Finite Group Databasebeta

← #10694886

Subgroups — #10694886

Order 768.

522 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 107 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×#5114C211yesmaximal, characteristic
2C3×#1810C211yesmaximal, characteristic, complemented
3C8×#217C211yesmaximal, characteristic, complemented
4#69980C211yesmaximal, characteristic, complemented
5#69977C211yesmaximal, characteristic
6#69972C211yesmaximal, characteristic, complemented
7#69970C211yesmaximal, characteristic
8#9813—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C22×#204C411yescharacteristic, complemented
13C2×C3×#108—21no—
14C3×#407C2211yescharacteristic
15C3×#407C2211yescharacteristic
16C4×#217C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5114—21no—
19#5114—21no—
20#5114—21no—
21#5114—21no—
22C2×#821C411yescharacteristic, complemented
23C2×#817C411yescharacteristic, complemented
24C2×#816C411yescharacteristic, complemented
25C2×C8×Dic3—21no—
26C3×#341—21no—
27#4550—21no—
28#4470—22no—
29#4469—22no—
30#4468—22no—
31#4467—22no—
32C2×#465—31no—
33#1810D311yescharacteristic, complemented
34C8×C22⋊C4—31no—
35#1508—31no—
36#1507—31no—
37#1472—31no—
38#1471—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12C2×C411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C22×C3×C8—42no—
46C22×C3×C8C2×C411yescharacteristic
47C22×C3×C8C2×C411yescharacteristic
48C22×C3×C8D411yescharacteristic
49C22×C3×C8D411yescharacteristic
50C22×C3×C8D411yescharacteristic
51C22×C3×C8D411yescharacteristic
52C2×#217—21no—
53C2×#217—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3C2×C411yescharacteristic
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68C2×#204—21no—
69C2×#204—21no—
70C2×#204—42no—
71C3×#108—42no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3⋊#95D412yes—
77#830—21no—
78#830D412yes—
79#821—21no—
80#821—21no—
81#817—21no—
82#817—21no—
83#816—21no—
84#816—21no—
85C8×Dic3—42no—
86C22×SD16—31no—
87C23×C8C2×D311yescharacteristic
88C2×C22⋊Q8—31no—
89C2×#121—31no—
90#465—61no—
91#465—61no—
92#465—61no—
93#465—61no—
94C2×#108—21no—
95C2×#106—31no—
96C2×#103—31no—
97C2×#102—31no—
98#407C2×D311yescharacteristic
99#407C2×D311yescharacteristic
100C2×C22⋊C8—31no—
101C2×C4×C8—61no—
102C4×C22⋊C4—31no—
103#341—21no—
104#336—61no—
105#328—62no—
106#327—62no—
107#324—62no—
108#323—62no—
109C24×C3#4111yescharacteristic
110C23×D3—21no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—41no—
120C2×#67—41no—
121C2×#67—82no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—41no—
129C2×D12—41no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—41no—
133C2×Dic6—41no—
134C3×M16—41no—
135C3×M16—41no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8C22⋊C411yescharacteristic
147C2×C3×C8C22⋊C411yescharacteristic
148C2×C3×C8C22⋊C411yescharacteristic
149C2×C3×C8C22⋊C411yescharacteristic
150#217—24no—
151#217—41no—
152#217—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166#204—42no—
167#204—42no—
168C22×Q8—31no—
169C22×D4—31no—
170C22×D4—61no—
171C23×C4C22×D311yescharacteristic
172C2×SD16—61no—
173C2×SD16—61no—
174C2×SD16—122no—
175C2×M16—21no—
176C2×M16—21no—
177C22×C8—42no—
178C22×C8—61no—
179C22×C8C4×D311yescharacteristic
180C22×C8C4×D311yescharacteristic
181C22×C8D1211yescharacteristic
182C22×C8D1211yescharacteristic
183C22×C8#6711yescharacteristic
184C22×C8#6711yescharacteristic
185C22⋊Q8—61no—
186C22⋊Q8—61no—
187C22⋊Q8—61no—
188C22⋊Q8—61no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200#108—42no—
201#106—61no—
202#106—61no—
203#103—61no—
204#103—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C4×C8—122no—
214#95—32no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3C4×D411yescharacteristic
219C23×C3#12111yescharacteristic
220C23×C3#12411yescharacteristic
221C22×D3—41no—
222C22×D3—42no—
223C22×D3—81no—
224C22×D3—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4C2×C22⋊C411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12711yescharacteristic
238#67—82no—
239#67—82no—
240C2×Dic3—24no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249D12—41no—
250D12—41no—
251D12—81no—
252Dic6—41no—
253Dic6—41no—
254Dic6—81no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8—81no—
259C3×C8—81no—
260C3×C8#9912yes—
261C3⋊C8—42no—
262C24—61no—
263C24#23911yescharacteristic
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—242no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—62no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4C2×D1211yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4#23911yescharacteristic
286C22×C4C2×#6711yescharacteristic
287SD16—122no—
288SD16—122no—
289M16—41no—
290M16—41no—
291M16—41no—
292M16—41no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—62no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8(C2×D3)⋊C411yescharacteristic
305C2×C8(C2×D3)⋊C411yescharacteristic
306C2×C8(C2×D3)⋊C411yescharacteristic
307C2×C8(C2×D3)⋊C411yescharacteristic
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C22⋊C4—64no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42—121no—
320C42—121no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—44no—
329C22×C3#39011yescharacteristic
330C22×C3#44312yes—
331C22×C3#47112yes—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4C2×#9911yescharacteristic
340C3×C4#41011yescharacteristic
341Dic3—44no—
342Dic3—81no—
343Dic3—81no—
344Dic3—81no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—121no—
349C23—122no—
350C23—241no—
351C23—241no—
352C23#89311yescharacteristic
353C23#89511yescharacteristic
354C23#94711yescharacteristic
355Q8—121no—
356Q8—121no—
357Q8—241no—
358D4—121no—
359D4—121no—
360D4—241no—
361D4—242no—
362D4—242no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—64no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—241no—
380C2×C4—241no—
381C2×C4#89311yescharacteristic
382C2×C4#89511yescharacteristic
383C2×C4C2×(C2×D3)⋊C411yescharacteristic
384C2×C4#94011yescharacteristic
385C8—21no—
386C8—21no—
387C8—21no—
388C8—81no—
389C8—81no—
390C8—122no—
391C8#80612yes—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—41no—
395C2×C3#196411yescharacteristic
396C2×C3#197411yescharacteristic
397C2×C3#197711yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—482no—
411C22#489512yes—
412C22#489812yes—
413C22#498811yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—124no—
418C4—241no—
419C4—241no—
420C4—241no—
421C4C2×#80611yescharacteristic
422C4#500611yescharacteristic
423C3#981311yescharacteristic, complemented, Sylow
424C2—22no—
425C2—22no—
426C2—41no—
427C2—481no—
428C2#6952811yescharacteristic
429C2#7069611yescharacteristic
430C2#7070811yescharacteristic
subgroupG/H
centerC22#4988
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1810C2
socleC22×C3#390
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1810D3
hypocenterC3#9813
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. C3×#1810C2semidirect
3. C8×#217C2semidirect
4. #69980C2semidirect
6. #69972C2semidirect
12. C22×#204C4semidirect
22. C2×#821C4semidirect
23. C2×#817C4semidirect
24. C2×#816C4semidirect
33. #1810D3semidirect
423. C3#9813semidirect

Maximal subgroups

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

subgroupindex
C2×#51142
C3×#18102
C8×#2172
#699802
#699772
#699722
#699702
#98133