Finite Group Databasebeta

← #13149409

Subgroups — #13149409

Order 1728.

748 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 131 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#11721760C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal, complemented
3#11721117C211yesmaximal, characteristic, complemented
4C4×#6196C211yesmaximal, characteristic, complemented
5#11721098C211yesmaximal, characteristic, complemented
6#11721096C211yesmaximal, characteristic
7#11721096C211yesmaximal, characteristic
8#11721092C212yesmaximal
9#11721088C212yesmaximal
10#11721085C212yesmaximal, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10594598—32nomaximal
13C2×C4×#272C2211yescharacteristic
14C2×#6196C2211yescharacteristic
15C2×#6196C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#88542C2211yescharacteristic
18#88542C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4×#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88451C2212yes—
29#88451C2212yes—
30#88449C2211yescharacteristic
31#88449C2212yes—
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#64098D312yescomplemented
37C3×#892—32no—
38C3×#892—61no—
39C3×#889—32no—
40#63918—32no—
41C4×#627—32no—
42#63867—32no—
43#63862—32no—
44#63861—32no—
45#63859—32no—
46#63858—32no—
47#63851—32no—
48#63851—32no—
49#63847—32no—
50#63846—32no—
51#63837—32no—
52C22×#272C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C4×#272—22no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#6196—22no—
65#6196—22no—
66#5696—92no—
67#5518—91no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—61no—
70C2×#627—32no—
71C2×#627—32no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4C2×D312yes—
74C4⋊(C3⋊Dic3)C2×D312yes—
75#3750C2×D312yes—
76#3750C2×D312yes—
77#3750C2×D312yes—
78#3750C2×D312yes—
79C4×C3⋊Dic3C2×D312yes—
80C3×(C2×D3)⋊C4—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—61no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96#3724—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—32no—
109Dic32—32no—
110C22×H(3,𝔽3)C2411yescharacteristic
111C2×#272—21no—
112C2×#272—21no—
113C2×#272—21no—
114C4×H(3,𝔽3)—21no—
115C4×H(3,𝔽3)—21no—
116C4×H(3,𝔽3)—21no—
117#1084—21no—
118#1084—21no—
119#1084#4112yes—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124C3×#125—92no—
125C3×#117—91no—
126#924—91no—
127#924—91no—
128#898—92no—
129#898—92no—
130#898—92no—
131#896—92no—
132#896—92no—
133#892—32no—
134#892—61no—
135#890—92no—
136#889—32no—
137#889—92no—
138#889—92no—
139#889—92no—
140#889—92no—
141#888—92no—
142#887—92no—
143#879—92no—
144#879—92no—
145#876—92no—
146#870—91no—
147#869—91no—
148C4×Dic6—91no—
149C4×Dic6—92no—
150C4×Dic6—91no—
151C22×C3×D3—32no—
152C22×C3×D3—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D312yes—
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C3×C4×D3—64no—
174C3×C4×D3—64no—
175#627—64no—
176#627—64no—
177#541—271noSylow
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179C2×H(3,𝔽3)2−1+411yescharacteristic
180C2×H(3,𝔽3)2−1+411yescharacteristic
181#272—41no—
182C22×C3×C4—91no—
183C2×C4×D3—32no—
184C2×C4×D3—61no—
185C2×Dic6—91no—
186C2×Dic6—91no—
187C2×Dic6—92no—
188C2×Dic6—92no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—92no—
192C3×C4⋊C4—92no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—92no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4D3211yescharacteristic
198C3×C22⋊C4—91no—
199C3×C22⋊C4—91no—
200C3×C42—92no—
201C3×C42—91no—
202C3×C42—91no—
203#217—92no—
204#217—92no—
205(C2×D3)⋊C4—32no—
206(C2×D3)⋊C4—32no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—61no—
209(C2×D3)⋊C4—92no—
210(C2×D3)⋊C4—92no—
211(C2×D3)⋊C4—92no—
212(C2×D3)⋊C4—92no—
213C4⋊Dic3—32no—
214C4⋊Dic3—32no—
215C4⋊Dic3—61no—
216C4⋊Dic3—92no—
217C4⋊Dic3—91no—
218C4⋊Dic3—91no—
219C4⋊Dic3—92no—
220C4⋊Dic3—92no—
221C4⋊Dic3—92no—
222Dic3⋊C4—32no—
223Dic3⋊C4—32no—
224Dic3⋊C4—32no—
225Dic3⋊C4—32no—
226Dic3⋊C4—92no—
227Dic3⋊C4—92no—
228Dic3⋊C4—92no—
229Dic3⋊C4—92no—
230Dic3⋊C4—92no—
231Dic3⋊C4—92no—
232Dic3⋊C4—92no—
233Dic3⋊C4—92no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238C4×Dic3—32no—
239C4×Dic3—32no—
240C4×Dic3—32no—
241C4×Dic3—61no—
242C4×Dic3—61no—
243C4×Dic3—92no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C4×Dic3—92no—
247C22×C32—21no—
248C22×C32C23×D312yes—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—22no—
259C32×C4—41no—
260C32×C4—41no—
261C3⋊Dic3—22no—
262C3⋊Dic3—22no—
263C3⋊Dic3—22no—
264C3⋊Dic3—22no—
265C3×Dic3—34no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—121no—
274C3×Dic3—121no—
275C4⋊Q8—271no—
276#126—272no—
277#126—272no—
278#125—92no—
279#125—271no—
280#125—272no—
281C22⋊Q8—271no—
282C22⋊Q8—271no—
283C4×Q8—271no—
284C4×Q8—271no—
285#117—91no—
286H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
287C23×C3—91no—
288C22×D3—32no—
289C22×D3—61no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—91no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—91no—
302C2×C3×C4—91no—
303C2×C3×C4—182no—
304C2×C3×C4C2×D3211yescharacteristic
305C2×C3×C4C2×D3211yescharacteristic
306C2×C3×C4C2×D3211yescharacteristic
307C2×Dic3—32no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C4×D3—64no—
322C4×D3—64no—
323Dic6—184no—
324Dic6—184no—
325Dic6—182no—
326Dic6—182no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32#100712yes—
331C2×C32D3×#4112yes—
332C2×C32#101012yes—
333C3×D3—122no—
334C3×D3—122no—
335C2×Q8—271no—
336C2×Q8—271no—
337C22×C4—91no—
338C4⋊C4—91no—
339C4⋊C4—92no—
340C4⋊C4—92no—
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4—92no—
344C4⋊C4—271no—
345C4⋊C4—271no—
346C4⋊C4—272no—
347C4⋊C4—272no—
348C4⋊C4—272no—
349C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C22⋊C4—91no—
351C22⋊C4—91no—
352C22⋊C4—272no—
353C22⋊C4—272no—
354C42—91no—
355C42—91no—
356C42—92no—
357C42—272no—
358C22×C3—32no—
359C22×C3—61no—
360C22×C3—181no—
361C22×C3—181no—
362C22×C3—181no—
363C22×C3C22×D3211yescharacteristic
364C2×D3—62no—
365C2×D3—62no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—121no—
369C2×D3—121no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—62no—
376C3×C4—62no—
377C3×C4—92no—
378C3×C4—121no—
379C3×C4—121no—
380C3×C4—182no—
381C3×C4—182no—
382C3×C4—182no—
383C3×C4—182no—
384C3×C4—181no—
385C3×C4—181no—
386Dic3—34no—
387Dic3—62no—
388Dic3—62no—
389Dic3—62no—
390Dic3—62no—
391Dic3—62no—
392Dic3—62no—
393Dic3—62no—
394Dic3—121no—
395Dic3—121no—
396Dic3—182no—
397Dic3—182no—
398Dic3—182no—
399Dic3—182no—
400C32—21no—
401C32#569612yescomplemented
402C23—91no—
403Q8—542no—
404Q8—542no—
405C2×C4—91no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—182no—
413C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C2×C3—32no—
417C2×C3—32no—
418C2×C3—32no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—361no—
423C2×C3#6430611yescharacteristic
424C2×C3#6430911yescharacteristic
425C2×C3#6431611yescharacteristic
426D3—122no—
427D3—122no—
428C22—181no—
429C22—181no—
430C22—181no—
431C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—92no—
436C4—182no—
437C4—181no—
438C4—181no—
439C4—182no—
440C4—182no—
441C4—182no—
442C3—32no—
443C3—61no—
444C3#1059459911yescharacteristic
445C2—361no—
446C2#1172249111yescharacteristic
447C2#1172250211yescharacteristic
448C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64098⋊C3C2semidirect
3. #11721117C2semidirect
4. C4×#6196C2semidirect
5. #11721098C2semidirect
10. #11721085C2semidirect
11. #11721078C2semidirect
36. #64098D3semidirect
286. H(3,𝔽3)#541semidirect
401. C32#5696semidirect

Maximal subgroups

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

subgroupindex
#117217602
#64098⋊C32
#117211172
C4×#61962
#117210982
#117210962
#117210922
#117210882
#117210852
#117210782
#105945983