Finite Group Databasebeta

← #10714448

Subgroups — #10714448

Order 768.

422 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 91 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#83742C211yesmaximal, characteristic
2C3×#2112C211yesmaximal, characteristic, complemented
3#71977C211yesmaximal, characteristic, complemented
4#71448C211yesmaximal, characteristic, complemented
5#71442C211yesmaximal, characteristic, complemented
6#71427C211yesmaximal, characteristic, complemented
7#67822C211yesmaximal, characteristic, complemented
8#13395—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10#5791—21no—
11C2×#904C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14#5207—21no—
15#5116—21no—
16#5115—21no—
17#5114—22no—
18#5113—21no—
19C2×#819C2211yescharacteristic
20C2×#819C2211yescharacteristic
21#5106C2211yescharacteristic
22C3×C22⋊C16—22no—
23#4552—21no—
24#4517—21no—
25#4516—21no—
26#4513—22no—
27#4510—22no—
28#3050—31no—
29#2155—31no—
30#2148—31no—
31#2143—31no—
32#2112D311yescharacteristic, complemented
33#1810—31no—
34#1389—31no—
35C23×C3×C4D411yescharacteristic
36C2×#235—21no—
37C22×C3×C8—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8D411yescharacteristic
44C22×C3×C8C2311yescharacteristic
45#953—41no—
46#930—22no—
47#929—41no—
48C4×#67—42no—
49#926—21no—
50#924—22no—
51C2×#208—21no—
52C2×#208—21no—
53C2×Dic12D411yescharacteristic
54#904—21no—
55#904—21no—
56#904—21no—
57#904—21no—
58C2×D24D411yescharacteristic
59C2×#204—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32—21no—
64C2×C3×C16—21no—
65C2×C3×C16D412yescomplemented
66#830—42no—
67#821—22no—
68#819—21no—
69#819—21no—
70#819—21no—
71#819—21no—
72#818—21no—
73#817D412yes—
74#816—22no—
75C2×#135—31no—
76C23×C8C2×D311yescharacteristic
77#525—61no—
78C2×M32C2×D311yescharacteristic
79C2×M32C2×D311yescharacteristic
80#467—61no—
81#466—61no—
82#465—62no—
83C2×#108—31no—
84C2×#108—31no—
85#427—31no—
86#416—61no—
87#407—61no—
88#365—62no—
89#362—61no—
90#361—61no—
91#359—62no—
92C22⋊C16—22no—
93#341—61no—
94C24×C3SD1611yescharacteristic
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4SD1611yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C2×#67—42no—
104C2×#67—41no—
105#235—41no—
106#235—41no—
107#235—41no—
108#235—41no—
109C2×D12—21no—
110C2×C4×D3—41no—
111C2×Dic6—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8C2×D411yescharacteristic
120C2×C3×C8C2×D411yescharacteristic
121C2×C3×C8C2×D411yescharacteristic
122C2×C3×C8#4111yescharacteristic
123C2×C3×C8#4111yescharacteristic
124C2×C3×C8#4111yescharacteristic
125#217—42no—
126#217—41no—
127(C2×D3)⋊C4—42no—
128C4⋊Dic3—22no—
129Dic3⋊C4—42no—
130C4×Dic3—41no—
131#208—41no—
132#208—41no—
133#208—41no—
134#208—41no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137Dic12—21no—
138Dic12—21no—
139D24—21no—
140D24—21no—
141#204—41no—
142#204—41no—
143C3×C16—22no—
144C3×C16—41no—
145C2×#41—61no—
146C23×C4D1211yescharacteristic
147#135—61no—
148#135—61no—
149#135—61no—
150#135—61no—
151C2×Q16—31no—
152C2×SD16—61no—
153C2×D8—31no—
154C2×M16—61no—
155C2×M16—61no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8D1211yescharacteristic
163C22×C8C22×D311yescharacteristic
164#123—121no—
165C22⋊Q8—62no—
166#121—62no—
167#120—121no—
168C4×D4—122no—
169#117—61no—
170M32—21no—
171M32—21no—
172M32—21no—
173M32—21no—
174C2×C16—21no—
175C2×C16#6712yescomplemented
176#108—61no—
177#108—61no—
178#108—61no—
179#108—61no—
180#107—61no—
181#106—32no—
182#103—62no—
183#102—62no—
184C22⋊C8—122no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—42no—
189C23×C3C2×SD1611yescharacteristic
190C22×D3—41no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4#13511yescharacteristic
202C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204#67—84no—
205#67—81no—
206#67—81no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209D12—41no—
210D12—41no—
211C4×D3—81no—
212C4×D3—81no—
213Dic6—41no—
214Dic6—41no—
215C3×C8—41no—
216C3×C8#12111yescharacteristic
217C3×C8#12111yescharacteristic
218C3×C8#12311yescharacteristic
219C3×C8#12411yescharacteristic
220C3⋊C8—81no—
221C3⋊C8—81no—
222C24#20411yescharacteristic
223#41—121no—
224#41—121no—
225#41—121no—
226#41—121no—
227C2×Q8—61no—
228C2×D4—61no—
229C2×D4—122no—
230C2×D4—121no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—121no—
238C22×C4#20411yescharacteristic
239C22×C4C2×D1211yescharacteristic
240Q16—61no—
241Q16—61no—
242SD16—121no—
243SD16—121no—
244D8—61no—
245D8—61no—
246M16—121no—
247M16—121no—
248M16—121no—
249M16—121no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—121no—
258C2×C8—121no—
259C2×C8#23511yescharacteristic
260C2×C8D3×D411yescharacteristic
261C2×C8D3×D411yescharacteristic
262C2×C8#23711yescharacteristic
263C2×C8#23711yescharacteristic
264C2×C8C2×#6711yescharacteristic
265C4⋊C4—62no—
266C4⋊C4—122no—
267C22⋊C4—122no—
268C22⋊C4—122no—
269C22⋊C4—121no—
270C42—121no—
271C16—22no—
272C16—41no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3#45011yescharacteristic
282C22×C3#46511yescharacteristic
283C22×C3#48111yescharacteristic
284C2×D3—82no—
285C2×D3—81no—
286C3×C4—41no—
287C3×C4#39411yescharacteristic
288C3×C4#44911yescharacteristic
289C3×C4#46911yescharacteristic
290C3×C4#48211yescharacteristic
291Dic3—82no—
292Dic3—81no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—42no—
297C23—121no—
298C23C2×#20411yescharacteristic
299Q8—121no—
300Q8—121no—
301D4—121no—
302D4—121no—
303D4—244no—
304D4—241no—
305D4—241no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—122no—
314C2×C4—121no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4#88211yescharacteristic
318C2×C4#88611yescharacteristic
319C2×C4#90411yescharacteristic
320C2×C4#90811yescharacteristic
321C2×C4#90811yescharacteristic
322C2×C4#93011yescharacteristic
323C8—41no—
324C8—241no—
325C8—241no—
326C8#88311yescharacteristic
327C8#88511yescharacteristic
328C8#93811yescharacteristic
329C8#93911yescharacteristic
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—42no—
334C2×C3#203411yescharacteristic
335D3—161no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—242no—
345C22—241no—
346C22#473111yescharacteristic
347C22#473511yescharacteristic
348C22#511411yescharacteristic
349C4—41no—
350C4—242no—
351C4—241no—
352C4#473411yescharacteristic
353C4#473711yescharacteristic
354C4#496111yescharacteristic
355C4#513511yescharacteristic
356C3#1339511yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—481no—
362C2#6993511yescharacteristic
subgroupG/H
centerC4#5135
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#2112C2
socleC2×C3#2034
prime radical#83742C2
hypercenter#2112D3
hypocenterC3#13395
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C3×#2112C2semidirect
3. #71977C2semidirect
4. #71448C2semidirect
5. #71442C2semidirect
6. #71427C2semidirect
7. #67822C2semidirect
32. #2112D3semidirect
65. C2×C3×C16D4semidirect
175. C2×C16#67semidirect
356. C3#13395semidirect

Maximal subgroups

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

subgroupindex
#837422
C3×#21122
#719772
#714482
#714422
#714272
#678222
#133953