Finite Group Databasebeta

← #10698873

Subgroups — #10698873

Order 768.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
2C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
3#71617C211yesmaximal, characteristic, complemented
4#70432C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic, complemented
6#69730C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#10457—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#833—21no—
15#5019—21no—
16#5016—22no—
17#5015—21no—
18#4966—21no—
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×D4⋊C8—21no—
23C3×D4⋊C8—21no—
24#4530—21no—
25#4486—21no—
26#4485—21no—
27#4459—21no—
28#4459—21no—
29#4457—22no—
30#3036—31no—
31#1800—31no—
32#1566—31no—
33#1547—31no—
34#1492—31no—
35C2×D4⋊C8D311yescharacteristic, complemented
36#1292—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#214—22no—
50#932—42no—
51#932—42no—
52C2×#213—22no—
53#930—21no—
54#930—21no—
55C2×#208—21no—
56C2×#208—21no—
57C2×#208—21no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8D411yescharacteristic
67C3×C4×C8D411yescharacteristic
68#833—42no—
69#810—22no—
70#809—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C2×AGL(1,ℤ/8)—61no—
78#532—31no—
79C2×C4×D4C2×D311yescharacteristic
80#463—62no—
81#461—61no—
82#459—61no—
83#423—31no—
84#423—31no—
85#423—31no—
86C2×C4⋊C8C2×D311yescharacteristic
87#417—61no—
88C2×#100—61no—
89C2×C4×C8C2×D311yescharacteristic
90#343—61no—
91#334—62no—
92#332—61no—
93#331—61no—
94D4⋊C8—21no—
95D4⋊C8—21no—
96D4⋊C8—61no—
97D4⋊C8—61no—
98C24×C3—21no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4SD1612yes—
105C22×C3×C4—21no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4—81no—
109C22×C3×C4—81no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—42no—
126C2×C3×C8—41no—
127C2×C3×C8—81no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#214—44no—
137#213—44no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142#208—41no—
143#208—41no—
144#208—41no—
145#208—41no—
146#208—41no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×#41—61no—
152C22×D4D1211yescharacteristic
153C23×C4—21no—
154AGL(1,ℤ/8)—122no—
155AGL(1,ℤ/8)—122no—
156C2×SD16—62no—
157C2×D8—62no—
158C2×M16—61no—
159C2×M16—61no—
160C2×M16—61no—
161C22×C8—21no—
162C22×C8—21no—
163C4⋊Q8—31no—
164#127—31no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—61no—
173C4×D4—61no—
174C2×C4⋊C4D1211yescharacteristic
175C2×C22⋊C4—21no—
176C2×C42C22×D311yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185#103—62no—
186#102—62no—
187#100—122no—
188C4×C8—21no—
189C4×C8#6711yescharacteristic
190C4×C8#6711yescharacteristic
191C23×C3—22no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—81no—
195C23×C3—81no—
196C23×C3#12311yescharacteristic
197C22×D3—41no—
198C3×D4—22no—
199C3×D4—22no—
200C3×D4—41no—
201C3×D4—41no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—82no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×C3×C4#12411yescharacteristic
216C2×C3×C4C2×SD1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220#67—81no—
221#67—81no—
222C2×Dic3—41no—
223D12—41no—
224D12—41no—
225D12—41no—
226D12—41no—
227D12—81no—
228C4×D3—81no—
229C4×D3—81no—
230Dic6—41no—
231Dic6—41no—
232C3×C8—22no—
233C3×C8—41no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C3⋊C8—81no—
238C24—21no—
239#41—121no—
240#41—121no—
241#41—121no—
242#41—121no—
243C2×Q8—61no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—121no—
253C2×D4#20412yes—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—121no—
260C22×C4C2×D1211yescharacteristic
261C22×C4#23711yescharacteristic
262C22×C4#23711yescharacteristic
263SD16—124no—
264D8—124no—
265M16—121no—
266M16—121no—
267M16—121no—
268M16—121no—
269M16—121no—
270M16—121no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—62no—
276C2×C8—81no—
277C2×C8—121no—
278C2×C8—121no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—82no—
295C22×C3—82no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3#39411yescharacteristic
299C22×C3#48111yescharacteristic
300C22×C3#48211yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—81no—
308C3×C4#44911yescharacteristic
309C3×C4#45011yescharacteristic
310C3×C4#46511yescharacteristic
311C3×C4#46911yescharacteristic
312C3×C4#47111yescharacteristic
313C3×C4#47211yescharacteristic
314Dic3—81no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—81no—
319C23—81no—
320C23—121no—
321C23#88611yescharacteristic
322Q8—121no—
323Q8—121no—
324D4—22no—
325D4—22no—
326D4—41no—
327D4—41no—
328D4—121no—
329D4—121no—
330D4—121no—
331D4—121no—
332D4—241no—
333D4—241no—
334D4—241no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—121no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4#88211yescharacteristic
347C2×C4#88311yescharacteristic
348C2×C4#88511yescharacteristic
349C2×C4C2×#20411yescharacteristic
350C2×C4#90411yescharacteristic
351C2×C4#90811yescharacteristic
352C2×C4#90811yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93811yescharacteristic
355C2×C4#93911yescharacteristic
356C8—22no—
357C8—41no—
358C8—81no—
359C8—122no—
360C8—241no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—82no—
365C2×C3#203411yescharacteristic
366C2×C3#205011yescharacteristic
367C2×C3#205111yescharacteristic
368D3—161no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#473511yescharacteristic
380C22#473711yescharacteristic
381C22#496111yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—241no—
387C4#473111yescharacteristic
388C4#473411yescharacteristic
389C4#511411yescharacteristic
390C4#513511yescharacteristic
391C4#516011yescharacteristic
392C4#517711yescharacteristic
393C3#1045711yescharacteristic, complemented, Sylow
394C2—21no—
395C2—21no—
396C2—82no—
397C2—481no—
398C2#6993511yescharacteristic
399C2#7019611yescharacteristic
400C2#7037911yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×D4⋊C8C2
socleC22×C3#394
prime radical#82307C2
hypercenterC2×D4⋊C8D3
hypocenterC3#10457
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×D4⋊C8C2semidirect
3. #71617C2semidirect
4. #70432C2semidirect
5. #69735C2semidirect
6. #69730C2semidirect
7. #67705C2semidirect
35. C2×D4⋊C8D3semidirect
393. C3#10457semidirect

Maximal subgroups

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

subgroupindex
#823072
C2×C3×D4⋊C82
#716172
#704322
#697352
#697302
#677052
#104573