Finite Group Databasebeta

← #74933

Subgroups — #74933

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#386C211yesmaximal, characteristic, complemented
2#5223C211yesmaximal, characteristic
3#5222C211yesmaximal, characteristic, complemented
4#5213C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic, complemented
6#4961C211yesmaximal, characteristic, complemented
7#4959C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4953C211yesmaximal, characteristic, complemented
10#4947C211yesmaximal, characteristic, complemented
11#4947C211yesmaximal, characteristic, complemented
12#4675C211yesmaximal, characteristic, complemented
13#4673C211yesmaximal, characteristic, complemented
14#4660C211yesmaximal, characteristic, complemented
15#4658C211yesmaximal, characteristic, complemented
16#2824—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
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#400—31no—
53#400—31no—
54#397—31no—
55#397—31no—
56#394—31no—
57#394—31no—
58#394—31no—
59#394—31no—
60#393—31no—
61#393—31no—
62#393—31no—
63#392—31no—
64#386—31no—
65#386D311yescharacteristic, complemented
66#385—31no—
67C24×C3C2311yescharacteristic
68C24×C3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C2×C3×D4—22no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C22×C3×C4—41no—
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—41no—
83C2×#67—41no—
84C2×#67—41no—
85C2×#67—41no—
86C2×#67—41no—
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C2×Dic6—41no—
95C2×Dic6—41no—
96C2×Dic6—41no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100#217—22no—
101#217—41no—
102#217—41no—
103#217—41no—
104#217—41no—
105#217—41no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111C4⋊Dic3—41no—
112Dic3⋊C4—41no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115C22×Q8—31no—
116C22×D4—31no—
117C22×D4—31no—
118C22×D4C2×D311yescharacteristic
119C23×C4C2×D311yescharacteristic
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊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⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4C2×D311yescharacteristic
136C2×C22⋊C4C2×D311yescharacteristic
137C2×C22⋊C4C2×D311yescharacteristic
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—21no—
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—21no—
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—
212Dic6—44no—
213C24—31no—
214C24C22×D311yescharacteristic
215C24C22×D311yescharacteristic
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—62no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D311yescharacteristic
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—62no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C2×#4111yescharacteristic
276C22×C3H(5,𝔽2)11yescharacteristic
277C22×C3H(5,𝔽2)11yescharacteristic
278C22×C3H(5,𝔽2)11yescharacteristic
279C22×C3H(5,𝔽2)11yescharacteristic
280C22×C3H(5,𝔽2)11yescharacteristic
281C22×C32−1+411yescharacteristic
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3—41no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—61no—
315C23—61no—
316C23—61no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23#23512yes—
322C23C23×D311yescharacteristic
323Q8—124no—
324D4—44no—
325D4—124no—
326D4—124no—
327C2×C4—21no—
328C2×C4—21no—
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—44no—
343C2×C4—61no—
344C2×C4—61no—
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×C3—42no—
365C2×C3—81no—
366C2×C3#53811yescharacteristic
367C2×C3#53811yescharacteristic
368C2×C3#53911yescharacteristic
369C2×C3#56011yescharacteristic
370C2×C3H(3,𝔽4)11yescharacteristic
371C2×C3#56211yescharacteristic
372C2×C3#56211yescharacteristic
373D3—81no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22C2×#23511yescharacteristic
396C22#100411yescharacteristic
397C22#100411yescharacteristic
398C22#100411yescharacteristic
399C22#100411yescharacteristic
400C22#100911yescharacteristic
401C22#101011yescharacteristic
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—121no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4—121no—
412C4—121no—
413C4—121no—
414C3#282411yescharacteristic, complemented, Sylow
415C2—42no—
416C2—81no—
417C2—241no—
418C2#549411yescharacteristic
419C2#556011yescharacteristic
420C2#556411yescharacteristic
421C2#559411yescharacteristic
422C2#562011yescharacteristic
423C2#562111yescharacteristic
424C2#566711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radical#5223C2
hypercenter#386D3
hypocenterC3#2824
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. #5222C2semidirect
4. #5213C2semidirect
5. #5210C2semidirect
6. #4961C2semidirect
7. #4959C2semidirect
8. #4959C2semidirect
9. #4953C2semidirect
10. #4947C2semidirect
11. #4947C2semidirect
12. #4675C2semidirect
13. #4673C2semidirect
14. #4660C2semidirect
15. #4658C2semidirect
65. #386D3semidirect
414. C3#2824semidirect

Maximal subgroups

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

subgroupindex
C3×#3862
#52232
#52222
#52132
#52102
#49612
#49592
#49532
#49472
#46752
#46732
#46602
#46582
#28243