Finite Group Databasebeta

← #13140216

Subgroups — #13140216

Order 1728.

727 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 117 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×#88542C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88451C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591315—31nomaximal
7#10591314—31nomaximal
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#88542—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15#88451—24no—
16C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
17C2×C3×(C2×D3)⋊C4—31no—
18C2×C3×(C2×D3)⋊C4—31no—
19C2×C3×(C2×D3)⋊C4—61no—
20C2×C3×C4⋊Dic3—31no—
21C2×(C2×D3)⋊Dic3—32no—
22C2×(C2×D3)⋊Dic3—32no—
23#63658D311yescharacteristic, complemented
24C3×C3⋊#95—31no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×#272—21no—
29C22×#272—21no—
30C22×#272—22no—
31C22×#272—22no—
32C22×#272D412yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)D412yes—
35C2×#1084—21no—
36C2×#1084—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43#4992—91no—
44#4953—91no—
45#4674—91no—
46C23×C3×D3—31no—
47C23×C3×D3—31no—
48C23×C3×D3—61no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D312yes—
53C22×C3⋊Dic3C2×D312yes—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—31no—
56C22×C3×Dic3—31no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—61no—
59C4⋊(C3⋊Dic3)—22no—
60C3×(C2×D3)⋊C4—64no—
61C3×(C2×D3)⋊C4—62no—
62C3×(C2×D3)⋊C4—62no—
63C3×C4⋊Dic3—62no—
64(C2×D3)⋊Dic3—64no—
65(C2×D3)⋊Dic3—64no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)#4111yescharacteristic
69C22×H(3,𝔽3)#4112yes—
70C22×H(3,𝔽3)#4112yes—
71C2×#272—22no—
72C2×#272—24no—
73C2×#272—41no—
74C2×#272—41no—
75C2×#272—42no—
76C4×H(3,𝔽3)—22no—
77#1084—41no—
78(C3⋊Dic3)⋊C3—42no—
79(C3⋊Dic3)⋊C3—42no—
80C2×C3×C4⋊C4—91no—
81C2×C3×C22⋊C4—91no—
82C2×#217—92no—
83C2×(C2×D3)⋊C4—31no—
84C2×(C2×D3)⋊C4—31no—
85C2×(C2×D3)⋊C4—61no—
86C2×(C2×D3)⋊C4—92no—
87C2×(C2×D3)⋊C4—92no—
88C2×C4⋊Dic3—31no—
89C2×C4⋊Dic3—91no—
90C3×#95—91no—
91C3⋊#95—31no—
92C3⋊#95—91no—
93C3⋊#95—92no—
94C3⋊#95—92no—
95C3⋊#95—92no—
96C23×C32—21no—
97C23×C32C22×D311yescharacteristic
98C23×C32C22×D311yescharacteristic
99C22×C3×D3—32no—
100C22×C3×D3—32no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C22×C3×D3—61no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C22×C3×D3—61no—
108C22×C3×D3—62no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—122no—
113C22×C3×D3—122no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—42no—
118C2×C32×C4D1212yes—
119C2×C32×C4#6712yes—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—22no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—122no—
138#397—271noSylow
139C2×H(3,𝔽3)#12111yescharacteristic
140C2×H(3,𝔽3)#12312yes—
141C2×H(3,𝔽3)#12311yescharacteristic
142C2×H(3,𝔽3)#12411yescharacteristic
143C2×H(3,𝔽3)#12612yes—
144#272—42no—
145C24×C3—91no—
146C23×D3—31no—
147C23×D3—31no—
148C23×D3—61no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—92no—
153C22×C3×C4—92no—
154C22×C3×C4—91no—
155C22×C3×C4D3211yescharacteristic
156C22×Dic3—31no—
157C22×Dic3—31no—
158C22×Dic3—32no—
159C22×Dic3—32no—
160C22×Dic3—61no—
161C22×Dic3—92no—
162C22×Dic3—92no—
163C3×C4⋊C4—182no—
164C3×C22⋊C4—182no—
165#217—184no—
166(C2×D3)⋊C4—62no—
167(C2×D3)⋊C4—62no—
168(C2×D3)⋊C4—64no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—184no—
171C4⋊Dic3—62no—
172C4⋊Dic3—182no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32C2×D1211yescharacteristic
179C22×C32#23512yes—
180C22×C32#23512yes—
181C22×C32D3×D411yescharacteristic
182C22×C32D3×D411yescharacteristic
183C22×C32#23711yescharacteristic
184C22×C32#23712yes—
185C22×C32#23911yescharacteristic
186C22×C32#23912yes—
187C22×C32C2×#6711yescharacteristic
188C2×C3×D3—64no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—124no—
194C2×C3×D3—124no—
195C2×C3×D3—122no—
196C2×C3×D3—122no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—122no—
200C2×C3×D3—122no—
201C2×C3×D3—121no—
202C2×C3×D3—121no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—42no—
206C3⋊Dic3—42no—
207C3⋊Dic3—42no—
208C3×Dic3—122no—
209C3×Dic3—121no—
210C3×Dic3—122no—
211C3×Dic3—122no—
212C3×Dic3—121no—
213C2×C4⋊C4—91no—
214C2×C22⋊C4—91no—
215C2×C22⋊C4—272no—
216#95—91no—
217#95—272no—
218H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—61no—
222C23×C3—92no—
223C23×C3—181no—
224C23×C3—182no—
225C23×C3—182no—
226C23×C3—181no—
227C23×C3C2×D3211yescharacteristic
228C22×D3—32no—
229C22×D3—32no—
230C22×D3—62no—
231C22×D3—61no—
232C22×D3—61no—
233C22×D3—61no—
234C22×D3—61no—
235C22×D3—62no—
236C22×D3—62no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—62no—
241C22×D3—122no—
242C22×D3—122no—
243C2×C3×C4—22no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—62no—
249C2×C3×C4—122no—
250C2×C3×C4—182no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4—182no—
256C2×C3×C4—182no—
257C2×C3×C4—181no—
258C2×C3×C4#62612yes—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—62no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—122no—
271C2×Dic3—182no—
272C2×Dic3—182no—
273C2×Dic3—182no—
274C2×Dic3—182no—
275C2×Dic3—182no—
276C2×Dic3—182no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32#87511yescharacteristic
283C2×C32#88312yes—
284C2×C32#88611yescharacteristic
285C2×C32#89412yes—
286C2×C32#89511yescharacteristic
287C2×C32#89812yes—
288C2×C32#89812yes—
289C2×C32#92911yescharacteristic
290C2×C32#93811yescharacteristic
291C2×C32#94711yescharacteristic
292C3×D3—124no—
293C3×D3—122no—
294C3×D3—122no—
295C24—91no—
296C22×C4—92no—
297C22×C4—91no—
298C22×C4—92no—
299C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C4⋊C4—182no—
301C22⋊C4—182no—
302C22⋊C4—544no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—32no—
306C22×C3—32no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—62no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—61no—
318C22×C3—184no—
319C22×C3—182no—
320C22×C3—362no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3#379611yescharacteristic
324C22×C3#379912yes—
325C22×C3#380211yescharacteristic
326C22×C3#380412yes—
327C22×C3C2×#62611yescharacteristic
328C2×D3—64no—
329C2×D3—64no—
330C2×D3—62no—
331C2×D3—62no—
332C2×D3—64no—
333C2×D3—124no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—122no—
337C2×D3—122no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—122no—
341C2×D3—124no—
342C2×D3—122no—
343C3×C4—22no—
344C3×C4—62no—
345C3×C4—62no—
346C3×C4—122no—
347C3×C4—362no—
348C3×C4—362no—
349C3×C4—361no—
350Dic3—121no—
351Dic3—121no—
352Dic3—122no—
353Dic3—122no—
354Dic3—122no—
355Dic3—362no—
356Dic3—362no—
357C32—21no—
358C32#467411yescharacteristic, complemented
359C32#499211yescharacteristic, complemented
360C23—92no—
361C23—181no—
362C23—182no—
363C23—182no—
364C23—181no—
365C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C2×C4—22no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4—182no—
372C2×C4—182no—
373C2×C4—181no—
374C2×C4—182no—
375C2×C4#620012yes—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—32no—
379C2×C3—32no—
380C2×C3—32no—
381C2×C3—32no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—62no—
388C2×C3—62no—
389C2×C3—61no—
390C2×C3—61no—
391C2×C3—362no—
392C2×C3#6384812yes—
393C2×C3#6385412yes—
394C2×C3#6386411yescharacteristic
395C2×C3#6391611yescharacteristic
396C2×C3#6396611yescharacteristic
397D3—122no—
398D3—122no—
399D3—124no—
400C22—184no—
401C22—182no—
402C22—361no—
403C22—362no—
404C22—361no—
405C22#8864911yescharacteristic
406C22#8865212yes—
407C22#8865511yescharacteristic
408C22#8867112yes—
409C22C2×#620011yescharacteristic
410C4—22no—
411C4—362no—
412C4—361no—
413C4—362no—
414C3—31no—
415C3—31no—
416C3—61no—
417C3#1059132611yescharacteristic
418C2—362no—
419C2#1172108812yes—
420C2#1172110212yes—
421C2#1172111411yescharacteristic
422C2#1172115911yescharacteristic
423C2#1172130111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#397
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
16. C2×C4⋊(C3⋊Dic3)D3semidirect
23. #63658D3semidirect
218. H(3,𝔽3)#397semidirect
358. C32#4674semidirect
359. C32#4992semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
#63658⋊C32
#117203342
#105913153
#105913143