Finite Group Databasebeta

← #14518834

Subgroups — #14518834

Order 1944.

945 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 945 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×C3⋊(C3⋊((C3⋊D3)⋊C3))C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#1084C211yesmaximal, characteristic, complemented
4#10620156—31nomaximal
5#10620152—31nomaximal
6#10619902—33nomaximal
7#10619884—31nomaximal
8C2×C32×H(3,𝔽3)C2211yescharacteristic
9C3⋊(C3⋊((C3⋊D3)⋊C3))—21no—
10C3×C3⋊((C3⋊D3)⋊C3)—21no—
11C2×C3⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
12C2×C3×C3⋊(C3⋊D3)—31no—
13C2×C3×C3⋊(C3⋊D3)D311yescharacteristic, complemented
14C2×C32×C3⋊D3—31no—
15C32×C3⋊Dic3—31no—
16C32×C3⋊Dic3—31no—
17C32×C3⋊Dic3—61no—
18C2×C3⋊((C3⋊D3)⋊C3)—31no—
19C2×C3⋊((C3⋊D3)⋊C3)—33no—
20C2×C3⋊((C3⋊D3)⋊C3)—63no—
21C2×C3⋊((C3⋊D3)⋊C3)—61no—
22C2×C3⋊((C3⋊D3)⋊C3)D313yescomplemented
23C2×C3×(C3⋊D3)⋊C3—31no—
24C2×C3×(C3⋊D3)⋊C3—61no—
25C3⋊#1084D311yescharacteristic, complemented
26C3×#1084—33no—
27C3×#1084—63no—
28C32×H(3,𝔽3)D411yescharacteristic, complemented, Sylow
29#6295—91no—
30#6295—93no—
31#6292—91no—
32#6292—93no—
33#6292—91no—
34#6291—91no—
35#6291—93no—
36#6198—93no—
37C2×C34—21no—
38C2×C34C2×D311yescharacteristic
39C2×C34C2×D311yescharacteristic
40C3⋊(C3⋊(C3⋊D3))—21no—
41C3×C3⋊(C3⋊D3)—21no—
42C3×C3⋊(C3⋊D3)—61no—
43C32×C3⋊D3—61no—
44C2×C3×H(3,𝔽3)—21no—
45C2×C3×H(3,𝔽3)—21no—
46C2×C3×H(3,𝔽3)—23no—
47C2×C3×H(3,𝔽3)—23no—
48C2×C3×H(3,𝔽3)—43no—
49C2×C3×H(3,𝔽3)—41no—
50C2×C3×H(3,𝔽3)C2×D311yescharacteristic
51C2×C3×H(3,𝔽3)C2×D313yes—
52C3⋊((C3⋊D3)⋊C3)—23no—
53C3⋊((C3⋊D3)⋊C3)—62no—
54C3⋊((C3⋊D3)⋊C3)—66no—
55C3⋊((C3⋊D3)⋊C3)—61no—
56C3⋊((C3⋊D3)⋊C3)—63no—
57C3×(C3⋊D3)⋊C3—62no—
58C3×(C3⋊D3)⋊C3—61no—
59C2×C3⋊(C3⋊D3)—31no—
60C2×C3⋊(C3⋊D3)—31no—
61C2×C3⋊(C3⋊D3)—33no—
62C2×C3⋊(C3⋊D3)—61no—
63C2×C3⋊(C3⋊D3)—63no—
64C2×C3⋊(C3⋊D3)—93no—
65C2×C3⋊(C3⋊D3)—183no—
66C2×C3⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
67C2×C3×C3⋊D3—33no—
68C2×C3×C3⋊D3—31no—
69C2×C3×C3⋊D3—31no—
70C2×C3×C3⋊D3—61no—
71C2×C3×C3⋊D3—93no—
72C2×C3×C3⋊D3—93no—
73C2×C3×C3⋊D3—91no—
74C2×C3×C3⋊D3—91no—
75C2×C3×C3⋊D3—181no—
76C2×C3×C3⋊D3—183no—
77C2×C32×D3—93no—
78C2×C32×D3—91no—
79C3×C3⋊Dic3—33no—
80C3×C3⋊Dic3—33no—
81C3×C3⋊Dic3—31no—
82C3×C3⋊Dic3—31no—
83C3×C3⋊Dic3—61no—
84C3×C3⋊Dic3—63no—
85C32×Dic3—93no—
86C32×Dic3—91no—
87C32×Dic3—91no—
88C32×Dic3—183no—
89C32×Dic3—181no—
90C2×(C3⋊D3)⋊C3—33no—
91C2×(C3⋊D3)⋊C3—63no—
92C2×(C3⋊D3)⋊C3—93no—
93C2×(C3⋊D3)⋊C3—183no—
94#1084—33no—
95#1084—63no—
96C34—21no—
97C34#6711yescharacteristic, complemented
98C34#6711yescharacteristic, complemented
99C3×H(3,𝔽3)—21no—
100C3×H(3,𝔽3)—21no—
101C3×H(3,𝔽3)—23no—
102C3×H(3,𝔽3)—23no—
103C3×H(3,𝔽3)—43no—
104C3×H(3,𝔽3)—41no—
105C3×H(3,𝔽3)D1211yescharacteristic, complemented
106C3×H(3,𝔽3)#6713yescomplemented
107#638—271no—
108C4⋊(C3⋊D3)—271no—
109#626—273no—
110#626—273no—
111#626—271no—
112#626—279no—
113#626—271no—
114#626—273no—
115#625—271no—
116C2×C33—23no—
117C2×C33—21no—
118C2×C33—21no—
119C2×C33—21no—
120C2×C33—23no—
121C2×C33—21no—
122C2×C33—21no—
123C2×C33—33no—
124C2×C33—33no—
125C2×C33—41no—
126C2×C33—43no—
127C2×C33—41no—
128C2×C33—63no—
129C2×C33—63no—
130C2×C33—63no—
131C2×C33—123no—
132C2×C33D3211yescharacteristic
133C2×C33D3213yes—
134C2×C33D3211yescharacteristic
135C2×C33D3211yescharacteristic
136C2×C33C2×C3⋊D311yescharacteristic
137C3⋊(C3⋊D3)—21no—
138C3⋊(C3⋊D3)—61no—
139C3⋊(C3⋊D3)—66no—
140C3⋊(C3⋊D3)—62no—
141C3⋊(C3⋊D3)—63no—
142C3⋊(C3⋊D3)—61no—
143C3⋊(C3⋊D3)—183no—
144C3⋊(C3⋊D3)—186no—
145C3×C3⋊D3—61no—
146C3×C3⋊D3—63no—
147C3×C3⋊D3—62no—
148C3×C3⋊D3—61no—
149C3×C3⋊D3—183no—
150C3×C3⋊D3—182no—
151C3×C3⋊D3—183no—
152C3×C3⋊D3—186no—
153C3×C3⋊D3—181no—
154C3×C3⋊D3—181no—
155C32×D3—183no—
156C32×D3—181no—
157C2×H(3,𝔽3)—23no—
158C2×H(3,𝔽3)—23no—
159C2×H(3,𝔽3)—23no—
160C2×H(3,𝔽3)—43no—
161C2×H(3,𝔽3)—43no—
162C2×H(3,𝔽3)—46no—
163C2×H(3,𝔽3)—43no—
164C2×H(3,𝔽3)D3213yes—
165(C3⋊D3)⋊C3—66no—
166(C3⋊D3)⋊C3—63no—
167(C3⋊D3)⋊C3—186no—
168(C3⋊D3)⋊C3—183no—
169C22×C32—271no—
170C2×C3⋊D3—31no—
171C2×C3⋊D3—31no—
172C2×C3⋊D3—61no—
173C2×C3⋊D3—93no—
174C2×C3⋊D3—91no—
175C2×C3⋊D3—93no—
176C2×C3⋊D3—91no—
177C2×C3⋊D3—93no—
178C2×C3⋊D3—181no—
179C2×C3⋊D3—183no—
180C2×C3⋊D3—183no—
181C2×C3⋊D3—271no—
182C2×C3⋊D3—279no—
183C2×C3⋊D3—541no—
184C2×C3⋊D3—541no—
185C2×C3⋊D3—549no—
186C2×C3⋊D3—542no—
187C2×C3×D3—99no—
188C2×C3×D3—93no—
189C2×C3×D3—93no—
190C2×C3×D3—91no—
191C2×C3×D3—271no—
192C2×C3×D3—271no—
193C2×C3×D3—273no—
194C2×C3×D3—271no—
195C2×C3×D3—541no—
196C2×C3×D3—541no—
197C2×C3×D3—543no—
198C32×C4—271no—
199C3⋊Dic3—31no—
200C3⋊Dic3—31no—
201C3⋊Dic3—61no—
202C3×Dic3—99no—
203C3×Dic3—93no—
204C3×Dic3—93no—
205C3×Dic3—91no—
206C3×Dic3—93no—
207C3×Dic3—91no—
208C3×Dic3—183no—
209C3×Dic3—181no—
210C3×Dic3—183no—
211C3×Dic3—189no—
212C33—21no—
213C33—23no—
214C33—21no—
215C33—21no—
216C33—23no—
217C33—21no—
218C33—21no—
219C33—33no—
220C33—33no—
221C33—41no—
222C33—43no—
223C33—41no—
224C33—63no—
225C33—63no—
226C33—63no—
227C33—123no—
228C33#62511yescharacteristic
229C33#62513yescomplemented
230C33#62611yescharacteristic, complemented
231C33#62611yescharacteristic, complemented
232C33#63811yescharacteristic, complemented
233H(3,𝔽3)—23no—
234H(3,𝔽3)—23no—
235H(3,𝔽3)—23no—
236H(3,𝔽3)—43no—
237H(3,𝔽3)—43no—
238H(3,𝔽3)—46no—
239H(3,𝔽3)—43no—
240H(3,𝔽3)#62613yescomplemented
241#67—811no—
242#67—811no—
243#67—813no—
244D12—813no—
245D12—811no—
246C2×C32—23no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—23no—
251C2×C32—23no—
252C2×C32—31no—
253C2×C32—39no—
254C2×C32—33no—
255C2×C32—31no—
256C2×C32—43no—
257C2×C32—41no—
258C2×C32—61no—
259C2×C32—69no—
260C2×C32—61no—
261C2×C32—63no—
262C2×C32—62no—
263C2×C32—61no—
264C2×C32—61no—
265C2×C32—63no—
266C2×C32—69no—
267C2×C32—61no—
268C2×C32—121no—
269C2×C32—123no—
270C2×C32—129no—
271C2×C32—121no—
272C2×C32—122no—
273C2×C32—121no—
274C2×C32—541no—
275C2×C32((C3⋊D3)⋊C3)⋊C213yes—
276C2×C32D3×C3⋊D311yescharacteristic
277C2×C32D3×C3⋊D311yescharacteristic
278C2×C32#111311yescharacteristic
279C2×C32#111313yes—
280C3⋊D3—61no—
281C3⋊D3—62no—
282C3⋊D3—61no—
283C3⋊D3—183no—
284C3⋊D3—182no—
285C3⋊D3—186no—
286C3⋊D3—183no—
287C3⋊D3—181no—
288C3⋊D3—181no—
289C3⋊D3—186no—
290C3⋊D3—183no—
291C3⋊D3—549no—
292C3⋊D3—541no—
293C3⋊D3—542no—
294C3⋊D3—5418no—
295C3⋊D3—544no—
296C3⋊D3—542no—
297C3×D3—189no—
298C3×D3—183no—
299C3×D3—181no—
300C3×D3—183no—
301C3×D3—541no—
302C3×D3—542no—
303C3×D3—543no—
304C3×D3—541no—
305C3×D3—542no—
306C3×D3—546no—
307C3×D3—541no—
308C22×C3—273no—
309C22×C3—271no—
310C22×C3—811no—
311C2×D3—93no—
312C2×D3—91no—
313C2×D3—271no—
314C2×D3—273no—
315C2×D3—271no—
316C2×D3—271no—
317C2×D3—541no—
318C2×D3—543no—
319C2×D3—541no—
320C2×D3—813no—
321C2×D3—1623no—
322C3×C4—273no—
323C3×C4—271no—
324Dic3—91no—
325Dic3—93no—
326Dic3—91no—
327Dic3—181no—
328Dic3—183no—
329C32—23no—
330C32—21no—
331C32—21no—
332C32—23no—
333C32—21no—
334C32—23no—
335C32—31no—
336C32—39no—
337C32—33no—
338C32—31no—
339C32—43no—
340C32—41no—
341C32—61no—
342C32—61no—
343C32—69no—
344C32—63no—
345C32—62no—
346C32—61no—
347C32—61no—
348C32—63no—
349C32—69no—
350C32—61no—
351C32—121no—
352C32—123no—
353C32—121no—
354C32—129no—
355C32—122no—
356C32—121no—
357C32#619813yescomplemented
358C32#629011yescharacteristic
359C32#629111yescharacteristic, complemented
360C32#629511yescharacteristic
361C32#629513yescomplemented
362D4—2431noSylow
363C2×C3—23no—
364C2×C3—21no—
365C2×C3—31no—
366C2×C3—33no—
367C2×C3—61no—
368C2×C3—63no—
369C2×C3—61no—
370C2×C3—63no—
371C2×C3—121no—
372C2×C3—123no—
373C2×C3—543no—
374C2×C3—541no—
375C2×C3—1621no—
376C2×C3((C3⋊D3)⋊C3)⋊D313yes—
377C2×C3#6643211yescharacteristic
378C2×C3C3⋊#111311yescharacteristic
379D3—183no—
380D3—181no—
381D3—541no—
382D3—542no—
383D3—546no—
384D3—542no—
385D3—543no—
386D3—541no—
387D3—541no—
388D3—1623no—
389D3—1626no—
390C22—271no—
391C22—811no—
392C4—271no—
393C3—23no—
394C3—21no—
395C3—31no—
396C3—33no—
397C3—61no—
398C3—63no—
399C3—61no—
400C3—63no—
401C3—121no—
402C3—123no—
403C3#1061988413yescomplemented
404C3#1061990211yescharacteristic, complemented
405C3#1062015711yescharacteristic
406C2—541no—
407C2—1621no—
408C2#1175809011yescharacteristic
subgroupG/H
center, hypercenterC2#11758090
commutator, Fitting, cosocleC2×C32×H(3,𝔽3)C22
FrattiniC2×C3C3⋊#1113
socleC2×C33D32
hypocenterC32×H(3,𝔽3)D4
LCSG ▷ C2×C32×H(3,𝔽3) ▷ C32×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(C3⋊((C3⋊D3)⋊C3))C2semidirect
2. C2×C3×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3×C3⋊#1084C2semidirect
11. C2×C3⋊(C3⋊(C3⋊D3))D3semidirect
13. C2×C3×C3⋊(C3⋊D3)D3semidirect
22. C2×C3⋊((C3⋊D3)⋊C3)D3semidirect
25. C3⋊#1084D3semidirect
28. C32×H(3,𝔽3)D4semidirect
66. C2×C3⋊(C3⋊D3)C3⋊D3semidirect
97. C34#67semidirect
98. C34#67semidirect
105. C3×H(3,𝔽3)D12semidirect
106. C3×H(3,𝔽3)#67semidirect
229. C33#625semidirect
230. C33#626semidirect
231. C33#626semidirect
232. C33#638semidirect
240. H(3,𝔽3)#626semidirect
357. C32#6198semidirect
359. C32#6291semidirect
361. C32#6295semidirect
403. C3#10619884semidirect
404. C3#10619902semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊((C3⋊D3)⋊C3))2
C2×C3×C3⋊((C3⋊D3)⋊C3)2
C3×C3⋊#10842
#106201563
#106201523
#106199023
#106198843