Finite Group Databasebeta

← #10698921

Subgroups — #10698921

Order 768.

452 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 93 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#82307C211yesmaximal, characteristic
2C3×#1804C211yesmaximal, characteristic, complemented
3#71601C211yesmaximal, characteristic, complemented
4#70431C211yesmaximal, characteristic, complemented
5#70422C211yesmaximal, characteristic, complemented
6#69545C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#10464—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#102—21no—
14C3×#382C2211yescharacteristic
15#5019—21no—
16#5016—22no—
17#5015—21no—
18#4966—21no—
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×#340—21no—
23#4525—22no—
24#4485—21no—
25#4482—21no—
26#4479—22no—
27#4451—22no—
28#3036—31no—
29#1804D311yescharacteristic, complemented
30#1571—31no—
31#1540—31no—
32#1517—31no—
33#1460—31no—
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C2×#235—21no—
38C22×C3×C8—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#214—22no—
48#932—42no—
49#932—42no—
50C2×#213—22no—
51#930—21no—
52#930—21no—
53C2×#208—21no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#102—42no—
62C3×#95—21no—
63C3×#95—21no—
64#830—21no—
65#830D412yes—
66#810—22no—
67#809—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C2×AGL(1,ℤ/8)—61no—
75#532—31no—
76C2×C4×D4C2×D311yescharacteristic
77#463—62no—
78#461—61no—
79#459—61no—
80#431—31no—
81#431—31no—
82#423—31no—
83C2×C4⋊C8C2×D311yescharacteristic
84#417—61no—
85C2×#102—21no—
86#382C2×D311yescharacteristic
87#340—21no—
88#335—62no—
89#332—61no—
90#331—61no—
91#327—62no—
92#323—62no—
93C24×C3—21no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4D812yes—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4—82no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×#67—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112#235—41no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#214—44no—
132#213—44no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137#208—41no—
138#208—41no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C2×#41—61no—
143C22×D4#6711yescharacteristic
144C23×C4—21no—
145AGL(1,ℤ/8)—122no—
146AGL(1,ℤ/8)—122no—
147C2×SD16—62no—
148C2×D8—62no—
149C2×M16—61no—
150C22×C8—21no—
151C4⋊Q8—31no—
152#127—31no—
153#124—61no—
154#121—61no—
155#121—61no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4#6711yescharacteristic
164C2×C22⋊C4—21no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—32no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—62no—
173#102—42no—
174#102—62no—
175C22⋊C8—32no—
176C22⋊C8—61no—
177C4×C8—61no—
178C4×C8—61no—
179#95—21no—
180#95—21no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3#12111yescharacteristic
187C22×D3—41no—
188C3×D4—22no—
189C3×D4—22no—
190C3×D4—41no—
191C3×D4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—82no—
196C2×C3×C4—82no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4#12411yescharacteristic
207C2×C3×C4C2×D811yescharacteristic
208C2×C3×C4#13511yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C3×C4#13711yescharacteristic
211#67—81no—
212#67—81no—
213C2×Dic3—41no—
214D12—41no—
215D12—41no—
216D12—81no—
217D12—81no—
218C4×D3—81no—
219C4×D3—81no—
220Dic6—41no—
221Dic6—41no—
222C3×C8—81no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225C3⋊C8—81no—
226C24—21no—
227#41—121no—
228#41—121no—
229#41—121no—
230#41—121no—
231C2×Q8—61no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—121no—
241C2×D4#21312yes—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—82no—
247C22×C4—121no—
248C22×C4D3×D411yescharacteristic
249C22×C4#23711yescharacteristic
250C22×C4C2×#6711yescharacteristic
251SD16—124no—
252D8—124no—
253M16—121no—
254M16—121no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C42#23511yescharacteristic
272C42#23511yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—42no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3#39411yescharacteristic
283C22×C3#46611yescharacteristic
284C22×C3#46811yescharacteristic
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—161no—
294C3×C4#44711yescharacteristic
295C3×C4#45211yescharacteristic
296C3×C4#48011yescharacteristic
297C3×C4#48211yescharacteristic
298Dic3—81no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—81no—
303C23—81no—
304C23—121no—
305C23#93911yescharacteristic
306Q8—121no—
307Q8—121no—
308D4—22no—
309D4—22no—
310D4—41no—
311D4—41no—
312D4—121no—
313D4—121no—
314D4—241no—
315D4—241no—
316D4—241no—
317D4—241no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4#88311yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#88511yescharacteristic
334C2×C4#92911yescharacteristic
335C2×C4C2×#21311yescharacteristic
336C2×C4#93711yescharacteristic
337C2×C4#94211yescharacteristic
338C2×C4#94911yescharacteristic
339C2×C4#95011yescharacteristic
340C8—81no—
341C8—122no—
342C8—122no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—82no—
347C2×C3#203311yescharacteristic
348C2×C3#204811yescharacteristic
349C2×C3#205311yescharacteristic
350D3—161no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—82no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22#495911yescharacteristic
362C22#503811yescharacteristic
363C22#504911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—161no—
370C4—241no—
371C4#496711yescharacteristic
372C4#500311yescharacteristic
373C4#503611yescharacteristic
374C4#504711yescharacteristic
375C3#1046411yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2—82no—
379C2—481no—
380C2#6952311yescharacteristic
381C2#7029611yescharacteristic
382C2#7033211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1804C2
socleC22×C3#394
prime radical#82307C2
hypercenter#1804D3
hypocenterC3#10464
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C3×#1804C2semidirect
3. #71601C2semidirect
4. #70431C2semidirect
5. #70422C2semidirect
6. #69545C2semidirect
7. #67631C2semidirect
29. #1804D3semidirect
375. C3#10464semidirect

Maximal subgroups

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

subgroupindex
#823072
C3×#18042
#716012
#704312
#704222
#695452
#676312
#104643