Finite Group Databasebeta

← #77118

Subgroups — #77118

Order 384.

610 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#938C211yesmaximal, characteristic
2C2×#67⋊C4C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5220C211yesmaximal, characteristic, complemented
5#4989C211yesmaximal, characteristic, complemented
6#4980C211yesmaximal, characteristic, complemented
7#4956C211yesmaximal, characteristic, complemented
8#4956C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10#4952C211yesmaximal, characteristic, complemented
11#4952C211yesmaximal, characteristic, complemented
12#4945C211yesmaximal, characteristic, complemented
13#4943C211yesmaximal, characteristic, complemented
14D3×#95C211yesmaximal, characteristic, complemented
15#4660C211yesmaximal, characteristic, complemented
16#2444—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34#938—24no—
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41#67⋊C4—24no—
42C3×#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#121—31no—
55C2×C4×D4—31no—
56#400—31no—
57#397—31no—
58#397—31no—
59#394D311yescharacteristic, complemented
60#388—31no—
61#387—31no—
62#386—31no—
63#386—31no—
64#386—31no—
65#386—31no—
66#385—31no—
67#381—31no—
68C2×#95—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—41no—
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—22no—
81C2×#67—22no—
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67D414yes—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C2×C4×D3—41no—
104C2×C4×D3—41no—
105C3×C4⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110#217—22no—
111#217—22no—
112#217—22no—
113#217—41no—
114#217—41no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117C4⋊Dic3—22no—
118Dic3⋊C4—22no—
119Dic3⋊C4—41no—
120C4×Dic3—22no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4C2×D311yescharacteristic
124C23×C4—31no—
125C23×C4—31no—
126C23×C4—31no—
127C23×C4—31no—
128#121—64no—
129C4×D4—64no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4C2×D311yescharacteristic
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4C2×D311yescharacteristic
142C2×C22⋊C4C2×D311yescharacteristic
143C2×C22⋊C4C2×D311yescharacteristic
144C2×C22⋊C4C2×D311yescharacteristic
145C2×C42—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95C2×D311yescharacteristic
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3C2×D412yes—
172C23×C3C2411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3C2×D412yes—
178C22×D3#4112yes—
179C22×D3#4112yes—
180C3×D4—44no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4#4112yes—
196#67—28no—
197#67—44no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—44no—
218C2×Dic3C2×D412yes—
219C2×Dic3#4112yes—
220C4×D3—44no—
221C4×D3—44no—
222C4×D3—44no—
223C24—31no—
224C24C22×D311yescharacteristic
225C24C22×D311yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—34no—
229C2×D4—41no—
230C2×D4—62no—
231C2×D4—62no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—62no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4—121no—
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—121no—
263C22⋊C4—22no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C42—62no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3C22×D411yescharacteristic
290C22×C3C2×#4111yescharacteristic
291C22×C3C2×#4111yescharacteristic
292C22×C3C2×#4111yescharacteristic
293C22×C3C2×#4111yescharacteristic
294C22×C3H(5,𝔽2)11yescharacteristic
295C22×C3H(5,𝔽2)11yescharacteristic
296C2×D3—22no—
297C2×D3—22no—
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—22no—
302C2×D3#12314yes—
303C3×C4—22no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308Dic3—22no—
309Dic3—22no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3—41no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—32no—
329C23—32no—
330C23—32no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23D3×D412yes—
336C23C23×D311yescharacteristic
337D4—44no—
338D4—68no—
339D4—124no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—124no—
376C2×C4—124no—
377C2×C4—124no—
378C2×C4—124no—
379C2×C4#23712yes—
380C2×C3—42no—
381C2×C3—81no—
382C2×C3C2×#12311yescharacteristic
383C2×C3#52911yescharacteristic
384C2×C3#53811yescharacteristic
385C2×C3#54611yescharacteristic
386C2×C3#54611yescharacteristic
387C2×C3#55311yescharacteristic
388C2×C3#55311yescharacteristic
389D3—24no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—34no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22C2×D3×D411yescharacteristic
412C22C2×#23711yescharacteristic
413C22#100411yescharacteristic
414C22#100411yescharacteristic
415C22D3×#4111yescharacteristic
416C22D3×#4111yescharacteristic
417C22D3×#4111yescharacteristic
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—62no—
424C4—62no—
425C4—121no—
426C4—121no—
427C4—121no—
428C4—121no—
429C4—121no—
430C3#244411yescharacteristic, complemented, Sylow
431C2—42no—
432C2—64no—
433C2—81no—
434C2#559111yescharacteristic
435C2#561011yescharacteristic
436C2#561311yescharacteristic
437C2#561411yescharacteristic
438C2D3×#12311yescharacteristic
439C2#566111yescharacteristic
440C2#568011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#938C2
hypercenter#394D3
hypocenterC3#2444
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#67⋊C4C2semidirect
3. C3×#394C2semidirect
4. #5220C2semidirect
5. #4989C2semidirect
6. #4980C2semidirect
7. #4956C2semidirect
8. #4956C2semidirect
9. #4956C2semidirect
10. #4952C2semidirect
11. #4952C2semidirect
12. #4945C2semidirect
13. #4943C2semidirect
14. D3×#95C2semidirect
15. #4660C2semidirect
59. #394D3semidirect
430. C3#2444semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#67⋊C42
C3×#3942
#52202
#49892
#49802
#49562
#49522
#49452
#49432
D3×#952
#46602
#24443