Finite Group Databasebeta

← #14518838

Subgroups — #14518838

Order 1944.

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

normal subgroups only

This group has 919 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⋊#272C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊((C3⋊Dic3)⋊C3))C211yesmaximal, characteristic, complemented
4#10620157—31nomaximal
5#10620151—31nomaximal
6#10619904—33nomaximal
7#10619886—31nomaximal
8C2×C32×H(3,𝔽3)C2211yescharacteristic
9C3×C3⋊#272—21no—
10C3×C3⋊((C3⋊D3)⋊C3)—21no—
11C2×C3×C3⋊(C3⋊D3)D311yescharacteristic, complemented
12C2×C32×C3⋊D3—31no—
13C2×C32×C3⋊D3—31no—
14C2×C32×C3⋊D3—31no—
15C2×C32×C3⋊D3—61no—
16C3⋊(C3⋊(C3⋊Dic3))D311yescharacteristic, complemented
17C3×C3⋊(C3⋊Dic3)—31no—
18C2×C3⋊#272D311yescharacteristic, complemented
19C2×C3×#272—33no—
20C2×C3×#272—63no—
21C2×C3⋊((C3⋊D3)⋊C3)D313yescomplemented
22C2×C3×(C3⋊D3)⋊C3—31no—
23C2×C3×(C3⋊D3)⋊C3—61no—
24C3⋊((C3⋊Dic3)⋊C3)—31no—
25C3⋊((C3⋊Dic3)⋊C3)—33no—
26C3⋊((C3⋊Dic3)⋊C3)—63no—
27C3⋊((C3⋊Dic3)⋊C3)—61no—
28C32×H(3,𝔽3)D411yescharacteristic, complemented, Sylow
29#6295—91no—
30#6295—93no—
31#6291—91no—
32#6290—93no—
33#6290—91no—
34#6290—91no—
35#6290—93no—
36#6200—93no—
37C2×C34—21no—
38C2×C34C2×D311yescharacteristic
39C2×C34C2×D311yescharacteristic
40C3×C3⋊(C3⋊D3)—21no—
41C32×C3⋊D3—62no—
42C32×C3⋊D3—61no—
43C32×C3⋊D3—61no—
44C32×C3⋊D3—61no—
45C2×C3×H(3,𝔽3)—21no—
46C2×C3×H(3,𝔽3)—21no—
47C2×C3×H(3,𝔽3)—23no—
48C2×C3×H(3,𝔽3)—23no—
49C2×C3×H(3,𝔽3)—43no—
50C2×C3×H(3,𝔽3)—41no—
51C2×C3×H(3,𝔽3)C2×D311yescharacteristic
52C2×C3×H(3,𝔽3)C2×D313yes—
53C3⋊#272—21no—
54C3×#272—66no—
55C3×#272—63no—
56C3⋊((C3⋊D3)⋊C3)—23no—
57C3×(C3⋊D3)⋊C3—62no—
58C3×(C3⋊D3)⋊C3—61no—
59C2×C3⋊(C3⋊D3)C3⋊D311yescharacteristic, complemented
60C2×C3×C3⋊D3—33no—
61C2×C3×C3⋊D3—33no—
62C2×C3×C3⋊D3—33no—
63C2×C3×C3⋊D3—31no—
64C2×C3×C3⋊D3—31no—
65C2×C3×C3⋊D3—31no—
66C2×C3×C3⋊D3—31no—
67C2×C3×C3⋊D3—61no—
68C2×C3×C3⋊D3—61no—
69C2×C3×C3⋊D3—63no—
70C2×C3×C3⋊D3—93no—
71C2×C32×D3—93no—
72C2×C32×D3—91no—
73C2×C32×D3—93no—
74C2×C32×D3—91no—
75C2×C32×D3—91no—
76C2×C32×D3—183no—
77C2×C32×D3—181no—
78C3⋊(C3⋊Dic3)—31no—
79C3⋊(C3⋊Dic3)—31no—
80C3⋊(C3⋊Dic3)—33no—
81C3⋊(C3⋊Dic3)—61no—
82C3⋊(C3⋊Dic3)—63no—
83C3⋊(C3⋊Dic3)—93no—
84C3⋊(C3⋊Dic3)—183no—
85C3×C3⋊Dic3—93no—
86C3×C3⋊Dic3—91no—
87C3×C3⋊Dic3—91no—
88C3×C3⋊Dic3—181no—
89C3×C3⋊Dic3—183no—
90C2×#272—33no—
91C2×#272—63no—
92C2×(C3⋊D3)⋊C3—33no—
93C2×(C3⋊D3)⋊C3—63no—
94(C3⋊Dic3)⋊C3—93no—
95(C3⋊Dic3)⋊C3—183no—
96C34—21no—
97C34D1211yescharacteristic, 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)#6711yescharacteristic, complemented
106C3×H(3,𝔽3)#6713yescomplemented
107#638—271no—
108#638—271no—
109#626—271no—
110#626—273no—
111#626—271no—
112#625—273no—
113#625—273no—
114#625—271no—
115#625—279no—
116C2×C33—23no—
117C2×C33—21no—
118C2×C33—21no—
119C2×C33—21no—
120C2×C33—21no—
121C2×C33—23no—
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—62no—
139C3×C3⋊D3—63no—
140C3×C3⋊D3—63no—
141C3×C3⋊D3—66no—
142C3×C3⋊D3—61no—
143C3×C3⋊D3—61no—
144C3×C3⋊D3—63no—
145C3×C3⋊D3—61no—
146C3×C3⋊D3—62no—
147C3×C3⋊D3—61no—
148C3×C3⋊D3—183no—
149C32×D3—182no—
150C32×D3—186no—
151C32×D3—183no—
152C32×D3—181no—
153C32×D3—181no—
154C32×D3—183no—
155C32×D3—181no—
156C2×H(3,𝔽3)—23no—
157C2×H(3,𝔽3)—23no—
158C2×H(3,𝔽3)—23no—
159C2×H(3,𝔽3)—43no—
160C2×H(3,𝔽3)—43no—
161C2×H(3,𝔽3)—46no—
162C2×H(3,𝔽3)—43no—
163C2×H(3,𝔽3)D3213yes—
164#272—66no—
165#272—63no—
166(C3⋊D3)⋊C3—66no—
167(C3⋊D3)⋊C3—63no—
168C22×C32—271no—
169C22×C32—271no—
170C2×C3⋊D3—31no—
171C2×C3⋊D3—31no—
172C2×C3⋊D3—31no—
173C2×C3⋊D3—31no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—93no—
177C2×C3×D3—99no—
178C2×C3×D3—99no—
179C2×C3×D3—93no—
180C2×C3×D3—93no—
181C2×C3×D3—93no—
182C2×C3×D3—91no—
183C2×C3×D3—93no—
184C2×C3×D3—93no—
185C2×C3×D3—91no—
186C2×C3×D3—91no—
187C2×C3×D3—183no—
188C2×C3×D3—181no—
189C2×C3×D3—183no—
190C2×C3×D3—189no—
191C2×C3×D3—271no—
192C2×C3×D3—541no—
193C3⋊Dic3—91no—
194C3⋊Dic3—93no—
195C3⋊Dic3—91no—
196C3⋊Dic3—93no—
197C3⋊Dic3—181no—
198C3⋊Dic3—183no—
199C3⋊Dic3—183no—
200C3⋊Dic3—271no—
201C3⋊Dic3—279no—
202C3⋊Dic3—541no—
203C3⋊Dic3—541no—
204C3⋊Dic3—549no—
205C3⋊Dic3—542no—
206C3×Dic3—271no—
207C3×Dic3—273no—
208C3×Dic3—271no—
209C3×Dic3—541no—
210C3×Dic3—543no—
211C33—21no—
212C33—23no—
213C33—21no—
214C33—21no—
215C33—21no—
216C33—23no—
217C33—21no—
218C33—33no—
219C33—33no—
220C33—41no—
221C33—43no—
222C33—41no—
223C33—63no—
224C33—63no—
225C33—63no—
226C33—123no—
227C33#62511yescharacteristic, complemented
228C33#62611yescharacteristic
229C33#62613yescomplemented
230C33#62611yescharacteristic, complemented
231C33#63811yescharacteristic, complemented
232H(3,𝔽3)—23no—
233H(3,𝔽3)—23no—
234H(3,𝔽3)—23no—
235H(3,𝔽3)—43no—
236H(3,𝔽3)—43no—
237H(3,𝔽3)—46no—
238H(3,𝔽3)—43no—
239H(3,𝔽3)#62513yescomplemented
240#67—813no—
241#67—811no—
242#67—811no—
243#67—813no—
244D12—811no—
245C2×C32—23no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—23no—
250C2×C32—23no—
251C2×C32—31no—
252C2×C32—39no—
253C2×C32—33no—
254C2×C32—31no—
255C2×C32—43no—
256C2×C32—41no—
257C2×C32—61no—
258C2×C32—69no—
259C2×C32—61no—
260C2×C32—63no—
261C2×C32—62no—
262C2×C32—61no—
263C2×C32—61no—
264C2×C32—63no—
265C2×C32—69no—
266C2×C32—61no—
267C2×C32—121no—
268C2×C32—123no—
269C2×C32—129no—
270C2×C32—121no—
271C2×C32—122no—
272C2×C32—121no—
273C2×C32—541no—
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—61no—
282C3⋊D3—62no—
283C3⋊D3—61no—
284C3⋊D3—62no—
285C3⋊D3—61no—
286C3⋊D3—183no—
287C3×D3—182no—
288C3×D3—186no—
289C3×D3—189no—
290C3×D3—183no—
291C3×D3—189no—
292C3×D3—186no—
293C3×D3—1818no—
294C3×D3—183no—
295C3×D3—183no—
296C3×D3—181no—
297C3×D3—181no—
298C3×D3—181no—
299C3×D3—183no—
300C3×D3—183no—
301C3×D3—541no—
302C3×D3—542no—
303C22×C3—273no—
304C22×C3—273no—
305C22×C3—271no—
306C22×C3—271no—
307C2×D3—91no—
308C2×D3—93no—
309C2×D3—93no—
310C2×D3—91no—
311C2×D3—91no—
312C2×D3—181no—
313C2×D3—183no—
314C2×D3—271no—
315C2×D3—541no—
316C3×C4—811no—
317Dic3—273no—
318Dic3—271no—
319Dic3—271no—
320Dic3—543no—
321Dic3—541no—
322Dic3—813no—
323Dic3—1623no—
324C32—23no—
325C32—21no—
326C32—21no—
327C32—23no—
328C32—21no—
329C32—23no—
330C32—31no—
331C32—31no—
332C32—39no—
333C32—33no—
334C32—43no—
335C32—41no—
336C32—61no—
337C32—61no—
338C32—69no—
339C32—63no—
340C32—62no—
341C32—61no—
342C32—61no—
343C32—61no—
344C32—63no—
345C32—69no—
346C32—121no—
347C32—123no—
348C32—121no—
349C32—129no—
350C32—122no—
351C32—121no—
352C32#620013yescomplemented
353C32#629011yescharacteristic, complemented
354C32#629111yescharacteristic
355C32#629511yescharacteristic
356C32#629513yescomplemented
357D4—2431noSylow
358C2×C3—23no—
359C2×C3—21no—
360C2×C3—33no—
361C2×C3—31no—
362C2×C3—63no—
363C2×C3—61no—
364C2×C3—63no—
365C2×C3—61no—
366C2×C3—121no—
367C2×C3—123no—
368C2×C3—543no—
369C2×C3—541no—
370C2×C3—543no—
371C2×C3—541no—
372C2×C3((C3⋊D3)⋊C3)⋊D313yes—
373C2×C3#6643211yescharacteristic
374C2×C3C3⋊#111311yescharacteristic
375D3—181no—
376D3—183no—
377D3—182no—
378D3—186no—
379D3—181no—
380D3—183no—
381D3—181no—
382D3—541no—
383D3—542no—
384C22—271no—
385C22—271no—
386C4—811no—
387C3—23no—
388C3—21no—
389C3—31no—
390C3—33no—
391C3—61no—
392C3—63no—
393C3—61no—
394C3—63no—
395C3—121no—
396C3—123no—
397C3#1061988613yescomplemented
398C3#1061990411yescharacteristic, complemented
399C3#1062015711yescharacteristic
400C2—541no—
401C2—541no—
402C2#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⋊#272C2semidirect
2. C2×C3×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3⋊(C3⋊((C3⋊Dic3)⋊C3))C2semidirect
11. C2×C3×C3⋊(C3⋊D3)D3semidirect
16. C3⋊(C3⋊(C3⋊Dic3))D3semidirect
18. C2×C3⋊#272D3semidirect
21. C2×C3⋊((C3⋊D3)⋊C3)D3semidirect
28. C32×H(3,𝔽3)D4semidirect
59. C2×C3⋊(C3⋊D3)C3⋊D3semidirect
97. C34D12semidirect
98. C34#67semidirect
105. C3×H(3,𝔽3)#67semidirect
106. C3×H(3,𝔽3)#67semidirect
227. C33#625semidirect
229. C33#626semidirect
230. C33#626semidirect
231. C33#638semidirect
239. H(3,𝔽3)#625semidirect
352. C32#6200semidirect
353. C32#6290semidirect
356. C32#6295semidirect
397. C3#10619886semidirect
398. C3#10619904semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2722
C2×C3×C3⋊((C3⋊D3)⋊C3)2
C3⋊(C3⋊((C3⋊Dic3)⋊C3))2
#106201573
#106201513
#106199043
#106198863