Finite Group Databasebeta

← #76751

Subgroups — #76751

Order 384.

450 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#392C211yesmaximal, characteristic
2#5222C211yesmaximal, characteristic, complemented
3#5222C211yesmaximal, characteristic, complemented
4#5222C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5221C211yesmaximal, characteristic, complemented
8#4952C211yesmaximal, characteristic, complemented
9#4952C211yesmaximal, characteristic, complemented
10#4951C211yesmaximal, characteristic, complemented
11#4949C211yesmaximal, characteristic, complemented
12#4948C211yesmaximal, characteristic, complemented
13#4948C211yesmaximal, characteristic, complemented
14#4657C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2829—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#399—31no—
53#398—31no—
54#397—31no—
55#397—31no—
56#396—31no—
57#396—31no—
58#394—31no—
59#394—31no—
60#394—31no—
61#394—31no—
62#394—31no—
63#392D311yescharacteristic
64#390—31no—
65#388—31no—
66#387—31no—
67C24×C3C2311yescharacteristic
68C24×C3C2311yescharacteristic
69C24×C3C2311yescharacteristic
70C2×C3×D4—22no—
71C2×C3×D4—41no—
72C2×C3×D4—41no—
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C3×C22⋊C4—22no—
92C3×C22⋊C4—41no—
93C3×C22⋊C4—41no—
94#217—41no—
95#217—41no—
96#217—41no—
97#217—41no—
98#217—41no—
99#217—41no—
100#217—41no—
101#217—41no—
102#217—41no—
103#217—41no—
104#217—41no—
105#217—41no—
106C4⋊Dic3—41no—
107C4⋊Dic3—41no—
108Dic3⋊C4—22no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113Dic3⋊C4—41no—
114C4×Dic3—41no—
115C22×D4C2×D311yescharacteristic
116C22×D4C2×D311yescharacteristic
117C22×D4C2×D311yescharacteristic
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊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⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4C2×D311yescharacteristic
139C2×C22⋊C4C2×D311yescharacteristic
140C2×C22⋊C4C2×D311yescharacteristic
141C2×C42—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
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—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3C2411yescharacteristic
172C3×D4—44no—
173C3×D4—44no—
174C3×D4—44no—
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#4112yes—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
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—
211C24C22×D311yescharacteristic
212C24C22×D311yescharacteristic
213C24C22×D311yescharacteristic
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235C4⋊C4—62no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—22no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C42—121no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
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×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3C2×#4111yescharacteristic
281C22×C3H(5,𝔽2)11yescharacteristic
282C22×C3H(5,𝔽2)11yescharacteristic
283C22×C3H(5,𝔽2)11yescharacteristic
284C22×C3H(5,𝔽2)11yescharacteristic
285C22×C3H(5,𝔽2)11yescharacteristic
286C22×C32−1+411yescharacteristic
287C3×C4—22no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299C23—21no—
300C23—21no—
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—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C23×D311yescharacteristic
321D4—44no—
322D4—44no—
323D4—44no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
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—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
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#23712yes—
360C2×C3—81no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3#53811yescharacteristic
364C2×C3#54011yescharacteristic
365C2×C3#54211yescharacteristic
366C2×C3#56011yescharacteristic
367C2×C3#56011yescharacteristic
368C2×C3#56211yescharacteristic
369C2×C3#56211yescharacteristic
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22C2×#23711yescharacteristic
392C22#100411yescharacteristic
393C22#100411yescharacteristic
394C22#100411yescharacteristic
395C22#100411yescharacteristic
396C22#100411yescharacteristic
397C22#101011yescharacteristic
398C4—22no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—121no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4—121no—
407C4—121no—
408C4—121no—
409C4—121no—
410C3#282911yescharacteristic, complemented, Sylow
411C2—81no—
412C2—81no—
413C2—81no—
414C2#559311yescharacteristic
415C2#559511yescharacteristic
416C2#559511yescharacteristic
417C2#560311yescharacteristic
418C2#562111yescharacteristic
419C2#562111yescharacteristic
420C2#571911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2829
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
2. #5222C2semidirect
3. #5222C2semidirect
4. #5222C2semidirect
5. #5222C2semidirect
6. #5222C2semidirect
7. #5221C2semidirect
8. #4952C2semidirect
9. #4952C2semidirect
10. #4951C2semidirect
11. #4949C2semidirect
12. #4948C2semidirect
13. #4948C2semidirect
14. #4657C2semidirect
15. #4652C2semidirect
410. C3#2829semidirect

Maximal subgroups

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

subgroupindex
C3×#3922
#52222
#52212
#49522
#49512
#49492
#49482
#46572
#46522
#28293