Finite Group Databasebeta

← #13180631

Subgroups — #13180631

Order 1728.

526 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 116 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
1#11724369C211yesmaximal, characteristic, complemented
2#11724369C212yesmaximal, complemented
3C2×C3×#3777C211yesmaximal, characteristic, complemented
4#67×#157C211yesmaximal, characteristic
5#67×#157C212yesmaximal
6#11724154C211yesmaximal, characteristic, complemented
7#11724152C212yesmaximal, complemented
8#11724151C212yesmaximal, complemented
9#11724150C211yesmaximal, characteristic, complemented
10#11724148C211yesmaximal, characteristic, complemented
11D3×#3777C211yesmaximal, characteristic, complemented
12#10596545—31nomaximal
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2212yes—
15C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
16C2×C3⋊PSU(3,𝔽2)C2212yes—
17C2×D3×#157C2211yescharacteristic
18C2×D3×#157C2212yes—
19C3⋊D3×#67C2211yescharacteristic
20#88994C2212yes—
21#88994C2211yescharacteristic
22#88946C2211yescharacteristic
23#88946C2212yes—
24#88945C2212yes—
25#88945C2212yes—
26#88945C2212yes—
27C3×#3777C2211yescharacteristic
28C3×#3777C2211yescharacteristic
29C3×#3777C2212yes—
30Dic3⋊#157C2211yescharacteristic
31Dic3⋊#157C2212yes—
32(C2×D3)⋊#157C2212yes—
33(C2×D3)⋊#157C2211yescharacteristic
34Dic3×#157C2212yes—
35Dic3×#157C2211yescharacteristic
36D4×#157—31no—
37D4×#157—32no—
38C22⋊PSU(3,𝔽2)—31no—
39C22⋊PSU(3,𝔽2)—32no—
40C22⋊PSU(3,𝔽2)—31no—
41C22⋊PSU(3,𝔽2)—32no—
42C2×#3777—31no—
43C2×#3777D311yescharacteristic, complemented
44C4⋊PSU(3,𝔽2)—31no—
45C4×PSU(3,𝔽2)—31no—
46#64243—32no—
47C22×C3×C3⋊D3C2311yescharacteristic
48C2×D3×C3⋊D3C2311yescharacteristic
49C2×C3⋊#157C2312yes—
50C2×C3⋊#157C2311yescharacteristic
51C2×C3⋊#157C2311yescharacteristic
52C2×C3⋊#157C2312yes—
53C2×C3×#157—22no—
54C2×C3×#157—42no—
55C2×C3×#157C2312yes—
56C2×C3×#157C2312yes—
57C2×C3×#157C2311yescharacteristic
58C2×C3×#157C2311yescharacteristic
59C3⋊PSU(3,𝔽2)—22no—
60C3⋊PSU(3,𝔽2)—42no—
61D3×#157—22no—
62D3×#157—42no—
63#6312Q811yescharacteristic, complemented
64C32×#67Q811yescharacteristic, complemented
65#6292Q811yescharacteristic, complemented
66#6290Q811yescharacteristic, complemented
67Dic3×C3⋊D3C2311yescharacteristic
68#5511—91nomaximal
69C22×#157—31no—
70C22×#157—32no—
71C22×#157C2×D311yescharacteristic
72C22×#157C2×D312yes—
73C2×PSU(3,𝔽2)—31no—
74C2×PSU(3,𝔽2)—32no—
75D4×C3⋊D3—31no—
76D3×#67—121no—
77C22⋊#157—31no—
78C22⋊#157—32no—
79C22⋊#157—31no—
80C22⋊#157—32no—
81C4⋊#157—31no—
82C4⋊#157—32no—
83C4×#157—32no—
84C4×#157—31no—
85#3777—32no—
86#3777—32no—
87#3777—32no—
88#3777—32no—
89#3777—31no—
90#3777C2×D312yes—
91#3777C2×D311yescharacteristic
92#3777C2×D311yescharacteristic
93C22×C33C2×Q811yescharacteristic
94C2×C3⋊(C3⋊D3)C2×Q811yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3C2×Q811yescharacteristic
98C2×C3×C3⋊D3C2411yescharacteristic
99C2×C32×D3C2×Q811yescharacteristic
100D3×C3⋊D3—21no—
101D3×C3⋊D3—21no—
102C3⋊#157—21no—
103C3⋊#157—22no—
104C3⋊#157—22no—
105C3⋊#157—21no—
106C3×#157—21no—
107C3×#157—22no—
108C3×#157—22no—
109C3×#157#4112yes—
110C3⋊(C3⋊Dic3)C2×Q811yescharacteristic
111C32×Dic3C2×Q811yescharacteristic
112C2×C3×C4⋊C4—91no—
113C4×#67—91no—
114C4×#67—92no—
115#924—92no—
116#924—91no—
117#897—91no—
118#896—92no—
119D3×C4⋊C4—91no—
120#889—92no—
121#888—91no—
122#887—91no—
123C22×C3⋊D3—31no—
124C22×C3⋊D3C22×D311yescharacteristic
125C22×C3×D3—121no—
126C2×D32—121no—
127C2×#157—22no—
128C2×#157—31no—
129C2×#157—32no—
130C2×#157—32no—
131C2×#157—31no—
132C2×#157—42no—
133C2×#157—62no—
134C2×#157—122no—
135C2×#157C22×D312yes—
136C2×#157C22×D312yes—
137C2×#157C22×D311yescharacteristic
138C2×#157C22×D311yescharacteristic
139PSU(3,𝔽2)—62no—
140PSU(3,𝔽2)—122no—
141C32×D4—31no—
142#638—31no—
143#638—31no—
144#638—121no—
145#638—241no—
146C4⋊(C3⋊D3)—31no—
147C4×C3⋊D3—31no—
148C3×#67—41no—
149#626—121no—
150#625—121no—
151D3×Dic3—121no—
152#554—271noSylow
153C2×C33—21no—
154C2×C33C22×Q811yescharacteristic
155C3⋊(C3⋊D3)—21no—
156C3×C3⋊D3—21no—
157C3×C3⋊D3C2×#4111yescharacteristic
158C3×C3⋊D3H(5,𝔽2)11yescharacteristic
159C32×D3—21no—
160C22×C3×C4—91no—
161C22×C3×C4—92no—
162C2×#67—91no—
163D3×D4—361no—
164C2×C4×D3—91no—
165C2×C4×D3—92no—
166C2×Dic6—91no—
167C2×Dic6—92no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4—92no—
171#217—92no—
172#217—91no—
173(C2×D3)⋊C4—91no—
174(C2×D3)⋊C4—92no—
175C4⋊Dic3—91no—
176C4⋊Dic3—92no—
177Dic3⋊C4—92no—
178Dic3⋊C4—91no—
179Dic3⋊C4—92no—
180Dic3⋊C4—92no—
181Dic3⋊C4—92no—
182C4×Dic3—91no—
183C4×Dic3—92no—
184C22×C32—31no—
185C22×C32—41no—
186C22×C32—81no—
187C22×C32D3×Q811yescharacteristic
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3⋊D3—31no—
191C2×C3⋊D3—61no—
192C2×C3⋊D3—61no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—241no—
195C2×C3⋊D3D3×Q811yescharacteristic
196C2×C3⋊D3C23×D311yescharacteristic
197C2×C3×D3—41no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C2×C3×D3—241no—
201C2×C3×D3—241no—
202D32—241no—
203D32—241no—
204#157—21no—
205#157—22no—
206#157—22no—
207#157—62no—
208#157—61no—
209#157—61no—
210#157—62no—
211#157#23512yes—
212C32×C4—31no—
213C3⋊Dic3—31no—
214C3⋊Dic3—121no—
215C3⋊Dic3—241no—
216C3×Dic3—41no—
217C4⋊Q8—271no—
218#125—272no—
219C22⋊Q8—272no—
220C22⋊Q8—271no—
221C22⋊Q8—272no—
222C22⋊Q8—271no—
223C4×Q8—271no—
224C4×D4—272no—
225C4×D4—271no—
226C2×C4⋊C4—91no—
227C2×C4⋊C4—271no—
228C33#55411yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C22×D3—91no—
231C22×D3—121no—
232C22×D3—361no—
233C3×D4—121no—
234C2×C3×C4—91no—
235C2×C3×C4—92no—
236C2×C3×C4—91no—
237C2×C3×C4—92no—
238C2×C3×C4—182no—
239C2×C3×C4—362no—
240#67—91no—
241#67—91no—
242#67—91no—
243#67—361no—
244#67—361no—
245#67—721no—
246#67PSU(3,𝔽2)11yescharacteristic, complemented
247C2×Dic3—91no—
248C2×Dic3—92no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—92no—
252D12—361no—
253C4×D3—182no—
254C4×D3—361no—
255C4×D3—362no—
256Dic6—182no—
257Dic6—362no—
258C2×C32—21no—
259C2×C32—41no—
260C2×C32—61no—
261C2×C32—81no—
262C2×C32—81no—
263C2×C32—161no—
264C2×C32C2×D3×Q811yescharacteristic
265C3⋊D3—21no—
266C3⋊D3—61no—
267C3⋊D3—241no—
268C3⋊D3—481no—
269C3⋊D3C2×#23511yescharacteristic
270C3⋊D3#100411yescharacteristic
271C3×D3—81no—
272C3×D3—121no—
273C3×D3—121no—
274C3×D3—241no—
275C2×Q8—271no—
276C2×Q8—272no—
277C2×D4—271no—
278C22×C4—91no—
279C22×C4—92no—
280C22×C4—271no—
281C22×C4—272no—
282C4⋊C4—91no—
283C4⋊C4—92no—
284C4⋊C4—91no—
285C4⋊C4—272no—
286C4⋊C4—271no—
287C4⋊C4—271no—
288C4⋊C4—272no—
289C4⋊C4—272no—
290C4⋊C4—272no—
291C4⋊C4—272no—
292C22⋊C4—271no—
293C22⋊C4—272no—
294C22⋊C4—272no—
295C22⋊C4—271no—
296C42—272no—
297C42—271no—
298C22×C3—41no—
299C22×C3—81no—
300C22×C3—91no—
301C22×C3—91no—
302C22×C3—121no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3C2×PSU(3,𝔽2)11yescharacteristic
306C2×D3—91no—
307C2×D3—121no—
308C2×D3—121no—
309C2×D3—181no—
310C2×D3—181no—
311C2×D3—241no—
312C2×D3—241no—
313C2×D3—361no—
314C2×D3—721no—
315C2×D3—721no—
316C2×D3—721no—
317C2×D3C2×PSU(3,𝔽2)11yescharacteristic
318C3×C4—92no—
319C3×C4—121no—
320C3×C4—182no—
321C3×C4—182no—
322C3×C4—181no—
323Dic3—91no—
324Dic3—182no—
325Dic3—181no—
326Dic3—182no—
327Dic3—181no—
328Dic3—361no—
329Dic3—721no—
330Dic3C2×PSU(3,𝔽2)11yescharacteristic
331C32—41no—
332C32—81no—
333C32#551111yescharacteristic, complemented
334C23—91no—
335C23—271no—
336Q8—542no—
337Q8—1082no—
338D4—31no—
339D4—271no—
340D4—271no—
341D4—271no—
342C2×C4—91no—
343C2×C4—92no—
344C2×C4—91no—
345C2×C4—92no—
346C2×C4—182no—
347C2×C4—271no—
348C2×C4—272no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—272no—
352C2×C4—362no—
353C2×C4—542no—
354C2×C4—1082no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—91no—
360C2×C3—91no—
361C2×C3—161no—
362C2×C3—181no—
363C2×C3—241no—
364C2×C3C22×PSU(3,𝔽2)11yescharacteristic
365D3—21no—
366D3—121no—
367D3—121no—
368D3—181no—
369D3—241no—
370D3—721no—
371D3—1441no—
372C22—31no—
373C22—91no—
374C22—91no—
375C22—181no—
376C22—181no—
377C22—271no—
378C22—541no—
379C22—541no—
380C22D3×PSU(3,𝔽2)11yescharacteristic
381C4—31no—
382C4—92no—
383C4—181no—
384C4—182no—
385C4—182no—
386C4—271no—
387C4—541no—
388C4—542no—
389C4—542no—
390C4—541no—
391C3—41no—
392C3—81no—
393C3#1059654511yescharacteristic, complemented
394C2—21no—
395C2—61no—
396C2—91no—
397C2—91no—
398C2—181no—
399C2—541no—
400C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FittingC22×C33C2×Q8
socleC2×C33C22×Q8
prime radicalC3⋊D3×#67C22
hypercenterC22D3×PSU(3,𝔽2)
hypocenterC33#554
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #11724369C2semidirect
2. #11724369C2semidirect
3. C2×C3×#3777C2semidirect
6. #11724154C2semidirect
7. #11724152C2semidirect
8. #11724151C2semidirect
9. #11724150C2semidirect
10. #11724148C2semidirect
11. D3×#3777C2semidirect
43. C2×#3777D3semidirect
63. #6312Q8semidirect
64. C32×#67Q8semidirect
65. #6292Q8semidirect
66. #6290Q8semidirect
228. C33#554semidirect
246. #67PSU(3,𝔽2)semidirect
333. C32#5511semidirect
393. C3#10596545semidirect

Maximal subgroups

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

subgroupindex
#117243692
C2×C3×#37772
#67×#1572
#117241542
#117241522
#117241512
#117241502
#117241482
D3×#37772
#105965453
#55119