Finite Group Databasebeta

← #76719

Subgroups — #76719

Order 384.

562 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#935C211yesmaximal, characteristic, complemented
2C2×#878C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5220C211yesmaximal, characteristic, complemented
7#5220C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#4952C211yesmaximal, characteristic, complemented
10#4947C211yesmaximal, characteristic, complemented
11#4947C211yesmaximal, characteristic, complemented
12#4945C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4659C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2696—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36#935—24no—
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43#878—24no—
44C3×#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123—31no—
55C2×C22⋊Q8—31no—
56#400—31no—
57#397—31no—
58#394—31no—
59#393—31no—
60#393—31no—
61#392D311yescharacteristic
62#390—31no—
63#388—31no—
64#387—31no—
65#386—31no—
66#386—31no—
67#386—31no—
68#380—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C2×Dic6—22no—
100C2×Dic6—41no—
101C2×Dic6—41no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105#217—22no—
106#217—22no—
107#217—22no—
108#217—22no—
109#217—22no—
110#217—41no—
111#217—41no—
112#217—41no—
113#217—41no—
114#217—41no—
115C4⋊Dic3—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—41no—
119Dic3⋊C4D414yescomplemented
120C4×Dic3—41no—
121C22×Q8—31no—
122C22×D4C2×D311yescharacteristic
123C22×D4C2×D311yescharacteristic
124C22×D4C2×D311yescharacteristic
125C23×C4—31no—
126C23×C4—31no—
127#123—64no—
128C22⋊Q8—64no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
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⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4C2×D311yescharacteristic
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C42—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×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3#4112yes—
178C23×C3#4112yes—
179C23×C3C2411yescharacteristic
180C3×D4—44no—
181C3×D4—44no—
182C3×D4—44no—
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×C4C2×D412yes—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
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×Dic3—44no—
219C2×Dic3C2×D412yes—
220C2×Dic3C2×D412yes—
221Dic6—44no—
222C24C22×D311yescharacteristic
223C24C22×D311yescharacteristic
224C24C22×D311yescharacteristic
225C2×Q8—62no—
226C2×Q8—121no—
227C2×Q8—121no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—121no—
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C4⋊C4—34no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—121no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C42—121no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
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×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3C22×D411yescharacteristic
296C22×C3C2×#4111yescharacteristic
297C22×C3C2×#4111yescharacteristic
298C22×C3H(5,𝔽2)11yescharacteristic
299C22×C3H(5,𝔽2)11yescharacteristic
300C22×C3H(5,𝔽2)11yescharacteristic
301C22×C32−1+411yescharacteristic
302C3×C4—22no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306Dic3—22no—
307Dic3—22no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314C23—21no—
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—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23#23712yes—
334C23#23712yes—
335C23C23×D311yescharacteristic
336Q8—124no—
337D4—44no—
338D4—44no—
339D4—44no—
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—32no—
352C2×C4—32no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
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×C4D3×D412yes—
378C2×C3—42no—
379C2×C3—42no—
380C2×C3—81no—
381C2×C3#53511yescharacteristic
382C2×C3#53811yescharacteristic
383C2×C3#53911yescharacteristic
384C2×C3#54611yescharacteristic
385C2×C3#54811yescharacteristic
386C2×C3#55111yescharacteristic
387C2×C3#56211yescharacteristic
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22C2×D3×D411yescharacteristic
410C22C2×#23711yescharacteristic
411C22C2×#23711yescharacteristic
412C22#100411yescharacteristic
413C22#100411yescharacteristic
414C22#100411yescharacteristic
415C22#101011yescharacteristic
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—62no—
421C4—62no—
422C4—121no—
423C4—121no—
424C4—121no—
425C4—121no—
426C4—121no—
427C4—121no—
428C3#269611yescharacteristic, complemented, Sylow
429C2—42no—
430C2—42no—
431C2—81no—
432C2#558711yescharacteristic
433C2#559111yescharacteristic
434C2#559311yescharacteristic
435C2#559811yescharacteristic
436C2#560211yescharacteristic
437C2#562111yescharacteristic
438C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2696
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#935C2semidirect
2. C2×#878C2semidirect
4. #5222C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #5220C2semidirect
8. #5220C2semidirect
9. #4952C2semidirect
10. #4947C2semidirect
11. #4947C2semidirect
12. #4945C2semidirect
13. #4942C2semidirect
14. #4659C2semidirect
15. #4652C2semidirect
119. Dic3⋊C4D4semidirect
428. C3#2696semidirect

Maximal subgroups

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

subgroupindex
C2×#9352
C2×#8782
C3×#3922
#52222
#52212
#52202
#49522
#49472
#49452
#49422
#46592
#46522
#26963