Finite Group Databasebeta

← #74851

Subgroups — #74851

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#386C211yesmaximal, characteristic, complemented
2#5223C211yesmaximal, characteristic
3#5220C211yesmaximal, characteristic, complemented
4#5213C211yesmaximal, characteristic, complemented
5#5211C211yesmaximal, characteristic, complemented
6#4958C211yesmaximal, characteristic, complemented
7#4956C211yesmaximal, characteristic, complemented
8#4956C211yesmaximal, characteristic, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4947C211yesmaximal, characteristic, complemented
11#4943C211yesmaximal, characteristic, complemented
12#4673C211yesmaximal, characteristic, complemented
13#4664C211yesmaximal, characteristic, complemented
14C3⋊#401C211yesmaximal, characteristic, complemented
15C3⋊#401C211yesmaximal, characteristic, complemented
16#2760—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C3×#95C2211yescharacteristic
37C3×#95C2211yescharacteristic
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C3⋊#95C2211yescharacteristic
41C3⋊#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#401—31no—
53#401—31no—
54#393—31no—
55#393—31no—
56#393—31no—
57#392—31no—
58#386—31no—
59#386—31no—
60#386—31no—
61#386—31no—
62#386—31no—
63#386D311yescharacteristic, complemented
64#381—31no—
65#381—31no—
66#381—31no—
67C24×C3C2311yescharacteristic
68C24×C3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4—41no—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—22no—
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—41no—
86C2×#67—41no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3—41no—
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C2×C4×D3—22no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×Dic6—41no—
101C2×Dic6—41no—
102C2×Dic6—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106#217—41no—
107#217—41no—
108#217—41no—
109#217—41no—
110#217—41no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115C22×Q8—31no—
116C22×D4—31no—
117C22×D4—31no—
118C22×D4C2×D311yescharacteristic
119C23×C4—31no—
120C23×C4—31no—
121C23×C4C2×D311yescharacteristic
122C2×C22⋊C4—31no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4C2×D311yescharacteristic
132C2×C22⋊C4C2×D311yescharacteristic
133C2×C22⋊C4C2×D311yescharacteristic
134#95—31no—
135#95—31no—
136#95—31no—
137#95—31no—
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95C2×D311yescharacteristic
149#95C2×D311yescharacteristic
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3#4112yes—
163C23×C3#4112yes—
164C23×C3C2411yescharacteristic
165C22×D3—21no—
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3#4112yes—
172C3×D4—44no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
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—44no—
189#67—44no—
190#67—44no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
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—44no—
213C4×D3—44no—
214Dic6—44no—
215C24—31no—
216C24C22×D311yescharacteristic
217C24C22×D311yescharacteristic
218C2×Q8—121no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—62no—
225C2×D4—62no—
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—121no—
229C2×D4—121no—
230C22×C4—22no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4—121no—
245C22×C4—121no—
246C22×C4C22×D311yescharacteristic
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C2×#4111yescharacteristic
278C22×C3C2×#4111yescharacteristic
279C22×C3C2×#4111yescharacteristic
280C22×C3H(5,𝔽2)11yescharacteristic
281C22×C3H(5,𝔽2)11yescharacteristic
282C22×C3H(5,𝔽2)11yescharacteristic
283C22×C3H(5,𝔽2)11yescharacteristic
284C2×D3—22no—
285C2×D3—22no—
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—32no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—61no—
322C23#23512yes—
323C23#23712yes—
324C23C23×D311yescharacteristic
325Q8—124no—
326D4—44no—
327D4—124no—
328D4—124no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—44no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
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—124no—
367C2×C4—124no—
368C2×C3—42no—
369C2×C3—42no—
370C2×C3#53811yescharacteristic
371C2×C3#53811yescharacteristic
372C2×C3#53811yescharacteristic
373C2×C3#55111yescharacteristic
374C2×C3#55111yescharacteristic
375C2×C3#55111yescharacteristic
376C2×C3#56011yescharacteristic
377D3—42no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—62no—
393C22—62no—
394C22—62no—
395C22—121no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22C2×#23511yescharacteristic
400C22C2×#23711yescharacteristic
401C22#100411yescharacteristic
402C22#100411yescharacteristic
403C22#100411yescharacteristic
404C22D3×#4111yescharacteristic
405C22#100911yescharacteristic
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—121no—
412C4—121no—
413C4—121no—
414C4—121no—
415C4—121no—
416C4—121no—
417C4—121no—
418C3#276011yescharacteristic, complemented, Sylow
419C2—42no—
420C2—42no—
421C2—122no—
422C2#549511yescharacteristic
423C2#555611yescharacteristic
424C2#556011yescharacteristic
425C2#559311yescharacteristic
426C2#561311yescharacteristic
427C2#562011yescharacteristic
428C2#565811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radical#5223C2
hypercenter#386D3
hypocenterC3#2760
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C3×#386C2semidirect
3. #5220C2semidirect
4. #5213C2semidirect
5. #5211C2semidirect
6. #4958C2semidirect
7. #4956C2semidirect
8. #4956C2semidirect
9. #4947C2semidirect
10. #4947C2semidirect
11. #4943C2semidirect
12. #4673C2semidirect
13. #4664C2semidirect
14. C3⋊#401C2semidirect
15. C3⋊#401C2semidirect
63. #386D3semidirect
418. C3#2760semidirect

Maximal subgroups

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

subgroupindex
C3×#3862
#52232
#52202
#52132
#52112
#49582
#49562
#49472
#49432
#46732
#46642
C3⋊#4012
#27603