Finite Group Databasebeta

← #69599

Subgroups — #69599

Order 384.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5804C211yesmaximal, characteristic
2C3×#386C211yesmaximal, characteristic, complemented
3C2×#834C211yesmaximal, characteristic, complemented
4#5213C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic, complemented
6C2×#806C211yesmaximal, characteristic, complemented
7#4527C211yesmaximal, characteristic, complemented
8#2044—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C23×C3×C4C2211yescharacteristic
11C22×#67C2211yescharacteristic
12C2×C3×C22⋊C4—21no—
13C2×C3×C22⋊C4C2211yescharacteristic
14#953—21no—
15C2×#217C2211yescharacteristic
16C2×#217C2211yescharacteristic
17C2×#217C2211yescharacteristic
18#939—21no—
19#939—21no—
20#937—21no—
21#935—21no—
22#929—21no—
23C2×(C2×D3)⋊C4—21no—
24C2×Dic3⋊C4—21no—
25C3×#95—21no—
26#834—22no—
27C3⋊#95—21no—
28C3⋊#95—21no—
29#806—22no—
30#534—31no—
31C2×#99—31no—
32C2×#99—31no—
33#386—31no—
34#386D311yescharacteristic, complemented
35#385—31no—
36#342—31no—
37C24×C3D411yescharacteristic
38C24×C3C2311yescharacteristic
39C23×D3D411yescharacteristic
40C2×C3×D4—21no—
41C2×C3×D4—21no—
42C2×C3×D4—21no—
43C2×C3×D4—21no—
44C2×C3×D4—21no—
45C2×C3×D4—21no—
46C22×C3×C4—21no—
47C22×C3×C4—21no—
48C22×C3×C4—21no—
49C22×C3×C4—21no—
50C22×C3×C4—42no—
51C22×C3×C4D411yescharacteristic
52C2×#67—21no—
53C2×#67—21no—
54C2×#67—21no—
55C2×#67—21no—
56C2×#67—21no—
57C2×#67—21no—
58C2×#67—21no—
59C2×#67—21no—
60C22×Dic3—21no—
61C22×Dic3—21no—
62C22×Dic3D411yescharacteristic
63C22×Dic3D411yescharacteristic
64C22×Dic3D411yescharacteristic
65C3×C22⋊C4—21no—
66C3×C22⋊C4—21no—
67C3×C22⋊C4—42no—
68#217—21no—
69#217—21no—
70#217—21no—
71#217—21no—
72#217—21no—
73#217—21no—
74#217—21no—
75#217D412yescomplemented
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—42no—
79Dic3⋊C4—21no—
80Dic3⋊C4—21no—
81Dic3⋊C4—42no—
82C22×D4—31no—
83C22×D4C2×D311yescharacteristic
84C23×C4C2×D311yescharacteristic
85#123—61no—
86#123—61no—
87#121—61no—
88#121—61no—
89#120—61no—
90#120—61no—
91C2×C4⋊C4—61no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—61no—
97C2×C22⋊C4C2×D311yescharacteristic
98#99—62no—
99#99—62no—
100#95—21no—
101#95—61no—
102#95—61no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3—21no—
106C23×C3—41no—
107C23×C3—41no—
108C23×C3—41no—
109C23×C3—41no—
110C23×C3C2×D411yescharacteristic
111C23×C3C2×D411yescharacteristic
112C23×C3C2×D411yescharacteristic
113C23×C3C2×D411yescharacteristic
114C23×C3#4111yescharacteristic
115C23×C3#4111yescharacteristic
116C23×C3#4111yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—41no—
121C22×D3—41no—
122C3×D4—41no—
123C3×D4—41no—
124C3×D4—41no—
125C3×D4—41no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139#67—41no—
140#67—41no—
141#67—41no—
142#67—41no—
143#67—81no—
144#67—81no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—41no—
152C2×Dic3—41no—
153C2×Dic3—41no—
154C2×Dic3—41no—
155C2×Dic3—41no—
156C2×Dic3—41no—
157C2×Dic3—41no—
158C24—31no—
159C24#6711yescharacteristic
160C24C22×D311yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—61no—
168C2×D4—61no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—61no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—42no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4#6711yescharacteristic
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—122no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—32no—
192C22⋊C4—42no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C22⋊C4—61no—
196C22⋊C4—61no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C22⋊C4—122no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—42no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—81no—
218C22×C3#12011yescharacteristic
219C22×C3#12111yescharacteristic
220C22×C3#12111yescharacteristic
221C22×C3#12111yescharacteristic
222C22×C3#12311yescharacteristic
223C22×C3#12311yescharacteristic
224C22×C3#12411yescharacteristic
225C2×D3—42no—
226C2×D3—41no—
227C2×D3—41no—
228C2×D3—41no—
229C2×D3—81no—
230C3×C4—41no—
231C3×C4—42no—
232C3×C4—81no—
233Dic3—42no—
234Dic3—41no—
235Dic3—41no—
236Dic3—41no—
237Dic3—81no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—61no—
246C23—61no—
247C23—61no—
248C23—121no—
249C23—121no—
250C23#23511yescharacteristic
251C23#23511yescharacteristic
252C23D3×D411yescharacteristic
253C23D3×D411yescharacteristic
254C23D3×D411yescharacteristic
255C23#23711yescharacteristic
256C23C2×#6711yescharacteristic
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—121no—
262D4—121no—
263D4—121no—
264D4—121no—
265D4—241no—
266D4—241no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—61no—
281C2×C4—62no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—81no—
300C2×C3#39411yescharacteristic
301C2×C3B(4,𝔽2)11yescharacteristic
302C2×C3#45811yescharacteristic
303D3—81no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22—122no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22—241no—
324C22#88311yescharacteristic
325C22#88411yescharacteristic
326C22#88411yescharacteristic
327C22#88511yescharacteristic
328C22#92911yescharacteristic
329C22#93711yescharacteristic
330C22#93911yescharacteristic
331C4—41no—
332C4—42no—
333C4—81no—
334C4—122no—
335C4—121no—
336C4—121no—
337C4—121no—
338C4—241no—
339C3#204411yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—241no—
348C2#495911yescharacteristic
349C2#503211yescharacteristic
350C2#505311yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC24×C3C23
FrattiniC24C22×D3
FittingC3×#386C2
socleC22×C3#120
prime radical#5804C2
hypercenter#386D3
hypocenterC3#2044
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C3×#386C2semidirect
3. C2×#834C2semidirect
4. #5213C2semidirect
5. #5210C2semidirect
6. C2×#806C2semidirect
7. #4527C2semidirect
34. #386D3semidirect
75. #217D4semidirect
339. C3#2044semidirect

Maximal subgroups

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

subgroupindex
#58042
C3×#3862
C2×#8342
#52132
#52102
C2×#8062
#45272
#20443