—62no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C2×C3⋊C8—32no—
174C2×C3⋊C8—92no—
175D24—184no—
176C22×C32—21no—
177C22×C32C2×D1211yescharacteristic
178C22×C32C2×#6711yescharacteristic
179C2×C3⋊D3—42no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—42no—
182C2×C3×D3—64no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—122no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C2×C3×D3—122no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—41no—
200C32×C4D2412yescomplemented
201C32×C4#21312yescomplemented
202C32×C4D3×D411yescharacteristic
203C32×C4D3×D411yescharacteristic
204C32×C4#23711yescharacteristic
205C32×C4#23911yescharacteristic
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C2×D8—272no—
210#127—91no—
211C4×D4—91no—
212C4⋊C8—91no—
213#102—272no—
214H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
215C23×C3—182no—
216C23×C3—181no—
217C22×D3—61no—
218C22×D3—61no—
219C22×D3—62no—
220C22×D3—62no—
221C22×D3—182no—
222C3×D4—92no—
223C3×D4—182no—
224C3×D4—182no—
225C3×D4—181no—
226C3×D4—362no—
227C3×D4—362no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—181no—
238C2×C3×C4—362no—
239C2×C3×C4#62611yescharacteristic
240C2×C3×C4#62611yescharacteristic
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—62no—
245D12—32no—
246D12—32no—
247D12—62no—
248D12—61no—
249D12—61no—
250D12—62no—
251D12—62no—
252D12—62no—
253D12—122no—
254D12—122no—
255D12—182no—
256D12—182no—
257D12—362no—
258D12—362no—
259C4×D3—122no—
260C4×D3—122no—
261C4×D3—124no—
262C3×C8—182no—
263C3⋊C8—62no—
264C3⋊C8—182no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32#89511yescharacteristic
269C2×C32C2×D2411yescharacteristic
270C2×C32#90811yescharacteristic
271C2×C32C2×#21311yescharacteristic
272C2×C32#93211yescharacteristic
273C2×C32#93811yescharacteristic
274C3⋊D3—82no—
275C3×D3—124no—
276C3×D3—122no—
277C3×D3—122no—
278C3×D3—242no—
279C2×D4—91no—
280C2×D4—92no—
281C2×D4—181no—
282C2×D4—181no—
283C22×C4—181no—
284D8—544no—
285C2×C8—92no—
286C4⋊C4—91no—
287C22⋊C4—181no—
288C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—61no—
292C22×C3—182no—
293C22×C3—362no—
294C22×C3—362no—
295C22×C3—362no—
296C22×C3—361no—
297C22×C3—361no—
298C22×C3C2×#62611yescharacteristic
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—64no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—122no—
307C2×D3—122no—
308C2×D3—122no—
309C2×D3—122no—
310C2×D3—122no—
311C2×D3—362no—
312C2×D3—362no—
313C2×D3—362no—
314C3×C4—21no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—31no—
319C3×C4—32no—
320C3×C4—32no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—62no—
326C3×C4—121no—
327C3×C4—361no—
328C3×C4#371412yes—
329C3×C4#379611yescharacteristic
330C3×C4#380211yescharacteristic
331Dic3—121no—
332Dic3—121no—
333Dic3—122no—
334C32—21no—
335C32#484311yescharacteristic, complemented
336C32#507211yescharacteristic, complemented
337C23—181no—
338C23—182no—
339D4—92no—
340D4—181no—
341D4—182no—
342D4—182no—
343D4—362no—
344D4—362no—
345C2×C4—181no—
346C2×C4—362no—
347C2×C4#620011yescharacteristic
348C2×C4#620011yescharacteristic
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C8—182no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—362no—
361C2×C3—722no—
362C2×C3C2×#371411yescharacteristic
363C2×C3#6383011yescharacteristic
364C2×C3#6391611yescharacteristic
365D3—122no—
366D3—122no—
367D3—124no—
368D3—242no—
369D3—722no—
370C22—182no—
371C22—361no—
372C22—361no—
373C22—362no—
374C22—362no—
375C22—362no—
376C22C2×#620011yescharacteristic
377C4—21no—
378C4—361no—
379C4#8843612yescomplemented
380C4#8864911yescharacteristic
381C4#8865511yescharacteristic
382C3—31no—
383C3—31no—
384C3—61no—
385C3#1059035511yescharacteristic
386C2—362no—
387C2—722no—
388C2C2×#8843611yescharacteristic
389C2#1172107111yescharacteristic
390C2#1172115911yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
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. #11721385C2semidirect
3. C2×#88436C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720308C2semidirect
18. #64083D3semidirect
25. C4⋊(C3⋊(C3⋊C8))D3semidirect
36. C4⋊#272D4semidirect
79. C4×H(3,𝔽3)D8semidirect
200. C32×C4D24semidirect
201. C32×C4#213semidirect
214. H(3,𝔽3)#459semidirect
335. C32#4843semidirect
336. C32#5072semidirect
379. C4#88436semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#117213852
C2×#884362
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203082
#105903493
#105903473