Finite Group Databasebeta

← #80230

Subgroups — #80230

Order 384.

700 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic
2C2×D3×Q16C211yesmaximal, characteristic, complemented
3C2×#915C211yesmaximal, characteristic, complemented
4#5628C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5239C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5173C211yesmaximal, characteristic, complemented
9#5050C211yesmaximal, characteristic, complemented
10#4809C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4772C211yesmaximal, characteristic, complemented
14#4739C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3011—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18#1010—24no—
19#1010—24no—
20C2×D3×Q8C2211yescharacteristic
21C2×D3×Q8C2211yescharacteristic
22C2×D3×Q8C2211yescharacteristic
23C2×#237—21no—
24C2×#237—21no—
25C2×#237C2211yescharacteristic
26C2×#237C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×Dic6—21no—
30C22×Dic6C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic
32C2×C3×SD16C2211yescharacteristic
33C3×C22⋊Q8C2211yescharacteristic
34#946C2211yescharacteristic
35C2×#216C2211yescharacteristic
36C2×#216C2211yescharacteristic
37C2×#216C2211yescharacteristic
38C2×#214C2211yescharacteristic
39#924C2211yescharacteristic
40D3×Q16—24no—
41#915—24no—
42C2×Dic12C2211yescharacteristic
43C2×Dic12C2211yescharacteristic
44C2×ΓL(1,𝔽25)C2211yescharacteristic
45C2×D3×C8C2211yescharacteristic
46#897C2211yescharacteristic
47#894—21no—
48#892C2211yescharacteristic
49#888—21no—
50#885—21no—
51#883—21no—
52C3×#103C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×C22⋊C8C2211yescharacteristic
55#835C2211yescharacteristic
56#832C2211yescharacteristic
57#830C2211yescharacteristic
58(C2×D3)⋊C8C2211yescharacteristic
59(C2×D3)⋊C8C2211yescharacteristic
60#816C2211yescharacteristic
61#816C2211yescharacteristic
62#810C2211yescharacteristic
63#810C2211yescharacteristic
64C2×2−1+4—31no—
65C2×#137—31no—
66C22×Q16—31no—
67#536—31no—
68#452—31no—
69#452—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#451—31no—
73#451—31no—
74#451—31no—
75#451—31no—
76#419—31no—
77#416—31no—
78#408—31no—
79C3×#41—22no—
80C3×#41—22no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—21no—
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90C2×#67D411yescharacteristic
91C22×Dic3—21no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94D3×Q8—22no—
95D3×Q8—22no—
96D3×Q8—22no—
97D3×Q8—22no—
98D3×Q8—22no—
99D3×Q8D414yes—
100#237—28no—
101#237—28no—
102#237—22no—
103#237—22no—
104#237—22no—
105#237D414yes—
106#235—28no—
107#235—22no—
108#235—22no—
109C2×D12D411yescharacteristic
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×Dic6—28no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6D411yescharacteristic
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2311yescharacteristic
125C2×Dic6C2311yescharacteristic
126C3×Q16—22no—
127C3×SD16—22no—
128C2×C3×C8C2311yescharacteristic
129C2×C3×C8C2311yescharacteristic
130C3×C4⋊C4—21no—
131C3×C4⋊C4C2311yescharacteristic
132C3×C22⋊C4—21no—
133#217—21no—
134#216—22no—
135#216—22no—
136#216—22no—
137#214—22no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3C2311yescharacteristic
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148Dic12—22no—
149Dic12—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
1522−1+4—64no—
1532−1+4—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—61no—
158C2×#41—61no—
159C2×#41—61no—
160C2×#41C2×D311yescharacteristic
161C22×Q8—31no—
162C22×Q8—31no—
163C22×Q8—31no—
164C22×Q8—31no—
165C22×Q8—61no—
166#137—64no—
167C2×Q16—31no—
168C2×Q16—31no—
169C2×Q16—31no—
170C2×Q16—31no—
171C2×Q16—31no—
172C2×Q16—64no—
173C2×Q16C2×D311yescharacteristic
174C2×SD16—31no—
175C2×SD16C2×D311yescharacteristic
176C2×M16—31no—
177C22×C8—31no—
178C4⋊Q8—61no—
179#124—61no—
180#123—61no—
181#123—61no—
182C22⋊Q8—31no—
183C22⋊Q8—31no—
184C22⋊Q8—31no—
185C22⋊Q8C2×D311yescharacteristic
186#117—31no—
187#103—31no—
188#103—31no—
189#103—31no—
190#103—31no—
191#103—31no—
192#103C2×D311yescharacteristic
193#102—31no—
194#102C2×D311yescharacteristic
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic
199C23×C3—21no—
200C23×C3C2×D411yescharacteristic
201C22×D3C2×D411yescharacteristic
202C22×D3C2×D411yescharacteristic
203C3×Q8—21no—
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8C2×D412yes—
207C3×D4—24no—
208C3×D4—21no—
209C3×D4C2×D412yes—
210C2×C3×C4—24no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4C2×D411yescharacteristic
217C2×C3×C4C2411yescharacteristic
218#67—24no—
219#67—22no—
220#67—24no—
221#67—22no—
222C2×Dic3—24no—
223C2×Dic3—24no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—24no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3C2×D411yescharacteristic
230C2×Dic3C2×D411yescharacteristic
231D12—22no—
232C4×D3—24no—
233C4×D3—24no—
234C4×D3—21no—
235C4×D3—21no—
236C4×D3—42no—
237C4×D3C2×D412yes—
238C4×D3C2×D412yes—
239Dic6—24no—
240Dic6—24no—
241Dic6—21no—
242Dic6—21no—
243Dic6—22no—
244Dic6—24no—
245Dic6—21no—
246Dic6—21no—
247Dic6—42no—
248Dic6C2×D412yes—
249Dic6C2×D412yes—
250C3×C8—21no—
251C3×C8—21no—
252C3⋊C8—21no—
253C3⋊C8—21no—
254#41—22no—
255#41—22no—
256#41—34no—
257#41—62no—
258#41—62no—
259#41—68no—
260#41—62no—
261#41—62no—
262#41—68no—
263#41—68no—
264#41—62no—
265C2×Q8—31no—
266C2×Q8—34no—
267C2×Q8—31no—
268C2×Q8—31no—
269C2×Q8—31no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—62no—
274C2×Q8—62no—
275C2×Q8—68no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—62no—
281C2×Q8—61no—
282C2×Q8C22×D311yescharacteristic
283C2×Q8C22×D311yescharacteristic
284C2×D4—21no—
285C2×D4—31no—
286C2×D4—31no—
287C2×D4—31no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4C22×D311yescharacteristic
291C22×C4—21no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4C22×D311yescharacteristic
301Q16—22no—
302Q16—62no—
303Q16—62no—
304Q16—62no—
305Q16—62no—
306Q16—62no—
307SD16—22no—
308SD16—62no—
309M16—62no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—62no—
313C2×C8C22×D311yescharacteristic
314C2×C8C22×D311yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—31no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4C22×D311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C42—61no—
327C22×C3—21no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3C22×D411yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—21no—
339C2×D3#12012yes—
340C3×C4—21no—
341C3×C4—22no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4C22×D411yescharacteristic
345C3×C4C22×D411yescharacteristic
346Dic3—21no—
347Dic3—22no—
348Dic3—22no—
349Dic3—41no—
350Dic3—41no—
351Dic3#12012yes—
352C23—21no—
353C23—31no—
354C23—31no—
355C23D3×D411yescharacteristic
356Q8—21no—
357Q8—21no—
358Q8—21no—
359Q8—32no—
360Q8—32no—
361Q8—62no—
362Q8—64no—
363Q8—64no—
364Q8—64no—
365Q8—61no—
366Q8—61no—
367Q8—61no—
368Q8—61no—
369Q8—122no—
370Q8D3×D412yes—
371D4—24no—
372D4—21no—
373D4—62no—
374D4—62no—
375D4—62no—
376D4—64no—
377D4—64no—
378D4D3×D412yes—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—31no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—31no—
389C2×C4—64no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—122no—
401C2×C4D3×D411yescharacteristic
402C2×C4C23×D311yescharacteristic
403C8—21no—
404C8—21no—
405C8—61no—
406C8—61no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3C2×#12011yescharacteristic
410C2×C3#57811yescharacteristic
411C2×C3#57811yescharacteristic
412D3—22no—
413D3—41no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—32no—
419C22—41no—
420C22—41no—
421C22—61no—
422C22—61no—
423C22—61no—
424C22—61no—
425C22—61no—
426C22C2×D3×D411yescharacteristic
427C4—21no—
428C4—22no—
429C4—32no—
430C4—41no—
431C4—41no—
432C4—61no—
433C4—62no—
434C4—62no—
435C4—121no—
436C4—121no—
437C4C2×D3×D411yescharacteristic
438C4C2×D3×D411yescharacteristic
439C3#301111yescharacteristic, complemented, Sylow
440C2—41no—
441C2—42no—
442C2—62no—
443C2—121no—
444C2D3×#12011yescharacteristic
445C2#577111yescharacteristic
446C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radicalC2×#1010C2
hypercenter#452D3
hypocenterC3#3011
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×D3×Q16C2semidirect
3. C2×#915C2semidirect
4. #5628C2semidirect
5. C3×#452C2semidirect
6. #5239C2semidirect
7. #5186C2semidirect
8. #5173C2semidirect
9. #5050C2semidirect
10. #4809C2semidirect
11. #4802C2semidirect
12. #4783C2semidirect
13. #4772C2semidirect
14. #4739C2semidirect
15. #4727C2semidirect
71. #452D3semidirect
439. C3#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C2×D3×Q162
C2×#9152
#56282
C3×#4522
#52392
#51862
#51732
#50502
#48092
#48022
#47832
#47722
#47392
#47272
#30113