Finite Group Databasebeta

← #31915002

Subgroups — #31915002

Order 18176.

676 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#14556047C211yesmaximal, characteristic
2C2×#17985798C212yesmaximal, complemented
3#22588264C211yesmaximal, characteristic
4#22586467C212yesmaximal, complemented
5C71×#2434C211yesmaximal, characteristic
6C22×#11765849—21no—
7C2×#14556047—24no—
8C2×#14556047—21no—
9C2×#14556047—21no—
10C2×#14556047C2212yes—
11C2×#14556047C2212yes—
12#17985963—21no—
13#17985798—24no—
14C23×Q8×C71C2211yescharacteristic
15C2×C4⋊Q8×C71C411yescharacteristic
16C2×C4⋊Q8×C71C2211yescharacteristic
17C2×C22⋊Q8×C71C411yescharacteristic
18#395×C71—21no—
19#395×C71—21no—
20#391×C71C411yescharacteristic
21#391×C71C2211yescharacteristic
22#385×C71—21no—
23#385×C71—21no—
24C4×C22⋊C4×C71C411yescharacteristic
25C2×#11765849—24no—
26C2×#11765849—22no—
27C2×#11765849—22no—
28C2×#11765849—41no—
29C2×#11765849—42no—
30C22×C71⋊C8—41no—
31#14556047—24no—
32#14556047—22no—
33#14556047—22no—
34#14556047—22no—
35#14556047—42no—
36#14556047D414yes—
37C22⋊(C71⋊C8)—41no—
38C22⋊(C71⋊C8)—41no—
39C22×Q8×C71—22no—
40C22×Q8×C71—22no—
41C22×Q8×C71—41no—
42C22×Q8×C71—41no—
43C22×Q8×C71—42no—
44C22×Q8×C71C2×C411yescharacteristic
45C22×Q8×C71C2×C411yescharacteristic
46C22×Q8×C71D411yescharacteristic
47C22×Q8×C71D411yescharacteristic
48C22×Q8×C71C2311yescharacteristic
49C23×C4×C71C2×C411yescharacteristic
50C23×C4×C71D411yescharacteristic
51C23×C4×C71D411yescharacteristic
52C4⋊Q8×C71—22no—
53C4⋊Q8×C71—22no—
54C4⋊Q8×C71—42no—
55C22⋊Q8×C71—41no—
56C22⋊Q8×C71—41no—
57C2×C4⋊C4×C71—21no—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71—21no—
61C2×C4⋊C4×C71—21no—
62C2×C4⋊C4×C71C2×C411yescharacteristic
63C2×C22⋊C4×C71—21no—
64C2×C22⋊C4×C71—21no—
65C2×C42×C71C2×C411yescharacteristic
66C2×C42×C71C2×C411yescharacteristic
67#95×C71—21no—
68#95×C71—21no—
69#95×C71—21no—
70#95×C71—21no—
71#95×C71—21no—
72#11765849—24no—
73#11765849—44no—
74#11765849—42no—
75#11765849—42no—
76#11765849—42no—
77C2×C71⋊C8—44no—
78C2×C71⋊C8—42no—
79C2×C71⋊C8—42no—
80C24×C71C22⋊C411yescharacteristic
81C2×Q8×C71—22no—
82C2×Q8×C71—22no—
83C2×Q8×C71—22no—
84C2×Q8×C71—21no—
85C2×Q8×C71—22no—
86C2×Q8×C71—21no—
87C2×Q8×C71—22no—
88C2×Q8×C71—41no—
89C2×Q8×C71—41no—
90C2×Q8×C71—41no—
91C2×Q8×C71—41no—
92C2×Q8×C71—41no—
93C2×Q8×C71—41no—
94C2×Q8×C71—44no—
95C2×Q8×C71—44no—
96C2×Q8×C71—44no—
97C2×Q8×C71—44no—
98C2×Q8×C71—41no—
99C2×Q8×C71—41no—
100C2×Q8×C71C2×D412yes—
101C2×Q8×C71#4112yes—
102C22×C4×C71—21no—
103C22×C4×C71—21no—
104C22×C4×C71—22no—
105C22×C4×C71—21no—
106C22×C4×C71—21no—
107C22×C4×C71—21no—
108C22×C4×C71—21no—
109C22×C4×C71—21no—
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71—21no—
113C22×C4×C71—22no—
114C22×C4×C71—22no—
115C22×C4×C71—21no—
116C22×C4×C71—21no—
117C22×C4×C71—41no—
118C22×C4×C71—41no—
119C22×C4×C71C22⋊C411yescharacteristic
120C22×C4×C71C22⋊C411yescharacteristic
121C22×C4×C71C22⋊C411yescharacteristic
122C22×C4×C71C22×C411yescharacteristic
123C22×C4×C71C2×D411yescharacteristic
124C22×C4×C71C2×D411yescharacteristic
125C4⋊C4×C71—41no—
126C4⋊C4×C71—42no—
127C4⋊C4×C71—42no—
128C4⋊C4×C71—42no—
129C4⋊C4×C71—41no—
130C4⋊C4×C71—42no—
131C4⋊C4×C71—41no—
132C22⋊C4×C71—24no—
133C22⋊C4×C71—41no—
134C22⋊C4×C71—41no—
135C42×C71—22no—
136C42×C71—41no—
137C23×C71—22no—
138C23×C71—21no—
139C23×C71—21no—
140C23×C71—41no—
141C23×C71#9911yescharacteristic
142C23×C71#9911yescharacteristic
143C23×C71C2×C22⋊C411yescharacteristic
144Q8×C71—42no—
145Q8×C71—42no—
146Q8×C71—44no—
147Q8×C71—44no—
148Q8×C71—42no—
149Q8×C71—42no—
150Q8×C71—82no—
151C2×C4×C71—22no—
152C2×C4×C71—22no—
153C2×C4×C71—22no—
154C2×C4×C71—22no—
155C2×C4×C71—21no—
156C2×C4×C71—21no—
157C2×C4×C71—21no—
158C2×C4×C71—21no—
159C2×C4×C71—21no—
160C2×C4×C71—21no—
161C2×C4×C71—21no—
162C2×C4×C71—21no—
163C2×C4×C71—22no—
164C2×C4×C71—22no—
165C2×C4×C71—21no—
166C2×C4×C71—21no—
167C2×C4×C71—21no—
168C2×C4×C71—21no—
169C2×C4×C71—21no—
170C2×C4×C71—21no—
171C2×C4×C71—42no—
172C2×C4×C71—42no—
173C2×C4×C71—42no—
174C2×C4×C71—42no—
175C2×C4×C71—41no—
176C2×C4×C71—41no—
177C2×C4×C71—41no—
178C2×C4×C71—41no—
179C2×C4×C71—42no—
180C2×C4×C71—41no—
181C2×C4×C71—41no—
182C2×C4×C71—41no—
183C2×C4×C71—41no—
184C2×C4×C71#9912yes—
185C2×C4×C71C4×D412yes—
186C2×C4×C71#12011yescharacteristic
187C2×C4×C71#12112yes—
188C2×C4×C71#12311yescharacteristic
189C71⋊C8—44no—
190C71⋊C8—82no—
191C22×C71—21no—
192C22×C71—21no—
193C22×C71—21no—
194C22×C71—21no—
195C22×C71—21no—
196C22×C71—21no—
197C22×C71—21no—
198C22×C71—42no—
199C22×C71—42no—
200C22×C71#35612yescomplemented
201C22×C71#38611yescharacteristic
202C22×C71C2×#9911yescharacteristic
203C22×C71C2×#9911yescharacteristic
204C4×C71—21no—
205C4×C71—22no—
206C4×C71—22no—
207C4×C71—21no—
208C4×C71—22no—
209C4×C71—42no—
210C4×C71—42no—
211C4×C71—41no—
212C4×C71—41no—
213C4×C71—81no—
214C4×C71—81no—
215#13094—711nomaximal, Sylow
216C2×C71—21no—
217C2×C71—21no—
218C2×C71—22no—
219C2×C71—41no—
220C2×C71#189811yescharacteristic
221C2×C71C2×#35611yescharacteristic
222C2×C71#213311yescharacteristic
223C22×#101—711no—
224#2434D7111yescharacteristic
225C2×#356—712no—
226#1784—711no—
227#1394—712no—
228C71#1309411yescharacteristic, complemented, Sylow
229C23×Q8C2×D7111yescharacteristic
230C22×M16—1421no—
231C2×C4⋊Q8Dic7111yescharacteristic
232C2×C4⋊Q8C2×D7111yescharacteristic
233C2×C22⋊Q8Dic7111yescharacteristic
234C2×#101—712no—
235C2×#101—712no—
236C2×#101—1421no—
237C2×#101—1424no—
238C2×#101—1421no—
239#407—1421no—
240#395—21no—
241#395—21no—
242#391Dic7111yescharacteristic
243#391C2×D7111yescharacteristic
244#385—21no—
245#385—21no—
246C4×C22⋊C4Dic7111yescharacteristic
247#356—1424no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—41no—
251C22×Q8—41no—
252C22×Q8—42no—
253C22×Q8C2×Dic7111yescharacteristic
254C22×Q8C2×Dic7111yescharacteristic
255C22×Q8C22⋊D7111yescharacteristic
256C22×Q8C22⋊D7111yescharacteristic
257C22×Q8C22×D7111yescharacteristic
258C23×C4C2×Dic7111yescharacteristic
259C23×C4C22⋊D7111yescharacteristic
260C23×C4C22⋊D7111yescharacteristic
261C2×M16—2841no—
262C2×M16—2842no—
263C2×M16—1422no—
264C2×M16—1424no—
265C2×M16—1422no—
266C22×C8—2841no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—42no—
270C22⋊Q8—41no—
271C22⋊Q8—41no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C2×Dic7111yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C42C2×Dic7111yescharacteristic
281C2×C42C2×Dic7111yescharacteristic
282#101—2842no—
283#101—714no—
284#101—1422no—
285#101—1422no—
286#101—1424no—
287#101—1422no—
288C22⋊C8—2841no—
289C22⋊C8—2841no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295C24C22⋊Dic7111yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8D4×D7112yes—
316C2×Q8#1176586412yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C22⋊Dic7111yescharacteristic
335C22×C4C22⋊Dic7111yescharacteristic
336C22×C4C22⋊Dic7111yescharacteristic
337C22×C4C22×Dic7111yescharacteristic
338C22×C4C2×C22⋊D7111yescharacteristic
339C22×C4C2×C22⋊D7111yescharacteristic
340M16—2844no—
341M16—2842no—
342M16—2842no—
343M16—2842no—
344M16—1424no—
345C2×C8—2844no—
346C2×C8—2842no—
347C2×C8—2842no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42—22no—
359C42—41no—
360C23—22no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23#1455604311yescharacteristic
365C23#1455604311yescharacteristic
366C23C2×C22⋊Dic7111yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—44no—
370Q8—44no—
371Q8—42no—
372Q8—42no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4#1455604312yes—
408C2×C4D4×Dic7112yes—
409C2×C4#1455609611yescharacteristic
410C2×C4#1455610012yes—
411C2×C4#1455610911yescharacteristic
412C8—2844no—
413C8—5682no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#1798579812yescomplemented
424C22#1798587911yescharacteristic
425C22C2×#1455604311yescharacteristic
426C22C2×#1455604311yescharacteristic
427C4—21no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#2258879411yescharacteristic
443C2C2×#1798579811yescharacteristic
444C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2434C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC22×Q8×C71C23
hypercenter#2434D71
hypocenterC71#13094
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#17985798C2semidirect
4. #22586467C2semidirect
200. C22×C71#356semidirect
228. C71#13094semidirect
423. C22#17985798semidirect

Maximal subgroups

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

subgroupindex
C22×#145560472
C2×#179857982
#225882642
#225864672
C71×#24342
#1309471