Finite Group Databasebeta

← #10697227

Subgroups — #10697227

Order 768.

550 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×C3×#474C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic, complemented
6#70426C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10188—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#474—22no—
14C3×#474—22no—
15C2×C3×#107—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#102—21no—
18#5088—21no—
19#5084—22no—
20#5080—21no—
21#5019—21no—
22#5016—22no—
23#5015—21no—
24#4968—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4525—22no—
31#4483—22no—
32#3227—31no—
33C2×#474D311yescharacteristic, complemented
34#3036—31no—
35#1548—31no—
36#1480—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×#235—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#214—22no—
58#932—42no—
59#932—42no—
60C2×#213—22no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×#107—24no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×#102—24no—
73#830—21no—
74#830D412yes—
75#810—22no—
76#809—22no—
77#809—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×C4×D4C2×D311yescharacteristic
89#498—61no—
90#486—61no—
91#481—62no—
92#476—62no—
93#474—22no—
94#474—22no—
95#463—62no—
96#461—61no—
97#459—61no—
98#431—31no—
99#431—31no—
100C2×#107—21no—
101#423—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103#417—61no—
104C2×#102—21no—
105#333—62no—
106D4⋊C8—62no—
107#323—62no—
108C24×C3—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4SD1612yes—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—82no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130#235—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#214—44no—
152#213—44no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×#41—61no—
163C22×Q8—21no—
164C22×D4#6711yescharacteristic
165C23×C4—21no—
166AGL(1,ℤ/8)—122no—
167AGL(1,ℤ/8)—122no—
168C2×SD16—62no—
169C2×D8—62no—
170C2×M16—61no—
171C22×C8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—31no—
176C4⋊Q8#6712yes—
177#127—31no—
178#124—61no—
179#121—61no—
180#121—61no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—61no—
186C4×D4—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C42C22×D311yescharacteristic
192#107—24no—
193#106—64no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8#6712yes—
200#103—62no—
201#102—24no—
202#102—62no—
203#102—64no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C4×C8—61no—
207C4×C8—61no—
208C23×C3—22no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3#12111yescharacteristic
214C22×D3—41no—
215C3×Q8—82no—
216C3×Q8—82no—
217C3×D4—22no—
218C3×D4—22no—
219C3×D4—41no—
220C3×D4—41no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
227C2×C3×C4—82no—
228C2×C3×C4—81no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12711yescharacteristic
235C2×C3×C4C2×SD1611yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×C3×C4#13711yescharacteristic
239#67—81no—
240#67—81no—
241C2×Dic3—41no—
242D12—42no—
243D12—41no—
244D12—41no—
245D12—42no—
246C4×D3—81no—
247C4×D3—81no—
248Dic6—41no—
249Dic6—41no—
250C3×C8—42no—
251C3⋊C8—42no—
252C3⋊C8—42no—
253C3⋊C8—81no—
254C24—21no—
255#41—121no—
256#41—121no—
257#41—121no—
258#41—121no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—61no—
262C2×Q8—82no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4#21512yes—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—121no—
279C22×C4D3×D411yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282SD16—124no—
283D8—124no—
284M16—121no—
285M16—121no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—121no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C42C2×#6711yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3#39211yescharacteristic
316C22×C3#46511yescharacteristic
317C22×C3#46811yescharacteristic
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—81no—
325C3×C4—82no—
326C3×C4#44811yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#45011yescharacteristic
329C3×C4#45211yescharacteristic
330C3×C4#47212yes—
331Dic3—81no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—81no—
336C23—81no—
337C23—121no—
338C23#93911yescharacteristic
339Q8—82no—
340Q8—82no—
341Q8—121no—
342Q8—121no—
343D4—22no—
344D4—22no—
345D4—41no—
346D4—41no—
347D4—122no—
348D4—122no—
349D4—121no—
350D4—121no—
351D4—241no—
352D4—241no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—121no—
362C2×C4—241no—
363C2×C4—241no—
364C2×C4#93711yescharacteristic
365C2×C4#93711yescharacteristic
366C2×C4#93811yescharacteristic
367C2×C4#93911yescharacteristic
368C2×C4#94011yescharacteristic
369C2×C4C2×#21511yescharacteristic
370C2×C4#94211yescharacteristic
371C2×C4#94911yescharacteristic
372C2×C4#95011yescharacteristic
373C2×C4#95311yescharacteristic
374C8—42no—
375C8—122no—
376C8—122no—
377C8—241no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—82no—
381C2×C3#200011yescharacteristic
382C2×C3#200911yescharacteristic
383C2×C3#201811yescharacteristic
384D3—161no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—82no—
389C22—82no—
390C22—81no—
391C22—81no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22#504811yescharacteristic
396C22#504911yescharacteristic
397C22#522311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—82no—
403C4—241no—
404C4#504611yescharacteristic
405C4#504711yescharacteristic
406C4#516012yes—
407C4#522711yescharacteristic
408C4#523811yescharacteristic
409C3#1018811yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—82no—
413C2—481no—
414C2#7169911yescharacteristic
415C2#7204611yescharacteristic
416C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#474C2
socleC22×C3#392
prime radical#82307C2
hypercenterC2×#474D3
hypocenterC3#10188
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C3×#474C2semidirect
2. #83216C2semidirect
4. #71774C2semidirect
5. #71601C2semidirect
6. #70426C2semidirect
7. #67707C2semidirect
33. C2×#474D3semidirect
409. C3#10188semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#832162
#823072
#717742
#716012
#704262
#677072
#101883