Finite Group Databasebeta

← #10698943

Subgroups — #10698943

Order 768.

420 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 95 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#82441C211yesmaximal, characteristic
2C3×#1461C211yesmaximal, characteristic, complemented
3#71627C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic
5#70772C211yesmaximal, characteristic
6#70749C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10467—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×#835—21no—
15#5031—21no—
16#5028—22no—
17#5027—21no—
18#4966—21no—
19#4936C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#324—22no—
23#4529—21no—
24#4486—21no—
25#4485—21no—
26#4470—22no—
27#4457—22no—
28#3037—31no—
29#1805—31no—
30#1570—31no—
31#1540—31no—
32#1517—31no—
33#1461D311yescharacteristic, complemented
34#1286—31no—
35C22×C3×Q8D411yescharacteristic
36C2×#235—21no—
37C3×C4×Q8—21no—
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#216—22no—
46#942—42no—
47#942—42no—
48C2×#215—22no—
49#930—21no—
50#930—21no—
51C2×C4⋊Dic3—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8D412yes—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#835—42no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#810—22no—
69#809—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C2×#137—61no—
75#532—31no—
76C2×C4×Q8C2×D311yescharacteristic
77#464—62no—
78#462—61no—
79#460—61no—
80#431C2×D311yescharacteristic
81#431C2×D311yescharacteristic
82#423—31no—
83C2×C4⋊C8—31no—
84#417—61no—
85C2×#103—61no—
86#382—31no—
87#340—61no—
88#334—62no—
89#332—61no—
90#331—61no—
91#327—62no—
92#324—22no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8SD1612yes—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—41no—
105C22×Dic3—41no—
106#235—41no—
107#235—41no—
108#235—41no—
109#235—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×C4×D3—41no—
114C2×Dic6—21no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127#216—44no—
128#215—44no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×#41—61no—
142C22×Q8D1211yescharacteristic
143#137—122no—
144#137—122no—
145C2×Q16—62no—
146C2×SD16—62no—
147C2×M16—61no—
148C22×C8—61no—
149C4⋊Q8—31no—
150#127—31no—
151#124—61no—
152#121—61no—
153#121—61no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×D4—61no—
159C4×D4—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4D1211yescharacteristic
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—32no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#103—62no—
172#103—122no—
173#102—62no—
174C22⋊C8—21no—
175C22⋊C8#6712yes—
176C4×C8—21no—
177C4×C8—21no—
178#95—61no—
179#95—61no—
180C23×C3#12111yescharacteristic
181C22×D3—41no—
182C3×Q8—22no—
183C3×Q8—22no—
184C3×Q8—41no—
185C3×Q8—41no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4#12311yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×C3×C4C2×SD1611yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
203#67—81no—
204#67—81no—
205C2×Dic3—41no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209D12—41no—
210D12—41no—
211D12—81no—
212D12—81no—
213C4×D3—81no—
214C4×D3—81no—
215Dic6—41no—
216Dic6—41no—
217C3×C8—42no—
218C3×C8—81no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221#41—121no—
222#41—121no—
223#41—121no—
224#41—121no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—61no—
231C2×Q8#20412yes—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—121no—
239C22×C4—121no—
240C22×C4C2×D1211yescharacteristic
241C22×C4#23511yescharacteristic
242C22×C4C2×#6711yescharacteristic
243Q16—124no—
244SD16—124no—
245M16—121no—
246M16—121no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—62no—
250C2×C8—121no—
251C2×C8—121no—
252C2×C8—121no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C42—41no—
264C42—41no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42#23711yescharacteristic
268C42#23711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3#39411yescharacteristic
272C22×C3#46511yescharacteristic
273C22×C3#46911yescharacteristic
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—81no—
283C3×C4#44911yescharacteristic
284C3×C4#45011yescharacteristic
285C3×C4#48111yescharacteristic
286C3×C4#48211yescharacteristic
287Dic3—81no—
288Dic3—161no—
289C23—121no—
290C23#93011yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295Q8—121no—
296Q8—121no—
297D4—121no—
298D4—121no—
299D4—241no—
300D4—241no—
301D4—241no—
302D4—241no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—121no—
311C2×C4—241no—
312C2×C4—241no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#88211yescharacteristic
317C2×C4#88311yescharacteristic
318C2×C4#88511yescharacteristic
319C2×C4#88611yescharacteristic
320C2×C4C2×#20411yescharacteristic
321C2×C4#90411yescharacteristic
322C2×C4#90811yescharacteristic
323C2×C4#90811yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C8—42no—
327C8—81no—
328C8—122no—
329C8—241no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#203411yescharacteristic
333C2×C3#204911yescharacteristic
334C2×C3#205311yescharacteristic
335D3—161no—
336C22—21no—
337C22—21no—
338C22—241no—
339C22—241no—
340C22—241no—
341C22#496111yescharacteristic
342C22#511411yescharacteristic
343C22#513511yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—81no—
350C4—241no—
351C4—481no—
352C4#473111yescharacteristic
353C4#473411yescharacteristic
354C4#473511yescharacteristic
355C4#473711yescharacteristic
356C3#1046711yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—481no—
360C2#6993511yescharacteristic
361C2#7004311yescharacteristic
362C2#7011011yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1461C2
socleC22×C3#394
prime radical#5477C22
hypercenter#1461D3
hypocenterC3#10467
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C3×#1461C2semidirect
3. #71627C2semidirect
6. #70749C2semidirect
7. #67738C2semidirect
33. #1461D3semidirect
356. C3#10467semidirect

Maximal subgroups

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

subgroupindex
#824412
C3×#14612
#716272
#707902
#707722
#707492
#677382
#104673