Finite Group Databasebeta

← #10697171

Subgroups — #10697171

Order 768.

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

normal subgroups only

This group has 562 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×#476C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic, complemented
6#70426C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10180—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#932—21no—
12#5477C2211yescharacteristic
13C3×#476—22no—
14C3×#476—22no—
15C2×C3×#106—21no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#102—21no—
18#5227—22no—
19#5085—21no—
20#5083—22no—
21#5079—21no—
22#5019—21no—
23#5016—22no—
24#5015—21no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4525—22no—
31#4483—22no—
32#3206—31no—
33C2×#476D311yescharacteristic, complemented
34#3036—31no—
35#1549—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×#215—24no—
58C2×#214—22no—
59#932—42no—
60#932—42no—
61C2×#213—22no—
62#930—21no—
63#930—21no—
64C2×#208—21no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869D411yescharacteristic
70C3×#106—24no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#102—24no—
74#835—24no—
75#830—21no—
76#830D412yes—
77#810—22no—
78#809—22no—
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#492—61no—
90#487—61no—
91#476—22no—
92#476—22no—
93#463—62no—
94#461—61no—
95#461—62no—
96#459—61no—
97#448—62no—
98#431—31no—
99#431—31no—
100C2×#106—21no—
101#423—31no—
102C2×C4⋊C8C2×D311yescharacteristic
103#417—61no—
104C2×#102—21no—
105#332—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×D4SD1612yescomplemented
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—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
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#215—84no—
152#214—44no—
153#213—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×#41—61no—
164C22×Q8—21no—
165C22×D4#6711yescharacteristic
166C23×C4—21no—
167AGL(1,ℤ/8)—122no—
168AGL(1,ℤ/8)—122no—
169C2×SD16—62no—
170C2×SD16—64no—
171C2×D8—62no—
172C2×M16—61no—
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—31no—
178C4⋊Q8#6712yes—
179#127—31no—
180#124—61no—
181#121—61no—
182#121—61no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—61no—
188C4×D4—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C42C22×D311yescharacteristic
194#106—24no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201#103—62no—
202#103—64no—
203#102—24no—
204#102—62no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207C4×C8—61no—
208C4×C8—61no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3#12111yescharacteristic
215C22×D3—41no—
216C3×Q8—44no—
217C3×Q8—82no—
218C3×D4—22no—
219C3×D4—22no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—82no—
227C2×C3×C4—82no—
228C2×C3×C4—82no—
229C2×C3×C4—81no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×C3×C4C2×SD1611yescharacteristic
237C2×C3×C4#13511yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4#13711yescharacteristic
240#67—81no—
241#67—81no—
242C2×Dic3—41no—
243D12—42no—
244D12—41no—
245D12—41no—
246D12—42no—
247C4×D3—81no—
248C4×D3—81no—
249Dic6—41no—
250Dic6—41no—
251C3×C8—42no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—21no—
256#41—121no—
257#41—121no—
258#41—121no—
259#41—121no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—61no—
263C2×Q8—82no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—121no—
273C2×D4#21512yescomplemented
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—121no—
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283SD16—124no—
284SD16—244no—
285D8—124no—
286M16—121no—
287M16—121no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42C2×#6711yescharacteristic
309C42C2×#6711yescharacteristic
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3#39211yescharacteristic
318C22×C3#46511yescharacteristic
319C22×C3#46811yescharacteristic
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—82no—
327C3×C4—81no—
328C3×C4#44811yescharacteristic
329C3×C4#44911yescharacteristic
330C3×C4#45011yescharacteristic
331C3×C4#45211yescharacteristic
332C3×C4#47112yes—
333Dic3—81no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—81no—
338C23—81no—
339C23—121no—
340C23#93911yescharacteristic
341Q8—44no—
342Q8—82no—
343Q8—121no—
344Q8—121no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—122no—
350D4—122no—
351D4—121no—
352D4—121no—
353D4—241no—
354D4—241no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—121no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4#93711yescharacteristic
367C2×C4#93711yescharacteristic
368C2×C4#93811yescharacteristic
369C2×C4#93911yescharacteristic
370C2×C4#94011yescharacteristic
371C2×C4C2×#21511yescharacteristic
372C2×C4#94211yescharacteristic
373C2×C4#94911yescharacteristic
374C2×C4#95011yescharacteristic
375C2×C4#95311yescharacteristic
376C8—42no—
377C8—122no—
378C8—122no—
379C8—241no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—82no—
383C2×C3#200011yescharacteristic
384C2×C3#200811yescharacteristic
385C2×C3#201911yescharacteristic
386D3—161no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—82no—
391C22—82no—
392C22—81no—
393C22—81no—
394C22—241no—
395C22—241no—
396C22—241no—
397C22#504811yescharacteristic
398C22#504911yescharacteristic
399C22#522311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—82no—
405C4—241no—
406C4#504611yescharacteristic
407C4#504711yescharacteristic
408C4#517712yes—
409C4#522711yescharacteristic
410C4#523811yescharacteristic
411C3#1018011yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—82no—
415C2—481no—
416C2#7169911yescharacteristic
417C2#7205911yescharacteristic
418C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#476C2
socleC22×C3#392
prime radical#82307C2
hypercenterC2×#476D3
hypocenterC3#10180
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×C3×#476C2semidirect
2. #83213C2semidirect
4. #71775C2semidirect
5. #71601C2semidirect
6. #70426C2semidirect
7. #67707C2semidirect
33. C2×#476D3semidirect
117. C2×C3×D4SD16semidirect
273. C2×D4#215semidirect
411. C3#10180semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4762
#832132
#823072
#717752
#716012
#704262
#677072
#101803